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Retrieval of Kinematic Fields from Dual-Beam Airborne Radar Data Gathered in Circular Trajectories during the FASTEX Experiment
1. Introduction
Since the beginning of modern meteorology, the description of the dynamical structure and organization of precipitating systems is a problem of crucial interest. This interest has been progressively
evolving from the large-scale and convective-scale motions toward mesoscale motions on the one hand, and to the scale interactions on the other hand. The feedback between current theory, experiment,
and modeling, along with the crucial role of the mesoscale phenomena, has initiated a considerable development of radar systems, and in particular, airborne radar systems, which are systems
particularly suited to this scale and to observing precipitation over oceans. Airborne radars with single-beam antennas were first developed (Jorgensen et al. 1983), followed by systems equipped with
dual-beam antennas (Hildebrand et al. 1994). Moreover, in addition to the development of new instrumentation, considerable efforts have been devoted to modeling.
The international field experiment Fronts and Atlantic Storm-Track Experiment (FASTEX) had both experimental and modeling scientific objectives (Joly et al. 1997). This experiment, devoted to frontal
waves and the frontal cyclogenesis in the Atlantic Ocean, took place in January–February 1997. During the FASTEX campaign, considerable experimental means associated with forecasting were deployed
throughout the Atlantic Ocean, while a great effort was made in the coupling of experimentally derived meteorological fields with models. The building of a FASTEX database, which is one of the main
objectives of the experiment, is intended to allow this coupling by documenting the observed phenomena (within frontal waves and secondary lows) by meteorological fields obtained through data
analysis in view of model validation and initialization.
An example of such fields is the vertical structure of the wind field obtained by analysis of the “purls”, that is, circular trajectories performed by the airborne dual-beam Doppler radars monitored
during FASTEX. Such purls were regularly performed along the straight lines flown by the aircraft within the FASTEX frontal cyclones. Performing purls all along the experiment allows the mesoscale
wind field and fields of related parameters (vertical vorticity, divergence, deformation, terminal fall velocity of the hydrometeors) to be retrieved on a regular basis. Purl processing was done
using a new analysis scheme named Dual-Beam Antenna Velocity Azimuth Display (DAVAD).
DAVAD is derived from the classical Velocity Azimuth Display (VAD) analysis of ground-based radar data, which relies upon conical scannings, and upon the hypothesis that the horizontal wind
components are linear in terms of the horizontal coordinates (Browning and Wexler 1968; Testud et al. 1980; Srivastava et al. 1986; Matejka and Srivastava 1991). The VAD analysis was adapted to dual
Doppler radar scanning by Scialom and Testud (1986) under the name of DVAD analysis. The VAD–DVAD ensemble was first adapted to single-beam airborne radars performing purls, an adaptation that is
straightforward but requires that many specific problems be solved, as pointed out by Protat et al. (1997). This method was implemented under the name of SAVAD analysis (single or double SAVAD,
depending on the number of radars), and applied to the data of TOGA–COARE (Tropical Ocean Global Atmosphere–Coupled Ocean Atmosphere Response Experiment; Webster and Lucas 1992). If the airborne
Doppler radar is dual-beam, the geometry of observation is slightly different. In this context, DAVAD is an adaptation of single SAVAD to a dual-beam geometry and thus allows the retrieval of the 3D
wind field and the associated physical parameters from the data gathered during the purls, and in particular of the vertical component of the vorticity, which is an advantage with respect to the
single SAVAD. Note that purls (either with single or dual-beam antenna) were first extensively exploited during the TOGA–COARE experiment by Mapes and Houze (1995) who analyzed statistically the
divergence profiles obtained throughout the experiment within these circular trajectories. They took advantage of (i) the short duration of purls, thus mitigating the effects of advection and
internal evolution; (ii) the short ranges, which minimize the effects of beam widening and attenuation, and of sea clutter contamination. Contrarily to Mapes and Houze method, in DAVAD, the global
volume is processed in order to retrieve not only the divergence profile, but also the other crucial kinematic parameters mentioned previously. Moreover, in the DAVAD approach, the vertical profiles
of horizontal divergence, mean vertical velocity and vertical vorticity are retrieved simultaneously through a linear regression allowing a separation of these three parameters, as specified in
section 2b(1). No assumption is thus needed for the value of the hydrometeor fall speed.
In section 2 of the present paper, the principle and mathematical formulation of the DAVAD analysis are detailed. In section 3, the main results of the performed simulations are given in order to
test the sensitivity of the method to effects of the various technical parameters of the aircraft, of the radar and of the radar sequence, in particular the radar noise, the data resolution in terms
of elevation and azimuth, and the radius of the purl. These simulations were particularly relevant in the context of FASTEX since their results were taken into account when achieving the final design
of this experiment. The application of the method to Intensive Observation Period (IOP) 12 extracted from the FASTEX dataset is given in section 4, along with elements of interpretation of the
derived 3D circulations.
2. Principle and mathematical formulation of the method
The present section is subdivided in two subsections in which the principle of the method (section 2a) and its refinements (section 2b) are successively developed. Indeed we shall see that if all the
first-order derivatives of the wind can be theoretically obtained by a least squares approach, their accuracy can be insufficient in real cases, and must be improved. Thus, since several
determinations of them can be obtained, it seems natural to present a variational approach of the problem in order to improve the accuracy and/or to resolve the problem even in the case of partially
available data.
a. Principle of the retrieval
Airborne Doppler radars have proved to be an essential tool in the field of radar meteorology, because of their ability to follow precipitating systems, in particular over oceans, and to get their
time evolution. The first radars possessed a single-beam antenna, scanning perpendicular to the aircraft track, offering a single view of the sampled system (Jorgensen et al. 1983; Hildebrand and
Mueller 1985). A second view (not colinear to the first one) allows, one using the continuity equation, to obtain the three-dimensional wind field. This can be achieved by performing two successive
paths nearly perpendicular to each other; however one needs to assume stationarity of the observed system, and in any case the time resolutions degraded using this technique. This is the reason why
Frush et al. (1986) proposed Fore–Aft Scanning Technique (FAST) scanning, in which the antenna is tilted with respect to the perpendicular to the track, offering a second viewing angle of the system.
This idea was improved by developing dual-beam antennas, scanning respectively fore and aft of the aircraft track (Hildebrand et al. 1994). Two airborne radars with such antennas were deployed during
FASTEX [i.e., the National Oceanic and Atmospheric Administration (NOAA) P3-43, and the U.S.–French ELDORA/ASTRAIA (Electra Doppler Radar–Analyse Stéréoscopique pour Radar à Impulsions Aéroporté)
radar onboard the National Center for Atmospheric Research (NCAR) Electra aircraft]. An illustration of this scanning is given in Fig. 1 in the case of a circular trajectory. This figure shows both
antennas (fore and aft) scanning circles. Without aircraft motion, the antennas would generate cones on which the black vertical circles would display the motion of a particular gate at a given range
from the radar. Due to aircraft motion, the sampling is helical along the circular aircraft trajectory. In the following to simply expose the DAVAD method, we use notations, in particular for angles,
close to that of ground-based Doppler radars and of VAD analysis. Of course, as indicated in Lee et al. (1994), the geometry of scanning in airborne radars needs several coordinate systems to be
used, in order to take into account the aircraft navigation angles (roll, pitch, and drift) as provided by the Inertial Navigation System. Simulations performed in the following take these
considerations into account. The effects of these angles are discussed in section 3g.
Let us consider one point
scanned successively by the fore and aft antennas (
Fig. 2
). The aircraft A flies along its circular trajectory on a horizontal plane at altitude
is the radius of the circular trajectory,
is the azimuth of the measured point with respect to the north,
′ is the azimuth of the aircraft (
Fig. 2a
is the elevation of the measured point with respect to the horizontal plane (
Fig. 2b
is the projection of
onto the vertical plane perpendicular to the aircraft track (containing radius
′ is the elevation (
Fig. 2b
) of the measured plane PAM with respect to the horizontal plane,
is the tilt angle of each antenna fore and aft of the plane perpendicular to the aircraft track,
is the distance of the sampled point from the radar, and finally angle
is the projection of angle
on the horizontal. With the convention used in this paper for the winds, angles are counted positive clockwise from the north, and speeds are positive toward the antenna. We may write for an aircraft
turning counterclockwise (
Fig. 2a
for the fore antenna and
for the aft antenna.
Equations (1)
are exchanged for an aircraft turning clockwise.
Considering the right triangles
AmM, ApP, APM,
(with right angles in
m, p, P,
respectively), it becomes (
Fig. 2b
Let us first consider the fore antenna in
Fig. 2b
. The measured point being at distance
from the aircraft [
is positive (negative) for gates describing circles external (internal) to the aircraft purl], its coordinates (
to the east,
to the north) can be written as
where the origin of the reference frame is point
on the vertical axis about which the aircraft performs its circular trajectory, and
is at altitude
of point
In the following, the presentation of the method will only concern external circles, that is, corresponding to gates toward the exterior of the purl. When considering internal gates, similar
equations may be written, and taking account of internal gates presents no additional difficulty.
An implicit hypothesis made in radar meteorology is that the horizontal velocity of hydrometeors is that of the surrounding air, while the measured vertical hydrometeor velocity
is the sum of the vertical air velocity −
and of the terminal fall velocity of the hydrometeors
. Since
) is the air particle speed measured along the viewing axis, and
are the horizontal wind components, we shall proceed as for the analysis from ground-based radars. We first write
Note that the geometry of the scanning is different from that of a ground-based radar performing a conical scanning. Thus, in order to use an analysis derived from the VAD analysis, the data must be
rearranged on cones in a way similar to that used in
Protat et al. (1997)
who consider the pseudo-conical scanning performed by a single-beam antenna.
As in the classical VAD analysis (
Browning and Wexler 1968
) at ground, we consider that the vertical air velocity is small with respect to the terminal fall velocity of the hydrometeors. As shown later, the vertical wind component neglected in
is computed by vertical integration of the continuity equation using the retrieved horizontal divergence, and this assumption is checked a posteriori. If the horizontal components of the wind are
assumed to be linear, we may write, using
(4) and (5)
are the horizontal components at point
Figs. 2a,b
), and ∂
, …, ∂
are the (assumed constant) horizontal derivatives of the horizontal components.
(7), and (8)
and using
along a horizontal circle performed by a given gate at given range
and elevation
we may write
) as a Fourier expansion up to order two as a function of azimuth
(as for the case of a ground-based radar):
In a way similar to the case of ground-based radars, Fourier coefficients are computed (i) by least squares fitting of the theoretical expansion
to its experimental values on each of the horizontal circles described by each sampling gate during the aircraft purl, and (ii) by deriving the associated kinematic parameters from these Fourier
coefficients (
Testud et al. 1980
Now consider both the fore and aft antennas. An additional index (f or a) is now necessary on a[i] and b[i] parameters and on related physical parameters to distinguish between information from the
fore (a[if], b[if], …) and the aft (a[ia], b[ia], …) antennas.
The Fourier coefficients can be written as
where DIV (=∂
+ ∂
), ROT (=∂
− ∂
), DET (=∂
− ∂
), and DES (=∂
+ ∂
) are, respectively, horizontal divergence, vertical component of the vorticity vector (denoted as vertical vorticity in the following), and stretching and shearing deformations.
It can be seen from (10)–(14) that using a single antenna or a single viewing angle [which is equivalent to retaining only one of the signs + or − in the terms with ± of (10), (13), and (14)] does
not allow for all parameters since the problem is underdetermined (five equations for seven unknowns, including V[f]). In order to solve this problem, measurements from two antennas are required in
order to obtain at least as many equations as unknowns (the case of V[f] will be discussed further in the next section). When comparing with the formulation of the ground-based traditional VAD
analysis, it is seen that for the airborne case, (i) coefficient a[0], which provides (after correcting for the fall velocity V[f]) the horizontal wind divergence DIV, is now contaminated by the
vertical vorticity ROT; (ii) terms a[1] and b[1] are unchanged; (iii) term a[2] (classically the stretching deformation DET) is now contaminated by the shearing deformation DES, while term b[2]
(classically the shearing deformation) is now contaminated by the stretching deformation. These “contaminating” terms are the expressions in parentheses following the minus sign in Eqs. (10), (13),
and (14) and are specific of the analysis of dual beam airborne radar data, while the other terms are similar to those obtained in the case of ground-based radars.
If both antennas have worked properly, adding and subtracting the two forms of (10) yields the divergence (with a contribution of the fall velocity) and the vorticity. Using (13) and (14) yields,
respectively, the stretching deformation and the shearing deformation for each antenna. Nevertheless, we cannot combine results deduced from the fore and the aft antennas in the same way as it was
done in the DVAD analysis from two ground-based radars: indeed, Eqs. (11)–(12) provide the horizontal components of the wind at the center vertical of rotation of the purl, but, contrary to the case
of ground-based radars, this provides twice the same information. Thus with a dual-beam antenna, the procedure to retrieve the 3D mesoscale wind field consists of the following.
• Process the data separately from each antenna, compute a[0f], a[0a], a[1f], a[1a], b[1f], b[1a], a[2f], a[2a], b[2f], b[2a] for each horizontal circle followed by each gate at given r and θ;
thus, we derive directly U[0] from b[1f] (b[1a]), V[0] from a[1f] (a[1a]), DET, and DES from a[2f] and b[2f] (or from a[2a] and b[2a]); since data are supposed to be representative of nearly
identical areas, it can be checked if U[0], V[0], DET, and DES derived from both antennas are almost equal. If this is the case, it provides some confidence in the results.
• Combine a[0f] and a[0a] in order to obtain DIV (contaminated by V[f]) and ROT.
• Calculate the horizontal derivatives ∂U/∂x, ∂U/∂y, ∂V/∂x, ∂V/∂y, from DIV, DET, DES, and ROT.
This procedure is straightforward and leads to good accuracy on the horizontal wind components but to moderate accuracy on the horizontal derivatives although the Fourier coefficients are accurate
(which is not true in the present case as developed in section 3). Thus, in order to improve this accuracy, the equations are processed using a variational approach, as detailed in the next section.
The vertical air velocity W is obtained by integrating the continuity equation (or mass conservation) by steps along the vertical using the horizontal divergence DIV at each altitude and starting
from the ground (with the boundary condition that W is zero at the ground). This integration process does not differ from that of the VAD and SAVAD analyses. Error on W increases with height since W
is built by adding ΔW variations calculated at each altitude slice starting from the ground and going upwards. In particular, due to the lack of measurements at lower heights, there is a systematic
error (some cm s^–1) on W. This error would be larger in convective cases.
b. Mathematical formulation of the problem and variational approach
The previous section discusses how the Fourier coefficients are derived from the processing of the radial winds for each radar gate. This processing is the first step of the DAVAD analysis. As shown
in section 2a, the horizontal wind components and their first order derivatives can be derived from various combinations of the Fourier coefficients, but with only moderate accuracy since in real
cases, for each gate and each elevation, there are at most 18 data points in the case of the P3-43 radar and 36 data points in the case of ELDORA-ASTRAIA radar, which is not sufficient to efficiently
reduce the impact of radar noise. Thus in a second step, data from the first step are merged in several altitude slices and processed as a function of elevation, which results in an increase of the
amount of data and better accuracy of the retrieved wind parameters.
1) Calculation of DIV, ROT, and V[f] from a[0] coefficient
This coefficient
contains contributions from DIV, ROT and
. As seen in
section 2a
, the respective contributions from ROT and DIV “contaminated” by
are separated in a first step for each gate by combining
The second step is similar to the traditional VAD analysis. It consists of separating quantities that could not be directly obtained, such as DIV and V[f], and on the other hand, improving the
accuracy of V[f], DIV, and ROT by processing as many data as possible in order to decrease the influence of the noise.
In what follows we describe this step in detail for the processing of
coefficients. The same procedure is used for the other Fourier coefficients. We merge the sets of (
) in altitude slices 300 m thick. These datasets correspond to various elevations
and radial distances
. We assume that DIV,
, and ROT are constant in each altitude layer
Instead of solving
(15) and (16)
directly for all measurements which would lead to an overdetermined system, we use a variational procedure in which
Eqs. (15) and (16)
must be approximately satisfied for the
available measurements of both
in each altitude layer
. This leads to minimization with respect to the unknowns DIV,
, and ROT, and the following expressions
in which sums Σ are over
measurements on each altitude layer
Thus, ROT, DIV, and
are estimated through a least squares fitting procedure leading to
= 2[cos
= (
= (
); and where
is a weighting factor automatically determined from the error matrix of the Fourier coefficients. This factor results from errors on Fourier parameters (thus indirectly from errors on measurements)
and quantifies the expected accuracy of the retrieved quantities (DIV, ROT, and
2) a[1] and b[1] coefficients
The coefficients
are, respectively, related to
. The procedure for their retrieval in each gate is similar to that used in the traditional VAD analysis. The estimate can also be improved by a least squares fitting procedure and a variational
approach using
, and
, respectively:
are weighting factors, and Σ
and Σ
are extended to
circles, respectively, corresponding to the fore and aft antenna. This in turn leads to
3) a[2] and b[2] coefficients
Both coefficients
include contributions from DET and DES. In the present case, several sets of equations are solved in order to retrieve DET and DES. As in
section 2b(1)
, the problem is generally overdetermined and the estimates are improved using a variational procedure to take this overdetermination into account using
Eqs. (13)–(14)
. Thus, we minimize the following cost functions with respect to DET and DES:
DET and DES are then given by the equations
are weighting factors determined as in
section 2b(1)
; Σ is as in
section 2b(1)
3. Capabilities of the DAVAD analysis
In this section, a wind field is simulated in order to evaluate the accuracy of the method on the retrieval of wind and its first-order derivatives, and the sensitivity of the method to contaminating
effects such as the radar noise, the data resolution and the radius of the purl. The simulated wind is assumed to vary linearly in the horizontal. In the first series of simulations, the wind has no
vertical variations, and the vertical wind is set to zero. The simulated wind at the radar vertical has the following characteristics, with the notations of
section 2
a. Software checking
The radius of the purl is 10 km, the aircraft flies at 360-m altitude (these parameters are not necessary realistic, they are just chosen for software checking). Data are simulated as regularly
organized on successive cones with elevations θ from −60° to +60°, with elevation resolution δθ (in °). On each cone, each gate from each antenna (fore and aft) describes a circle with azimuth
resolution δβ (in °)—each circle is defined by 360/δβ measurements. Let us recall (see section 2) that the wind retrieval consists of two steps: the first step is an analysis of the radial wind in
each circle in terms of Fourier coefficients, and the second is an analysis of the Fourier coefficients by altitude slices in order to retrieve the wind components, horizontal divergence, deformation
terms, and vertical vorticity. The altitude slice is set to 300 m thick.
The simulation of the wind field previously defined without noise, with δβ = 1° and δθ = 0.5°, leads to perfect recovery of the simulated wind; that is, the software checks out within errors of the
order 1 × 10^–6 m s^–1 for U[0], V[0], 10^–10 s^–1 for ∂U/∂x, ∂V/∂x, ∂U/∂y, ∂V/∂y, and 2 × 10^–5 m s^–1 for V[f].
b. Effect of the noise
If a Gaussian white noise 1.5 m s^–1 standard deviation is added to the simulated radial wind (this noise is typical of airborne radars), the accuracy of the retrieved parameters depends on the
elevation resolution retained for the analysis.
For an elevation range of −60° to +60° in 0.5° increments, and an azimuthal resolution of δβ = 1°, the results are of the order 1 × 10^–3 m s^–1 for U[0], V[0], 10^–7 to 10^–8 s^–1 for ∂U/∂x, ∂V/∂x,
∂U/∂y, ∂V/∂y, 2 × 10^–3 m s^–1 for V[f] and are still very accurate. Nevertheless, the real data resolution is much lower than this one as discussed in the next paragraph.
c. Effect of the data resolution
In this section, we study the effect on the wind retrieval of the data resolution and of the elevation range on simulated data including a noise 1.5 m s^–1 standard deviation.
• i) For an elevation range of −60° to +60° in 2° increments, and an azimuthal resolution of δβ = 3°, the results are of the order 2 × 10^–2 m s^–1 for U[0], V[0], 10^–6 to 10^–7 s^–1 for ∂U/∂x, ∂V
/∂x, ∂U/∂y, ∂V/∂y, 2 × 10^–2 m s^–1 for V[f] and are satisfactory.
• ii) If the azimuth resolution is degraded down to δβ = 15°, then the accuracy becomes 7 × 10^–2 m s^–1 for U[0], V[0], 10^–5 to 10^–6 s^–1 for ∂U/∂x, ∂V/∂x, ∂U/∂y, ∂V/∂y, 7 × 10^–2 m s^–1 for V[f
] and are still satisfactory.
• iii) If the elevation range is only −20° to +20° in 2° increments, and if the azimuth resolution is kept at δβ = 15°, then the accuracy is not very different from that of case ii, that is, 1 × 10
^–1 m s^–1 for U[0], V[0], 10^–5 to 10^–6 s^–1 for ∂U/∂x, ∂V/∂x, ∂U/∂y, ∂V/∂y, except for V[f] for which the error is around 0.25 m s^–1, which is attributed to the fact that high elevations are
excluded in the analysis.
• iv) If the elevation range is kept at −20° to +20° but is in 0.5° increments and if the azimuth resolution is only δβ = 60°, then the accuracy is close to that of case iii, since the improvement
of the resolution by a factor 4 on elevation is compensated by a degradation by the same factor on the azimuth resolution.
• v) For the P3 radar in the conditions of the FASTEX experiment and for when data with a resolution in elevation 0.5°, the resolution in azimuth is at most 20°; that is, the processing of data
from each circle to calculate the five Fourier coefficients is performed on at most, 18 data points. For an elevation range −20° to +20°, the accuracy is then 0.05 m s^–1 for U[0], V[0], 10^–5
−10^–6 s^–1 for ∂U/∂x, ∂V/∂x, ∂U/∂y, ∂V/∂y, 0.15 m s^–1, for V[f]. The accuracy for V[f] can be improved up to 0.05 m s^–1 when the elevation range is −60° to +60°.
d. Effect of the radius of the purl and of the altitude of the aircraft
Three radii of the purl, R, were tested, namely 2.5, 5, and 10 km. The corresponding times to perform these purls increase from 3 min up to 12 min, respectively. There is no effect of R on the
accuracy of the mean horizontal wind components because they are determined from (11)–(12) through a[1f,a] and b[1f,a]. Errors on wind derivatives decrease when the radius increases [see (10), and
(13)–(14)], in a way inversely proportional to the radius down to a limit of about 2 × 10^–7 s^–1. For example, the relative error on ROT (which corresponds to the worst case since ROT is calculated
from a residual factor) is 10% for R = 10 km, 20% for R = 5 km, 40% for R = 2.5 km. However, choosing R = 10 km corresponds to a purl duration of 12 min; this would either imply that a linear flight
pattern 60 km long be suppressed during the aircraft flight (which would change the sampling strategy), or if not, that the total flight duration would increase too much regarding the flight aircraft
range. Thus despite the moderate accuracy of the wind derivatives when using small radii, the trade-off was to fly purls of diameter 3 km during FASTEX.
Four aircraft altitudes, Z[A], were also tested—namely, 0.36, 1.3, and 5 km—the parameters of the simulation being those of section 3c, test ii [elevation range −60° to +60° in 2° increments, and
azimuth resolution δβ = 15°]. Errors on wind components and their derivatives are not very dependent on Z[A], because changing Z[A] does not change significantly the statistical distribution of
e. Retrieval of winds with vertical gradients
In realistic situations, winds vary, sometimes strongly, with height. In order to simulate such situations, the wind previously defined analytically has been retained, except that ∂U/∂y and ∂V/∂x
change with height. So, instead of a single volumetric value S[1] for vertical vorticity ROT we simulate S[1] = 10^–4 [integer part(altitude/300) − 3000] s^–1 in order that a large set of (positive
and negative) values for ROT be explored (ROT increases by 10^–4 s^–1 increments from −3 × 10^–4 s^–1 in the 0–300-m layer up to 8 × 10^–4 s^–1 in the 3300–3600-m layer). A similar variation S[2] (S
[2] = 0.4 × 10^–4 s^–1) is simulated for shearing deformation DES. It is permissible to work in this way, since with DAVAD like with VAD, the retrieval of the horizontal wind is performed in each
horizontal plane (The vertical wind is then integrated vertically using the horizontal divergence obtained previously). The result is that ROT and DES are retrieved with an accuracy of about 5–6 × 10
^–4 s^–1, in agreement with the previous results.
f. Recovery of quadratic winds with vertical gradients
In real applications, winds can significantly depart from the assumption of linearity, so that the retrieval of the wind components as if they were linear may lead to errors. Retrieving such a wind
through a method like DAVAD (or VAD) will linearize it, that is, provide mean values for the first order derivatives that should be linear, as was shown in
Scialom and Lemaître (1994)
. Up to a certain threshold, nonlinearities will lead to acceptable errors. Beyond this threshold, we will consider DAVAD to be no more robust with respect to nonlinearities. Thus a quadratic wind
was simulated in which the constant and linear terms are as in
section 3e
, while the second order derivatives terms are ∂
= 5 × 10
, ∂
= 3 × 10
, ∂
= −2 × 10
, ∂
= 3.4 × 10
, ∂
= 1.23 × 10
, and ∂
= −7.6 × 10
, respectively. The simulations were performed first without noise, then with a gaussian or white noise ±1.5 m s
standard deviation. Second order derivative terms lead to
variations of the wind about 11 m s
at 50-km distance from the radar (4 m s
at 30-km distance), which can be thought of as strong nonlinearities. Without noise, errors on the retrieved set of wind parameters are essentially biases:
The vertical component of vorticity and the terminal fall velocity suffer in this case strong errors while the other parameters are acceptable. This provides the limit for avoiding to use DAVAD, this
is, when the terminal fall velocity begins to depart from values expected in stratiform areas. Previous results hold when adding the noise to the quadratic wind. So, as discussed previously, this
indicates again that noise is well filtered out by the analysis.
g. Other sources of errors; in particular, navigation angles
In real cases, other factors must be taken in to account. As shown in Protat et al. (1997), the navigation angles—roll, pitch, and drift (see their Fig. 5)—suffer errors which in turn increase errors
on the kinematic retrieved parameters. For example, an error on the roll contaminates the elevation angles θ and θ′. In the same way, pitch and drift errors contaminate both the elevation and azimuth
angles. In addition, owing to aircraft drift, purls are not perfectly circular but look like ellipses (see Fig. 4), inducing variations of the purl radius up to 30% about its mean value. Protat et
al. (1997) first study a systematic error of 1° (which is huge) on roll, pitch, and drift, and show that it results in maximum errors of about 2 × 10^–5 m s^–1 for DIV, DET, and DES, and 0.5 m s^–1
for the horizontal wind retrieved by SAVAD. This is also the case for DAVAD. Moreover for DAVAD, the maximum error on ROT is 4 × 10^–5 s^–1 since ROT is obtained as a residual term. However,
systematic errors are corrected in the preprocessing of the airborne Doppler radar data using the echo returned from the earth’s surface (Testud et al. 1995). So the only remaining errors on
navigation and pointing angles are fluctuation errors which are well filtered out by the processing. The error on DIV and ROT are 3 × 10^–6 s^–1 and 6 × 10^–6 s^–1, respectively, for a noise with
standard deviation 0.2° for the roll angle.
The final accuracy (including all the previous sources of errors but nonlinearities) on the retrieved parameters can be estimated as 0.2 m s^–1 for U[0], V[0], 4 × 10^–5 to 4 × 10^–6 s^–1 for ∂U/∂x,
∂U/∂y, ∂V/∂x, ∂V/∂y (1.5 × 10^–4 for ROT), 0.1 m s^–1 for V[f].
In summary, the practical application of DAVAD in real cases must be done carefully due to the various sources of errors encountered in the data processing. It must be noted that although the
retrieval of the wind components from coefficients a[1] and b[1] and of the stretching and shearing terms from coefficients a[2] and b[2] is relatively straightforward, this is not the case for DIV
and V[f], on the one hand, and ROT on the other hand [see section 2b(1)]. ROT is calculated as a residual term [see Eq. (18)], the accuracy of which is lower. As for the main term in (17) from which
DIV and V[f] are retrieved, it has to be analyzed in terms of radial distance: effective determination of V[f] must be done by retaining measurements at high elevations while DIV will be determined
by selecting preferentially low-elevation measurements.
4. Analysis of the FASTEX IOP 12 purls using DAVAD
The performance of the DAVAD analysis was examined in section 3 using simulated data. The present section is devoted to an application of this analysis to real data. Owing to the fairly large number
of purls performed during this IOP, the selected case is the secondary low sampled during FASTEX IOP 12. This low has been documented by Lemaître et al. (1999). Data were collected in the so-called
Mesoscale Sampling Area of FASTEX (see the FASTEX Operations Plan; Jorgensen et al. 1996). The main experimental facilities for that study were the UK-C130 aircraft able to launch dropsondes, the
French–U.S. ELDORA-ASTRAIA radar onboard the Electra aircraft, and the P3-42 NOAA aircraft Doppler radar equipped with the dual-beam French antenna. The two airborne Doppler radars performed various
types of trajectories such as straight lines regularly interrupted by purls (FASTEX Operations Plan; Jorgensen et al. 1996). The straight lines allow multiscale description of the lows using the
Mutiple Analytical Doppler (MANDOP) analysis (Scialom and Lemaître 1990; Dou et al. 1996), while mesoscale profiles of the wind and its first derivatives are derived from the purl processing, as
shown previously. During this IOP12, only the P3-42 radar data were available. This IOP is the most explosive low deepening, roughly −54 mb in 24 hours (9 February 1997) observed during FASTEX. The
overall synoptic description of this situation is given in Lemaître et al. (1999). The explosive pressure deepening is associated with the low 34A on which the measurements are conducted. The
Meteosat satellite IR picture (Figs. 3a–d) shows the evolution of the synoptic situation between 1500 and 1800 UTC, with the explosive deepening of a secondary low and the progressive development of
the so-called cloud head and a dry slot features, which are the main signatures of a frontal cyclone. The low 34A first moves northeastward at 40 m s^–1, deepens and then tracks toward Iceland. A
conceptual scheme of the secondary low and associated circulation (as described in Browning et al. 1997) is given in Fig. 3e, describing the main flows involved in the internal dynamics, namely, the
warm conveyor belt (WCB) the main part of which (W1) is parallel to the cold front while the branch W2 wraps around the frontal cyclone, the cold conveyor belt (CCB) along the secondary warm front
associated with the secondary low. This low was sampled on the northwestern corner of the MSA. Figure 3c also shows the aircraft trajectory in the relative frame of reference linked to the sampled
system. It can be seen that the secondary low has been crossed over by the P3-42 aircraft in order to sample the precipitating systems at several scales.
The box in Fig. 3c corresponds to the 430 km × 430 km area in which the mesoscale retrieval of the dynamics presented in Lemaître et al. (1999) was performed using the MANDOP analysis. In MANDOP,
each component of the wind is expressed as the product of three functions, each of them being an expansion along an axis (x, y, or z, respectively) in series of orthogonal functions of a given base
(e.g., Legendre polynomials, Fourier …) up to a given order, so that the radial wind along a given direction can also be expressed analytically. The analytical radial wind is then adjusted (in the
least squares sense) to the experimental radial wind, for all experimental data in order to retrieve the coefficients of expansion of the wind. This process thus consists in solving a linear system,
whose matrix is calculated using the orthogonal functions of the base. The system is generally ill-conditioned, since the vertical component of the wind is generally one order of magnitude less than
the horizontal ones. So the continuity equation (or mass conservation) is also introduced in the process, along with a boundary condition, both expressed analytically under the form of matrices to be
added to the main matrix of the problem. An important point concerns the scale selection: a given order of expansion corresponds to the scales retained in the analysis, so that smaller scales are
filtered out. MANDOP presents numerous advantages associated with its analytical form—analytical form of physical parameters linked to the wind, possibility to process data from any source of radar
data, both ground-based and airborne, … (see Scialom and Lemaître 1990 for more details)—but needs independent determination of the terminal fall velocity of hydrometeors. While under stratiform
conditions, DAVAD, as seen in section 2, is able to provide it together with the vertical profile of the wind and associated first space derivatives. This is a decisive advantage of the present
approach over the dual Doppler analyses, including MANDOP when one is interested in simple methods likely to be used under (quasi)-real time conditions. Moreover the terminal fall velocity vertical
profile may help identify hydrometeors, and the wind vertical profile (including its first derivatives) may be used in assimilation schemes.
Figure 4 (adapted from Lemaître et al. 1999) shows the horizontal cross section of the 3D reflectivity and wind fields at 0.5-km altitude derived from the P3 tail-radar measurements. Figure 4 also
displays the aircraft track, with the main purls flown along the track between 1500 and 1800 UTC. Note that during the 2 h of radar observations, the cyclone moved and evolved noticeably (Figs. 3a–d
). However, since each purl lasts three minutes, it may be considered that advection and internal evolution do not affect individual profile calculations. As specified in Lemaître et al. (1999),
three flows can be identified around the secondary low in Fig. 4, in agreement with the conceptual scheme of Browning et al. (1997): the CCB in the northern part of the figure (north of the
convective band, with air at intermediate temperature); just south of this flow, the WCB that wraps around the frontal cyclone. The convective area around the cyclone is the east part of the cloud
head. The flow in the northwest part is possibly of polar origin, and the flow in the southwest part is associated with the dry intrusion. The purls were mostly performed in the cloud head area as
shown in Fig. 3. The wind pattern displayed in Fig. 4 suggests that the vorticity at mesoscale, is cyclonic at flight altitude in the secondary low area. We now turn to a description of the main
results obtained with DAVAD.
Regarding the horizontal wind components using DAVAD [displayed Figs. 5a–k (left) W–E and (right) S–N components], three regimes are identified: the first one [Figs. 5a–c or purls 1–3] is mainly
southerly and exhibits strong winds 25–35 m s^–1 in magnitude in the lower troposphere, which is typical of the WCB area, then a northeasterly component appears first below 1-km altitude (Fig. 5f,
purl 6) then up to 2–2.5 km [Figs. 5e–g, purls 5 and 7] and their magnitudes decrease; then they turn to be more northerly [Figs. 5h–i, purls 8 and 9], and finally turn southwesterly [Figs. 5j, k,
purls 10 and 11]. The latter flow is typical of the dry intrusion area while the previous ones associated with the secondary low, are probably related to the CCB. The wind derived from DAVAD is in
good agreement with the wind derived from the sensor onboard the aircraft and with the wind from MANDOP (see Fig. 5). Small discrepancies with MANDOP wind fields are due to the different conditions
in which both analyses are applied: MANDOP analysis was applied in a whole area 430 × 430 km^2 at once on data gathered during two hours and correspond to scales resolved of around 20–30 km, while
each purl processed independently with DAVAD is representative of a three minutes time lag, which corresponds to a larger scale of about 60 km.
The vertical profiles of the terminal fall velocity V[f] in stratiform or moderately convective areas are generally characterized by values of about 5–12 m s^–1 in rain, below the 0°C isotherm, and
0–2 m s^–1, in snow, above. In the present case, the eleven purls between 1530 and 1732 UTC show V[f] profiles of similar shapes [see Figs. 6(a)–(k)], within the error bars (±0.1 m s^–1). On the same
figure, each V[f] profile is associated with a mean reflectivity profile.
Purls located in the northeastern part [Figs. 6(a)–(d), purls 1–4] exhibit stronger values in rain (6–8 m s^–1, associated with strong mean reflectivity values of 20–25 dBZ) than purls in the
southern part of the domain (4–5 m s^–1; Figs. 6h–k, purls 8–11, characterized by lower mean reflectivities of around 15 dBZ). This suggests that strong evaporation occurs in the cloud head and dry
air intrusion. Moreover the transition altitude between rain and snow is around 1000 m in the first case, 700 m in the second, which confirms that the air in the northeastern part is warmer (WCB)
than the air in the southern part. Finally, a detailed examination of the V[f] and reflectivity profiles above the 0°C isotherm (i.e., in solid precipitation) may in turn be split into a lower part
with V[f] velocity of about 2 m s^–1 generally associated with a secondary reflectivity maximum, and an upper part, with V[f] values ranging from 0 to 1 m s^–1, associated with lower reflectivities.
This may be either the signature of a graupel layer overlaid by a snow layer or the signature of aggregation occuring (e.g., Locatelli and Hobbs 1974; Heymsfield 1978).
It must be noted that the consistent results obtained on the V[f] profiles are an additional indirect a posteriori validation for the whole DAVAD analysis since V[f] is the most difficult parameter
to obtain.
Vertical profiles of vertical vorticity are displayed in Figs. 7(a)–(k). The vertical vorticity appears to be positive—cyclonic, in the southern part of the domain, which is consistent with the
existence of a secondary low [Figs. 7h–k, purls 8–11]. Values increase from 4 × 10^–4 s^–1, at 1701 UTC to 25 × 10^–4 s^–1 at 1732 UTC. The vertical vorticity profile at 1732 UTC located within the
low center corresponds to the absolute maximum of all purls at low levels, below 500-m altitude. This vorticity maximum also corresponds to a strong shearing deformation (Fig. 9k). In the northern
part of the domain, vertical vorticity is generally negative (anticyclonic), consistent with the horizontal wind shear (Fig. 4), except at low altitudes at 1558 and 1610 UTC (Figs. 7c and d, purls 3
and 4, toward the low center) and between 1 and 2 km at 1701 UTC (Fig. 7h close to the absolute maximum). Figure 7 also displays some MANDOP-derived vertical vorticity results. The comparison with
the DAVAD results is generally good: the sign (generally positive, i.e., cyclonic) is preserved, but DAVAD vorticities are often greater than MANDOP vorticities, which may be attributed to the fact
that, although the mean horizontal scale for DAVAD is greater, it also includes smaller-scale motions filtered out in MANDOP applied in such a large domain.
Vertical profiles of the vertical wind velocity obtained by upward integration of the horizontal wind divergence are displayed Figs. 8a–k. These profiles show that the vertical wind velocity is
generally weak, in good agreement with the stratiform or moderately convective character of the precipitation, and also with the results of Lemaître et al. (1999) in which vertical motions are
moderate. Note that as explained in section 2a, there is a systematic error ΔW (some cm s^–1) due to the lack of data at lower altitudes. Moreover vertical wind velocities are generally negative
(subsiding) when located within anticyclonic areas. However, W is positive (ascending) at low altitude within the region of convergence between the WCB and the CCB (Fig. 8a), and within the vortex
maximum at 1732 UTC (Fig. 8k), which is consistent with the current observation that cyclonic areas are also convergence areas. Vertical velocity is also positive between 1- and 2-km altitude at 1633
UTC in the cloud head region (Fig. 8f) in agreement with the analysis of particle trajectories of Lemaître et al. (1999).
The stretching and shearing deformations (
Browning and Wexler 1968
) also provide information on the dynamics of the flows [
Figs. 9a–h
]. Stretching dominates at 1530 and 1701 UTC below 1.5 km and 1732 UTC (
Figs. 9a, h, and k
). Shearing dominates at 1610, 1701, above 1.5 km and at 1710 above 0.5 km (
Figs. 9d, h, and i
). The other profiles show both shearing and stretching. The effect of these deformations on the horizontal potential temperature
gradient can be estimated following
Bluestein (1986)
, expressing the frontogenetic function
in terms of these deformations (denoted DET and DES) and of divergence DIV, tilting and diabatic function, and restricting the discussion to the adiabatic part of the frontogenesis:
the angle between the
axis (toward the east) and the baroclinic zone corresponding to the secondary low (
= −30°). Application of this equation to the present data leads to
Figs. 10a–k
, which exhibits frontogenetic terms due to deformation alone, and to deformation plus divergence (denoted as DEF FRONTOGEN and TOTAL FRONTOG, respectively, on the figures):
Purls 2–5 are performed in frontolytic areas [
Figs. 10b–e
], while purls 8, 10, and 11 are rather within frontogenetic areas (
Figs. 10h, j, and k
). The conclusion is that, in the active part of the cloud head, the observed stretching deformation is frontolytic (the stretching term of
is −4 × 10
K m
), and therefore tends to reduce the horizontal gradient of temperature, whereas on the western part of the cyclone, the observed shearing deformation tends to produce a new baroclinic zone (the
shearing term of
is 3 × 10
K m
). These results are consistent with the temperature pattern retrieved by
Lemaître et al. (1999)
who found a relatively uniform field on the east side of the cloud head and a well-defined baroclinic region on its west side.
5. Conclusions
The DAVAD analysis described previously is devoted to the retrieval of the wind field, its divergence, deformations, and vertical vorticity, and the terminal fall velocity V[f] within the stratiform
area of convective precipitating systems. The analysis takes advantage of the VAD approach that it is derived from, since the data are rearranged in the same scanning geometry (conical). It benefits
from the additional information brought by the two viewing angles under which the dual-beam antenna samples the precipitating system. This allows the retrieval of all the derivatives, in particular
the vertical vorticity, which was not the case in the traditional VAD analysis, nor in the single beam approach of Protat et al. (1997). Simulations have shown that the method was able to retrieve
the wind and its first-order derivatives, including the terminal fall velocity of the hydrometeors with satisfying accuracy when the fore and aft antennas form an equal angle with respect to the
perpendicular to the aircraft track—that is, with small drift and when the purl is roughly circular and large enough. When this is not the case, errors increase on all parameters. When all sources of
errors are considered, typical error values for FASTEX data are 0.2 m s^–1 on U[0], V[0], 4 × 10^–5 to 4 × 10^–6 on U[x], U[y], V[x], V[y], (up to 1.5 × 10^–4 s^–1, i.e., 40% on ROT), 0.1 m s^–1 on V
The DAVAD analysis has then been applied to a selected FASTEX case, IOP 12 (9 February 1997), characterized by an explosive deepening of a secondary low. A systematic application of DAVAD on all the
purls flown during this IOP was performed. Application to real data was possible in spite of the somewhat poor resolution in azimuth of the P3-42 data. Consistent retrieved features are nevertheless
identified. The vertical profiles of terminal fall velocity of the hydrometeors were obtained for all the processed purls. In stratiform precipitation areas, they can be used as an a posteriori
validation for the whole DAVAD analysis and can be included as input and validation for the models. The retrieved wind field is representative of the area in which it is obtained (warm conveyor belt,
cold conveyor belt, dry intrusion) in good agreement with the wind sensor onboard the aircraft, and with the MANDOP-derived winds. The corresponding first-order derivative terms are also consistent:
the vertical vorticity, cyclonic, or anticyclonic depending on the location of the purl in the system, is reasonably consistent with that obtained independently using the MANDOP analysis applied at
the scale of the cyclone as a whole. Despite the fact that vertical vorticity is determined through a residual term with moderate accuracy (owing mostly to the small purl radii during FASTEX),
cyclonic vorticity is found, as expected, with a maximum value at the center of the observed secondary low. The various flows around the secondary low are identified from the horizontal and vertical
velocity components and from the deformation profiles, depending on the considered area. The description of the system is consistent with the analysis of Lemaître et al. (1999) done using MANDOP.
Moreover, an analysis of the deformation parameters indicates that frontogenesis occurs on the west side of the cloud head, and frontolysis occurs on its east side. This is in agreement with the
temperature pattern of Lemaître et al. (1999) who find a baroclinic zone on the western side of the cloud head and a relatively uniform field on its eastern side. The application of DAVAD to other
FASTEX IOPs is presently under progress, using particular the ASTRAIA-ELDORA dataset, the resolution of which is twice better than that of the P3-42. Results from IOPs 12 and 16 are readily available
to the scientific community in the FASTEX database.
FASTEX has been supported by the Programme Atmosphère et Océan à Moyenne Echelle of the Institut National des Sciences de l’Univers under Contract 97/01 and by the European Commission under Contract
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Fig. 1.
Scans performed by the dual-beam antenna on an airborne Doppler radar flying a “purl” (circular trajectory). The vertical circles display the trajectory that a gate at a given range from the radar
would follow without aircraft motion. Including the aircraft motion, these circles become helices
Citation: Journal of Atmospheric and Oceanic Technology 20, 5; 10.1175/1520-0426(2003)20<630:ROKFFD>2.0.CO;2
Fig. 2.
Airborne Doppler radar flying a purl: (a) azimuth angles of the dual-beam antenna (view from top), and (b) elevation angles of the dual-beam antenna. The variable M is a point at distance r from the
radar A successively seen by the fore and aft antennas; P is the projection of M onto the plane perpendicular to the aircraft trajectory; m and p are projections of M and P onto the horizontal,
respectively; R is the purl radius; and O is the origin of horizontal axes at altitude z of M
Citation: Journal of Atmospheric and Oceanic Technology 20, 5; 10.1175/1520-0426(2003)20<630:ROKFFD>2.0.CO;2
Fig. 3.
Meteosat satellite IR picture at (a) 1500, (b) 1600, (c) 1700, and (d) UTC 1800, and (e) a conceptual scheme of the secondary low and associated circulation as proposed in Browning et al. (1997).
Superimposed in (c) are the aircraft trajectory in the frame of reference linked to the whole precipitation area and the box for mesoscale retrieval using the MANDOP analysis (Lemaître et al. 1999)
Citation: Journal of Atmospheric and Oceanic Technology 20, 5; 10.1175/1520-0426(2003)20<630:ROKFFD>2.0.CO;2
Fig. 4.
Horizontal cross section of radar reflectivity and 3D absolute wind field at 0.5-km altitude as derived by Lemaître et al. (1999) using MANDOP. Superimposed is the aircraft track with the purls flown
Numbers 1–11 refer to purls at 1531, 1544, 1557, 1609, 1625, 1634, 1645, 1659, 1709, 1720, and 1732 UTC, respectively
Citation: Journal of Atmospheric and Oceanic Technology 20, 5; 10.1175/1520-0426(2003)20<630:ROKFFD>2.0.CO;2
Fig. 5.
Vertical profiles using DAVAD (diamonds) of the (left) W–E and (right) N–S horizontal components of the wind at (a) 1531, (b) 1544, (c) 1557, (d) 1609, (e) 1625, (f) 1634, (g) 1645, (h) 1659, (i)
1709, (j) 1720, and (k) 1732 UTC, respectively. Triangles are from MANDOP analysis, and asterisks are measurements from the wind sensor onboard the aircraft
Citation: Journal of Atmospheric and Oceanic Technology 20, 5; 10.1175/1520-0426(2003)20<630:ROKFFD>2.0.CO;2
Fig. 10.
Same as Fig. 5 for the frontogenetic terms due to (left) deformation alone, and to (right) deformation and divergence
Citation: Journal of Atmospheric and Oceanic Technology 20, 5; 10.1175/1520-0426(2003)20<630:ROKFFD>2.0.CO;2 | {"url":"https://journals.ametsoc.org/view/journals/atot/20/5/1520-0426_2003_20_630_rokffd_2_0_co_2.xml","timestamp":"2024-11-11T19:58:12Z","content_type":"text/html","content_length":"849229","record_id":"<urn:uuid:bf405e25-e6c1-4abb-8583-251b66ef04d9>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00866.warc.gz"} |
Function Spaces - PDF Free Download (2024)
Function Spaces Fourth Conference on Function Spaces May 14-19,2002 Southern Illrnois University at Edwardsville Krzysztof Jarosz Editor
Function Spaces Fourth Conference on Function Spaces May 14-19,2002 Southern Illinois University at Edwardsville
Andy R. Magid
This volwne contains the proceedings of the Fourth Conference on Function Spaces, held May 14-19, 2002, at Southern Illinois University at Edwardsville. 2000 Mathematics Subject Classification.
Primary 32H02, 46E25, 46H05, 46JlO, 46J15, 46L07, 47AlO, 47B38, 47LlO, 54D05.
Library of Congress Cataloging-in-Publication Data Conference on Function Space (4th: 2002 : Southern Illinois University at Edwardsville) Function spaces : Fourth Conference on Function Spaces, May
14-19, 2002, Southern Illinois University at Edwardsville / Krzysztof Jarosz, editor. p. cm. - (Contemporary mathematics, ISSN 0271-4132 j 328) Includes bibliographical references. ISBN 0-8218-3269-7
(softcover : alk. paper) 1. Function spaces-Congresses. I. Jarosz, Krzysztof, 1953- II. Title. III. Contemporary mathematics (American Mathematical Society) j v. 328 QA323.C66 2002 515'.73-dc21
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Components of resolvent sets and local spectral theory PIETRO AlENA AND FERNANDO VILLAFANE
A Cauchy-Green formula on the unit sphere in C 2 JOHN T. ANDERSON AND JOHN WERMER
A connected metric space that is not separably connected RICHARD M. ARON AND MANUEL MAESTRE
Weighted Chebyshev centres and intersection properties of balls in Banach spaces PRADIPTA BANDYOPADHYAY AND S. DUTTA
Complete isometries - an illustration of noncommutative functional analysis DAVID P. BLECHER AND DAMON M. HAY
Some recent trends and advances in certain lattice ordered algebras KARIM BOULABIAR, GERARD BUSKES, AND ABDELMAJID TRIKI
An extension of a theorem of Wermer, Bernard, Sidney and Hatori to algebras of functions on locally compact spaces 135
The unique decomposition property and the Banach-Stone theorem AUDREY CURNOCK, JOHN HOWROYD, AND NGAI-CHING WONG
Some more examples of subsets of Co and L1 [0, 1] failing the fixed point property P. N. DOWLING, C. J. LENNARD, AND B. TURETT
Homotopic composition operators on HOC) (Bn) PAMELA GORKIN, RAYMOND MORTINI, AND DANIEL SUAREZ
Characterization of conditional expectation in terms of positive projections J. J. GROBLER AND M. DE KOCK
Characterizations and automatic linearity for ring homomorphisms on algebras of functions OSAMU HATORI, TAKASHI ISHII, TAKESHI MIURA, AND SIN-EI TAKAHASI
Weak *-extreme points of injective tensor product spaces KRZYSZTOF JAROSZ AND T. S. S. R. K. RAO
Determining sets and fixed points for holomorphic endomorphisms KANG-TAE KIM AND STEVEN G. KRANTZ
Localization in the spectral theory of operators on Banach spaces T. L. MILLER, V. G. MILLER, AND M. M. NEUMANN
Abstract harmonic analysis, homological algebra, and operator spaces VOLKER RUNDE
Uniform algebras generated by unimodular functions STUART J. SIDNEY
Analytic functions on compact groups and their applications to almost periodic functions THOMAS TONEV AND S. A. GRIGORYAN
Preface The Fourth Conference on Function Spaces was held at Southern Illinois Universityat Edwardsville from May 14 to May 19, 2002. It was attended by over 100 participants from 25 countries. The
lectures covered a broad range of topics, including spaces and algebras of analytic functions of one and of many variables (and operators on such spaces), LP-spaces, spaces of Banach-valued
functions, isometries of function spaces, geometry of Banach spaces, and other related subjects. The main purpose of the conference was to bring together mathematicians interested in various problems
within the general area of function spaces and to allow a free discussion and exchange of ideas with people working on exactly the same problems as well as with people working on related questions.
Hence, most of the lectures, and therefore the papers in this volume, have been directed to non-experts. A number of articles contain an exposition of known results (known to experts) and open
problems; other articles contain new discoveries that are presented in a way that should be accessible also to mathematicians working in different areas of function spaces. The conference was the
fourth in a sequence of conferences on function spaces at SlUE, with the first held in the spring of 1990, the second in the spring of 1994, and the third one in the spring of 1998. The proceedings
of the first two conferences were published with Marcel Dekker in the Lecture Notes in Pure and Applied Mathematics series (#136 and #172); the proceedings of the third conference were published by
the AMS in the Contemporary Mathematics series (#232). The abstracts, the schedule of the talks, and other information, as well as the pictures of the participants, are available on the conference
Web page at http://www.siue.edu/MATH/conference/. The conference was sponsored by grants from Southern Illinois University and from the National Science Foundation. The editor would like to thank
everyone who contributed to the proceedings: the authors, the referees, the sponsoring institutions, and the American Mathematical Society. The editor would also like to express very special thanks
to his wife, Dorota, for her active professional help during all of the stages of the organization - without her help the conference and the proceedings would not have been possible. Krzysztof Jarosz
Components of resolvent sets and local spectral theory Pietro Aiena and Fernando Villafane ABSTRACT. In this paper we shall study the components of various resolvent sets associated with some spectra
originating from Fredholm theory. In particular, we obtain a classification of these components by using, in the case of operators of Kato type, the equivalences between the single valued extension
property at a point and some kernel-type and range type conditions established in [2], [6], [3] and [5]. We also show that certain subspace valued mappings coincide on the components of the Kato type
resolvent and give a precise description of the operators having empty Kato type spectrum.
1. Single valued extension property
Throughout this paper, T is assumed to be a bounded linear operator on a complex Banach space X and L( X) will denote the algebra of all bounded linear operators on L(X). If x E X, the local
resolvent set of T at x E X, denoted by PT(X), is defined as the union of all open subsets U of C such that there is an analytic function f : U - X which satisfies the equation (1.1)
x for all >.. E U .
The local spectrum aT (x) of T at x is defined by aT (x) := C \ PT (x) and obviously aT(x) ~ a(T), where a(T) denotes the spectrum of T. The operator T E L(X) is said to have the single-valued
extension property at >"0 E C ( SVEP at >"0 for brevity) if, for every neighborhood U of >"0, the only analytic function f : U - X which satisfies the equation (>.1 - T)f(>..) = 0 for all >.. E U is
the function f == O. The operator T E L(X) is said to have SVEP if T has SVEP at every point>" E C. Clearly, if T has SVEP at >"0, then the analytic solution of (1.1) in a neighborhood U of >"0 is
uniquely determined. The SVEP was first introduced by Dunford [8], [9] and has later received a systematic treatment in Dunford-Schwartz [10]. The SVEP at a point was first introduced by Finch [11]
and successively investigated by several authors, see [20], [28], [2], [6], [3] and [5] . The basic role of SVEP arises in local spectral theory, since every operator which satisfies the so-called
Bishop's property (13) enjoys this 1991 Mathematics Subject Classification. Primary 47AlO, 47A11. Secondary 47A53, 47A55. Key words and phrases. Single valued extension property, semi-regular
operators, Kato decomposition property . The research was supported by the International Cooperation Project between the University of Palermo (Italy) and the University of Barquisimeto.
property, see [18] for definition and results. Recall that an operator T E L(X) on a Banach space X is said to be decomposable if for every open cover {UI , U2 } of C there exist T-invariant closed
linear subspaces Xl and X 2 of X for which X = Xl + X 2 , a(T IXd ~ Ul and a(T IX2 ) ~ U2 • The class of decomposable operators contains, for instance, all normal operators, all spectral operators,
all operators with a non-analytic functional calculus and all compact operators, or more generally all operators with a totally disconnected spectrum. Note that T is decomposable if and only if T and
its dual T* have property ((3), see Theorem 2.5.19 [18]. Consequently, if T is decomposable, then both T and T* have SVEP. For an arbitrary subset F of C let XT(F) be the local spectral subspace
associated with F, defined by XT(F) := {x EX: aT(x) ~ F}. If F is a closed subset of C, let XT(F) be the glocal spectral subspace associated with F, defined as the set of all x E X for which there
exists an analytic function f : C \ F ---+ X which satisfies (>.J - T)f(A) = x for all A E C \ F. Clearly, XT(F) and XT(F) are (not necessarily closed) linear subspaces of X with XT(F) ~ XT(F) for
all closed sets F ~ C. Note that, by Proposition 3.3.2 of [18], the identity XT(F) = XT(F) holds for all closed sets F ~ C precisely when Thas SVEP, and this is the case if and only if X T (0) = {O},
see Proposition 1.2.16 of [18]. The SVEP at a point Ao may be characterized in a similar way: T has SVEP at Ao if and only if ker (AoI - T) n X T (0) = {O}, cf. [1, Theorem 1.9]. Two important
subspaces in Fredholm theory are the hyperrange of T, defined by TOO(X) = n:'=l Tn(X), and the hyperkernel of T defined by N°O(T) = U:'=l ker Tn. Recall that the ascent of an operator T is the
smallest non-negative integer p := p(T) such that ker TP = ker TP+l. If such integer does not exist, we put p(T) = 00. Obviously, if T has finite ascent p then N°o (T) = ker TP. Analogously, the
descent q := q(T) of an operator T is the smallest non-negative integer q such that Tq(X) = Tq+l(X). If such integer does not exist we put q(T) = 00. Also, if T has finite descent q then TOO(X) = Tq
(X). It is well-known that if p(T) and q(T) are both finite then p(T) = q(T), see [15, Proposition 38.3]. Furthermore, p(AoI - T) = q(AoI - T) < 00 if and only if Ao is a pole of the resolvent R(A,
T) := (>.J - T)-l, [15, Proposition 50.23]. Associated with T E L(X) there is another linear subspace of X, the quasinilpotent part of T defined as
Ho(T) := {x EX: lim IITnxll l / n = O}. n-->oo
Evidently, N°O(T) ~ Ho(T). Moreover, Ho(T) = X if and only if T is quasinilpotent, i.e. a(T) = {O}, see [20, Theorem 1.5]. The following decomposition property studied by Mbekhta [20], [19], Mbekhta
and Ouahab [21], has its origin in the classical treatment of perturbation theory due to Kato [17], who showed an important decomposition for semi-Fredholm operators: DEFINITION 1.1. An operator T E
L(X) is called semi-regular if T(X) is closed and N°O(T) ~ TOO(X). An operator T E L(X) is said to be of Kato type if there exists a pair of T-invariant closed subspaces (M, N) such that X = M EB N,
the restriction T 1M is semi-regular and T IN is nilpotent. The pair (M, N) is called a generalized Kato decomposition ( GKD, for brevity) for T.
If, additionally, in the definition above we assume that N is finite-dimensional then T is said to be essentially semi-regular, see Rakocevic [25] and Miiller [24]. By Proposition 3.1.6 of [18], T is
semi-regular if and only if T* is semi-regular. Analogously, by Corollary 3.4 of [24], T is. essentially semi-regular if and only if T* is essentially semi-regular. Moreover, if T is essentially
semi-regular then Tn is essentially semi-regular for all n E N, again by Corollary 3.4 of [24]. It should be noted that the range of an essentially semi-regular operator is always closed. In fact, if
(!vI, N) is a GKD for T with N finite-dimensional, then T(X) is the direct sum of T(!vI), which is closed because T I !vI is semi-regular, and of T(N), which is finite-dimensional.
Two very important class of essentially semi-regular operators is given by the class of upper semi-Fredholm operators, defined by ~+(X) := {T E L(X) :
see Kato [17, Theorem 4] or West [31]. The class of Fredholm operators is defined by ~(X) := ~+(X)n~_(X). Note that a semi-Fredholm operator T is semi-regular if and only if its jump j(T) is zero,
see ([31, Proposition 2.2]. Moreover, if T is of Kato type, and in particular if T is semi-regular, then TOO(X) is closed with T(TOO(X)) = TOO(X), see Theorem 2.3 and Theorem 2.4 of [2], and TOO(X)
coincides with the analytical core K(T) := {x E X: there exist a constant c> 0 and a sequence of elements Xn E X such that Xo = x, TX n = Xn-l, and Ilxnll ::; cnllxll for all n EN}. It should be
noted that both subspaces K(T) and Ho(T) admit a local spectral characterization. In fact, (1.2)
= {x EX: 0 f/. O"T(X)},
see Mbekhta [20], Vrbova [30] and also Propositions 3.3.7 and 3.3.13 of [18], and Ho(T) = XT({O}, see Propositions 3.3.7 and 3.3.13 of [18]. Therefore, if T has SVEP, then Ho(T) = X T ( {O}). In the
following lemma by A.L and its proof we denote the annihilator of a subset A ~ X, and by .L B the pre-annihilator of a subset B ~ X*. LEMMA
1.2. For every T E L(X), the following statements hold
(i) Ho(T) ~.L K(T*) and K(T) ~.L Ho(T*). (ii) If T is a Kato type operator and the pair (!vI, N) is a GKD for T then
Ho(T) = Ho(T I !vI) EB Ho(T I N) = Ho(T I !vI) EB N.
PROOF. (i) See Proposition 4.1 of [5]. (ii) The proof of the equality K(T) = K(T I M) may be found in [1]. The second equality in (1.3) is clear, since the nilpotency of TIN implies Ho(T I N) = N.
The inclusion Ho(T I M) + Ho(T I N) £;;; Ho(T) is evident. To show the opposite inclusion, consider an arbitrary element x E Ho(T) and set x = y + z, with y E M, zEN. Since TIN is quasi-nilpotent
then N = Ho(T) £;;; Ho(T). Therefore y = x - z E Ho(T) n M = Ho(T I M) and consequently Ho(T) £;;; Ho(T I M) + Ho(T IN). (iii) Assume that T is essentially semi-regular and hence also T* essentially
semi-regular. Then T*n is essentially semi-regular for all n E N, so that T*n(X*) is closed for all n E N. From part (i) we know that N°O(T) £;;; Ho(T) £;;;1. K(T*), so that, to show the first two
equalities of (1.4), we need only to prove the inclusion 1. K(T*) £;;; N°O(T). For every T E L(X) and every n E N we have ker Tn £;;; N°O(T) and hence N°O(T)1. £;;; ker Tn1. = T*n(X*), because the
last subspace is closed for -:--.------:-::::,-1.
all n E N. From this we easily obtain that N°O(T) £;;; T*OO(X*) = K(T*), where the last equality holds since T* is essentially semi-regular and hence of Kato type. Consequently, 1. K(T*) £;;; N°O(T).
Thus the first two equalities of (1.4) are proved. The equality K(T) =1. Ho(T*) is proved in a similar way. (iv) The semi-regularity of T entails that N°O(T) £;;; TOO(X) = K(T), so that, by part
(ii), Ho(T) = Noo(T) £;;; K(T) = K(T) , and this concludes the proof. 0 We have already observed that an operator T E L(X) has SVEP at >'0 precisely when ker (>'01 - T) n K(>'ol - T) = {a}. From the
inclusion ker (>'01 - T) £;;; N°O(>'ol - T) £;;; Ho(>'ol - T) it then follows that the condition Ho(>'ol - T) n K(>'ol - T) = {a} implies that T has SVEP at >'0' Example 2.5 of [5] shows that SVEP
for T at a point does not necessarily imply that Ho(>'ol - T) n K(>'ol - T) = {a}. In [2] it has been shown that also the condition N°O(>'ol - T) n (>'01 - T)OO(X) = {a} implies SVEP at >'0 for T.
The next result shows that these implications are actually equivalences in the case that >'01 - T is of Kato type. THEOREM 1.3. If >'01 - T is of Kato type then the following properties are
equivalent: (i) T has the SVEP at >'0; (ii) Ho(>'ol - T) n K(>'ol - T) = {a}; (iii) Ho(>'ol - T) is closed; (iv) >'01 - T has finite ascent; (v) N°O(>'ol - T) n (>'01 - T)OO(X) = {a}. Furthermore, if
>'01 - T E L(X) is essentially semi-regular, the assertions (i)(v) are equivalent to the following conditions: (vi) Ho(>'ol - T) is finite-dimensional. (vii) N°O(>'ol - T*) + (>'01 - T*)(X*) is weak
*-dense in X*; (viii) Ho(>'ol - T*) + (>'01 - T*)(X*) is weak *-dense in X*; (ix) Ho(>'ol - T*) + K(>'ol - T*) is weak *-dense in X*. In this case >'01 - T E iP+(X).
PROOF. The equivalence of (i), (ii), (iii) and (iv) has been established in [3, Theorem 2.6 and Corollary 2.7]. The equivalence (i) {:} (v) has been proved in Theorem 2.6 of [2], see also Theorem 2.1
of the present paper. Assume now that AoI - T is essentially semi-regular. We may assume that AO = O. (i) {:} (vi) Obviously, if Ho(T) is finite-dimensional then Ho(T) is closed, so T has SVEP at 0,
by the equivalence (i) {:} (iii). Conversely, if T has SVEP at 0 then also TIM has SVEP at 0, since the local SVEP is inherited by the restrictions to closed invariant subspaces. The semiregularity
of TIM then implies that TIM is injective, see Theorem 2.14 of [1] and therefore N°O(T) = {O}. By part (iii) of Lemma 1.2 we then conclude that Ho(T I M) = N°O(T 1M) = {O}. From part (ii) of Lemma
1.2 it follows that Ho(T) = {O} EB N = N is finite-dimensional. Finally, if T is essentially semi-regular then also Tn is essentially semi-regular and therefore, the ranges Tn(x) are closed for all n
E N. From Theorem 4.3 of [5] it then follows that the condition (i) is equivalent to each one of the conditions (vii), (viii) and (ix). It remains to establish that (i) implies that AoI - T E ~+(X).
Clearly, if Ho(AoI - T) is finite- dimensional then also its subspace ker (AoI - T) is finitedimensional. Since (AoI -T)(X) is closed we then conclude that AoI -T E ~+(X).
The next Theorem 2.1 will show that, if AoI - T of Kato type, then Ho(AoI T) n K(AoI - T) = N°O(AoI - T) n (AoI - T)OO(X). The following characterizations of SVEP for the dual T* are dual, in a
sense, to those given in Theorem 1.3. THEOREM 1.4. Suppose that AoI - T is of Kato type. Then the following statements are equivalent: (i) T* has SVEP at AO; (ii) X = Ho(AoI - T) + K(AoI - T); (iii)
AoI - T has finite descent; (iv) X = N°O(AoI - T) + (AoI - T)OO(X); (v) Ho(AoI - T) + K(AoI - T) is norm-dense in X; (vi) N°O(AoI - T) + (AoI - T)OO(X) is norm-dense in X; Furthermore? if AoI - T E L
(X) is a essentially semi-regular then the assertions (i)-(vi) are equivalent to the following conditions: (vii) K(AoI - T) is finite-codimensional; (viii) N°o (AoI - T) + (AoI - T) (X) is norm-dense
in X; (ix) Ho(AoI - T) + (AoI - T)(X) is norm-dense in X. In this case AoI - T E ~_(T). PROOF. Also here we may assume that AO = 0 and T is of Kato type. The equivalence of (i), (ii) and (iii) has
been established in Theorem 2.9 of [3]. The equivalence of (i) and (iv) has been proved in Theorem 2.9 of [2], see also Theorem 2.1 of the present paper. Clearly, (ii) ::::} (v), (iv) ::::} (vi). The
implications (v) ::::} (i) and (vi) ::::} (i) have been proved in Corollary 4.2 of [5], so that the statements (i)-(vi) are equivalent. Now, assume that T is essentially semi-regular. Then Tn(x) is
closed for
all n E N, so that, by Theorem 4.3 of [5], the statements (i), (viii) and (ix) are equivalent. To conclude the proof note first that if (M, N) is a GKD for T then the pair (Nl.,Ml.) is a GKD for T*.
Now, if T* has SVEP at 0, then, as observed in the proof Theorem 1.3, T* I Nl. is injective and therefore, see Lemma 2.8 of [2], TIM is surjective. Therefore K(T) = K(T I M) = M is
finite-codimensional, so that the implication (i) => (vii) is proved. Conversely, suppose that the analytical core K(T) is finite-co dimensional. From K(T) = TOO(X) ~ Tn(x) we deduce that q(T) < 00,
so that (vii) implies (iii), and the proof of the equivalences is complete . Finally, from the inclusion K(>"ol -T) ~ (>"01 -T)(X) we infer that, if K(>"o/T) finite-co dimensional, then also (>"01 -
T)(X) is finite-codimensional, so that >"01 - T E
.. E C : >..I - T is not bounded below} does not cluster at >"0, then T has SVEP at >"0 and, dually, if the surjectivity spectrum l7 su (T) = {>.. E C : >..I - T is not surjective} does not cluster
at >"0, then T* has SVEP at >"0' The next result shows that for Kato type operators these implications may be reversed. THEOREM
1.5. If >"01 - T is of Kato type, then the following equivalences hold:
(i) T has the SVEP at >"0 precisely when l7ap(T) does not cluster at >"0, [6, Theorem 2.2]; (ii) T* has the SVEP at >"0 precisely when l7su(T) does not cluster at >"0,' [6, Theorem 2.5].
2. Components In this section we shall take a closer look at the components of some resolvent sets associated with the various spectra originating from Fredholm theory. In particular, we shall obtain
a classification of these components, by using the constancy of some mappings and the equivalences between the SVEP at a point and the kernel-type and range type conditions, established in the
previous section. For an operator T E L(X), we consider the following parts of the ordinary spectrum: the Kato spectrum l7k(T) := {>.. E C : >..I - T is not semi-regular},
and the essential Kato spectrum l7ke(T) := {>.. E C : >..I - T is not essentially semi-regular}.
Moreover, we define l7kt(T) := {>.. E C : >..I - T is not of Kato type}.
It is known that the three sets O'k(T), O'kt(T) and O'ke(T) are closed, for the first set see [18, Proposition 3.1.9], for the other two sets see [4, Corollary 1]. Moreover, O'k(T) and O'ke(T) are
nonempty, since the first spectrum contains the boundary of O'(T), see [18, Proposition 3.1.6], while the second spectrum contains the boundary of the Fredholm spectrum O'f(T) := p, E C : >.J - T ~
cJ>(X)}, see [24, Theorem 3.8]. Next we shall show that O'kt(T) is non-empty precisely when O'(T) is not a finite set of poles. Let Pk(T) := C \ O'k(T), Pkt(T) := C \ O'kt(T) and Pke(T) := C \ O'ke
(T) be the resolvents associated with these spectra. The sets Pk(T), Pkt(T) and Pke(T) are open subsets of C, so they may be decomposed in connected disjoint open non-empty components. Clearly, (2.1)
Note that for every T E L(X) we have Pk(T) In [29]
= Pk(T*)
and Pke(T)
= Pke(T*).
6 Searcoid and West showed the constancy of the mappings
on the components of the semi-Fredholm resolvent Psf(T) := C \ O'sf(T), where O'sf(T) is the semi-Fredholm spectrum defined by
O'sf(T) := {A E C: >.J - T
cJ>+(X) U cJ>_(X)}.
From the Kato decomposition for semi-Fredholm operators we easily obtain the following inclusions (2.3) The work of 6 Searcoid and West [29] extended previous results established by Homer [16], by
Goldmann and Kraekovskii [13], [14], and by Saphar [26], which have established the constancy of the functions
on a component of the semi-Fredholm resolvent Psf(T), except for the discrete subset of points for which AI - T is not semi-regular. In the same vein, Forster [12] showed that the mappings
are constant as A ranges through a component of the Kato resolvent Pk(T). The constancy of these mappings has also been studied by Mbekhta and Ouahab [22], which showed the constancy of the mappings
A ~ Ho(>.J - T)
+ K(>.J -
A ~ Ho(>.J - T) n K(>.J - T)
on the components of Pkt(T). The next result shows that the mappings (2.2) and (2.4) coincide, respectively, on the components of Pkt(T), so that the Mbekhta and Ouahab result extends the previous
result of 6 Searcoid and West. 2.1. Let >.J - T be of Kato type. Then (i) N°O(>.J - T) + (>.J - T)OO(X) = Ho(>.J - T) + K(>.J - T).
(ii) N°o(>.J - T)
n (>.J -
T)OO(X) = Ho(>.J - T)
n K(>.J -
PROOF. (i) Throughout this proof we may take>. = O. Let (M, N) be a GKD for T such that (T I N)d = 0 for some integer dEN. By part (ii) of Lemma 1.2 we know that K(T) = K(T I M) = K(T) n M. Moreover,
by part (iv) of Lemma 1.2, the semi-regularity of TIM implies that Ho(T I M) ~ K(T I M) = K(T). From this we obtain
n K(T) = Ho(T) n (K(T) n M) = (Ho(T) n M) n K(T) = Ho(T I M) n K(T) = Ho(T 1M), and therefore Ho(T) n K(T) = Ho(T 1M). Ho(T)
We claim that Ho(T) + K(T) = NEB K(T). From N ~ ker Td ~ Ho(T) we obtain that NEB K(T) ~ Ho(T) + K(T). Conversely, from part (ii) of Lemma 1.2 we have
Ho(T) = NEB Ho(T I M) = NEB (Ho(T) n K(T))
NEB K(T),
so that
Ho(T) + K(T)
(N EB K(T)) + K(T)
NEB K(T),
so our claim is proved. Since K(T) = TOO(X) for every operator of Kato type, we obtain from the inclusion N ~ ker Td ~ N°O(T), that
+ K(T) =
so the equality N°O(T)
NEB K(T)
+ TOO (X)
= Ho(T)
+ TOO (X)
+ K(T),
+ K(T) is proved.
(ii) Suppose again that>. = O. Let (M, N) be a GKD for T such that, for some dEN, we have (T IN)d = O. Then ker Tn = ker (T IM)n for every natural n :2: d. Since ker Tn ~ ker Tn+! for all n E N we
then have
N°O(T) =
U ker Tn = U ker(T I M)n = N°O(T 1M).
The semi-regularity of TIM then implies, by part (iii) of Lemma 1.2, that (2.5)
N°O(T) = N°O(T I M) = Ho(T I M) = Ho(T)
n M.
Next we show that the equality Ho(T) n M = Ho(T) n M holds. The inclusion Ho(T) n M ~ Ho(T) n M is evident. Conversely, suppose that x E Ho(T) n M. Then there is a sequence (xn) C Ho(T) such that Xn
-+ x as n -+ 00. Let P be the projection of X onto M along N. Then PX n -+ Px = x and PX n E Ho(T) n M. Therefore x E Ho (T) n M. Finally, from (2.5) and taking into account that K(T) n M = K(T) =
TOO(X), we then obtain
n TOO (X) = Ho(T) n M n K(T) = Ho(T) n (M n K(T)) = Ho(T) n K(T),
so the proof is complete.
From the constancy of the mappings>. -+ Ho(>.J - T)nK(>.J -T), or, which is the same, of the mappings>. -+ Noo(>.J - T) n (>.J - T)OO(X), on the components of Pkt(T) and the results established in
the previous section we now obtain the following classification.
THEOREM 2.2. LetT E L(X) and 0. a component ofpkt(T). Then the following alternative holds: either (i) T has SVEP for every point ofn. In this case p(>.J - T) < 00 for all A E n. Moreover, aap(T)
does not have limit points in 0.; every point of 0., except possibly for at most countably many isolated points, is not an eigenvalue of T.
or (ii) T has SVEP at no point of n. In this case p(>.J - T) = Every point of 0. is an eigenvalue of T, PROOF. (i) Suppose that T has SVEP at AO E Ho(>.J - T) is closed and
for all A E
Then, by Theorem 1.3,
n K(AoI - T) = HO(AoI - T) n K(AoI - T) = {O}. Since the mapping A --+ Ho(>.J - T) n K(>.J - T) is constant on the component 0., then Ho(>.J - T) n K(>.J - T) = {O} for all A E 0. and this implies,
again by HO(AoI - T)
Theorem 1.3, that T has SVEP at every A E n. This is equivalent, also by Theorem 1.3, to saying that p(AI - T) < 00 for all A E n. Moreover, from Theorem 1.5, aap(T) does not cluster in 0. and,
consequently, every point of 0. is not an eigenvalue of T, except a subset of 0. which consists of at most count ably many isolated points.
(ii) This is clear, again by Theorem 1.3.
Recall that A E C is said to be a deficiency value for if >.J - T is not surjective. THEOREM 2.3. Let T E L(X) and 0. a component of Pkt(T). Then the following alternative holds: either (i) T* has
SVEP for every point of n. In this case q(>.J - T) < 00 for all A E n. Moreover, asu(T) does not have limit points in 0.; every point ofn, except possibly for at most countably many isolated points,
is not a deficiency value of T.
or (ii) T* has the SVEP at no point of n. In this case q(>.J - T) = A E 0. and every A E 0. is a deficiency value of T.
for all
PROOF. Proceed as in the proof of Theorem 2.2, combining the constancy on the components of Pkt(T) of the mapping A E 0. --+ K(>.J - T) + Ho(>.J - T) (or, equivalently, the constancy of the mapping A
E 0. --+ N°O(>.J -T) +(>.J -T)OO(X)), with Theorem 1.4 and Theorem 1.5. 0 The previous results lead to a precise description of the operators whose Kato type spectrum akt(T) is empty. Most of the
results of the following theorem may be found in Mbekhta [23] in the context of operators on Hilbert spaces. However, our proofs, involving local spectral theory, are considerably simpler and are
established in the more general context of operators acting on Banach spaces. Recall first that T E L(X) is algebraic if there exists a non-trivial polynomial h such that h(T) = O. THEOREM 2.4. For
an operator T E L(X) the following statements are equivalent: (i) akt(T) is empty;
(ii) AI - T has finite descent for every A E C; (iii) AI - T has finite descent for every A E 8a(T), where 8a(T) is the topological boundary of a(T); (iv) a(T) is a finite set of poles of R(A, T);
(v) T is algebraic.
PROOF. (i) (ii) Suppose that akt(T) = 0. Then Pkt(T) has an unique component n = C and therefore, by Theorem 2.2, T has SVEP at every point of C, since T has SVEP at the point ofthe resolvent p(T).
On the other hand, if >.1 -T is of Kato type, then also Al* - T* is of Kato type, see [4, Proposition 1). Therefore, C = Pkt(T) = Pkt(T*) and consequently, by Theorem 2.3, also T* has SVEP. Since >.1
- T is of Kato type, by Theorem 1.4, we then conclude that q(>.1 - T) < 00 for every A E C.
(ii)* (iii) Obvious. (iii) * (iv) Since T has SVEP at every A E 8a(T) then the condition q(>.1 T) < 00 entails that every A E 8a(T) is a pole of R(A, T), see Corollary 1 of [27), and hence an
isolated point of a(T). Clearly, this implies that a(T) = 8a(T), so a(T) is a finite set of poles. (iv) (i) It suffices to prove that >.1 - T is of Kato type for all A E a(T). Suppose that a(T) is a
finite set of poles of R(A, T). If A E a(T), let P be the spectral projection associated with the singleton {A}. Then X = M EB N, where M := K(>.1 - T) = ker P and N := Ho(>.1 - T) = P(X), see the
proof of Theorem 1.6 of [19) or also Theorem 1 of [27). Since A is a pole of R(A, T), by Proposition 50.2 of [15), >.1 - T has positive finite ascent and descent, and if p := p(Aol - T) = q(>.1 - T),
then N = ker (>.1 - T)P. From the classical Riesz functional calculus we know that a(T I M) = a(T) \ {A}, [15, Theorem 49.1), so that (>.1 - T) I M is bijective, while (>.1 - T I N)P = O. Therefore
>.1 - T is of Kato type for every A E C. (iv) (v) Assume that a(T) is a finite set of poles {Al,··· ,An}, where for every i = 1,· .. ,n with Pi we denote the order of Ai. Let h(A) := (AI - A)Pl ...
(An - A)Pn. Then, see Lemma 3.1.15 of [18),
n(Ail - T)Pi (X) = nK(Ai1 - T), n
where the last equality follows since T has SVEP and Ail - T is of Kato type, see Theorem 2.9 of [3). But the last intersection is {O}, since, by the local spectral characterization of the analytical
core (1.2), if x E K(Ai1 - T) n K(Ajl - T), with Ai =1= Aj, then aT(x) ~ {Ai} n {Aj} = 0 and hence x = 0, since T has SVEP. Therefore h(T) = O. (v) (i) As in the proof of (iv) (i) it suffices to show
that AI - T is of Kato type for all A E a(T). Let h be a polynomial such that h(T) = O. From the spectral mapping theorem we easily deduce that a(T) is a finite set {AI,··· ,An}. The points AI,···
,An are zeros of finite multiplicities of h, say k 1 ,··· ,kn' respectively, so that h(A) = (AI - A)kl ... (An - A)kn and hence
X = ker h(T) = $ker (Ail - T)ki, i=l
see Lemma 3.1.15 of [18]. Now, suppose that A = Ai for some j and define ho(A) :=
II(Ai -
We have M := ker ho(T) =
EB ker (Ail -
and if N := ker (Ail - T)kj, then X = M EEl Nand M, N are invariant under Ail - T. From the inclusion ker (Ail - T) ~ ker (Ail - T)k j = N, we infer that the restriction of Ail - T on M is injective.
It is easily seen that (Ail - T)(ker (AJ - T)ki) = ker (AJ - T)ki,
i =I- j,
so that (Ai 1- T)( M) = M. Hence the restriction of Ai 1- T on M is also surjective and therefore bijective. Obviously, (Ajl - T) I N)k j = 0, so that Ajl - T is of Kato type, as desired. 0 A bounded
operator on a Banach space X is said to satisfy a polynomial growth condition, if there exists a K > 0, a 8 > for which
IIexp(iAT)II ::; K(1
+ IAleS)
for all A E JR,
Examples of operators which satisfy a polynomial growth condition are hermitian operators on Hilbert spaces, nilpotent and projection operators, algebraic operators with real spectra, see Barnes [1].
In Laursen and Neumann [18, Theorem 1.5.19] it is shown that the class P(X) of operators which satisfy a polynomial growth condition coincides with the class of all generalized scalar operators
having real spectra. As noted in Barnes [1], if T E P(X) and Aol - T has closed range for some Ao E C then q(Aol - T) is finite. From Theorem 2.4 it follows that, if T E P(X), then the condition (AI
- T)(X) closed for all A E C implies that O"kt(T) = 0. Other classes of operators for which O"kt(T) = 0 may be found in [23]. The classification of the components of Pes(T) may be easily obtained
from Theorem 2.2 and Theorem 2.3, once it has been observed, that the two sets Pes (T) and Pkt(T) may be different only for a denumerable set, see for instance Corollary 1 of [4]. We now look at the
components of PSf(T). Recall that for a semi-Fredholm operator T E
The eigenvalues do not have a limit point in n and every point of value. (iii) T* has SVEP at the points of n, while T fails to have SVEP at the points of n. In this case we have ind (AI - T) > 0, p
(AI - T) = 00 and q(AI - T) < 00 for every A E n. The deficiency values do not have a limit point in n, while every point of n is an eigenvalue. (iv) Neither T or T* has SVEP at the points of n. In
this case we have p(AI - T) = q(AI - T) = 00 for every A E n. The index may assume every value in Z; all the points of n are eigenvalues and deficiency values. for every A E
n is a deficiency
PROOF. The case (i) is clear from the results established in the previous section, Theorem 2.2 and Theorem 2.3. The index ind (AI - T) = 0 by Proposition 38.6 of [15J. In the case (ii) the condition
p(AI - T) < 00 implies that AI - T has index less or equal to 0, while the condition q(AI - T) = 00 excludes that ind (AI - T) = 0, see Proposition 38.5 of [15J. A similar argument shows in the case
(iii) that ind (AI - T) > O. The statements of (iv) are clear. 0 The following corollary establishes that a very simple classification of the components of semi-Fredholm resolvent may be obtained in
the case that T, or T* has SVEP. Recall that the case that both T and T* have SVEP applies in particular to the decomposable operators. COROLLARY 2.6. Let T E L(X) and
any component of PSf(T). If T has
SVEP then only the case (i) and (ii) of Theorem 2.5 are possible, while if T* has SVEP only the case (i) and (iii) are possible. Finally, if both T and T* have SVEP then only the case (i) is
In the next result we consider the components of Pk(T), which is the smallest of the resolvent sets that we have considered. THEOREM 2.7. Let T E L(X) and n any component of Pk(T). Then one of the
following possibilities occurs: (i) Both T and T* have SVEP at every point of n. In this case we have n ~ p(T). (ii) T has SVEP at the points of n, while T* fails to have SVEP at every point of n. In
this case we have n n aap(T) = 0 and n ~ asu(T). (iii) T* has SVEP at the points of n, while T* fails to have SVEP at the points of n. In this case we have n n asu(T) = 0 and n ~ aap(T). (iv) Neither
T or T* have SVEP at the points of n. In this case we have n ~ aap(T) n asu(T). PROOF. (i) Let Ao E n. The subspaces M:= X and N := {O} give a GKD for T and the subspaces ..L N = X and ..L M = {O}
give a GKD for T*. As observed in the proof of Theorem 1.3 and Theorem 1.4 if T has SVEP at Ao then Ho(AoI - T) = N = {O} and if T* has SVEP at Ao then K(AoI - T) = M = X. Therefore Ao E p(T). (ii)
In this case Ho(AI - T) = {O} and (AI - T)(X) is closed for every A E n. If A ~ asu(T) then A E p(T) = p(T*) and this is impossible, since T* does not have SVEP at A.
(iii) In this case K(M - T) = X for every A E n, so A f/. O'su(T). If A f/. O'ap(T) then A E p(T) and this is impossible, since T does not have SVEP at the point A. (iv) Use the same arguments as in
part (ii) and (iii).
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extension property and the generalized Kato decomposition property. Acta Sci. Math. (Szeged) 67, (2001), 461-477. [3J P. Aiena, M. L. Colasante, M. Gonzalez Operators which have a closed
quasi-nilpotent part, Proc. Amer. Math. Soc. 130, (2002), 2701-2710. [4J P. Aiena, M. Mbekhta Characterization of some classes of operators by means of the Kato decomposition. (1996), Boll. Un. Mat.
It. 10-A, 609-21. [5J P. Aiena, T. L. Miller, M. M. Neumann On a localized single valued extension property, (2001), to appear on Proc. Royal Irish Acad. [6J P. Aiena, E. Rosas The single valued
extension property at the points of the approximate point spectrum. to appear on J. Math. Anal. Appl. [7J B. A. Barnes Operators which satisfy polynomial growth conditions. Pacific. J. Math. 138
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Dunford, J. T. Schwartz Linear operators, Part Ill. (1971), Wiley, New York. [I1J J. K. Finch The single valued extension property on a Banach space Pacific J. Math. 58 (1975), 61-69. [12J K. H.
Forster Uber die Invarianz eineger Riiume, die zum Operator T - >'A gehoren. Arch. Math. 17 (1966), 56-64. [13J M. A. Goldman, S. N. Kraekovskii Invariance of certain subspaces associated with A - >
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M. Neumann Introduction to local spectral theory, Clarendon Press, Oxford 2000. [19J M. Mbekhta Sur I 'unicite de la decomposition de Kato generalisee. Acta Sci. Math. (Szeged) 54 (1990), 367-77.
[20J M. Mbekhta Sur la theorie spectrale locale et limite des nilpotents. Proc. Amer. Math. Soc. 110 (1990), 621-631. [21J M. Mbekhta, A. Ouahab Operateur s-regulier dans un espace de Banach et
theorie spectrale. Acta Sci. Math. (Szeged) 59 (1994), 525-43. [22J M. Mbekhta, A. Ouahab Perturbation des operateurs s-reguliers . Topics in operator theory, operator algebras and applications,
Timisoara (1994), Rom. Acad. Bucharest, 239-249. [23J M. Mbekhta Ascent, descent et spectre essential quasi-Fredholm., Rendiconti Circ. Mat. Palermo (2), 46, (1997), 175-196. [24J V. Miiller On the
regular spectrum., J. Operator Theory 31 (1994), 363-380. [25J V. Rakocevic Generalized spectrum and commuting compact perturbation. Proc. Edinburgh Math. Soc. 36 (2), (1993), 197-209. [26J P. Saphar
Contribution a l'etude des applications lineaires dans un espace de Banach. Bull. Soc. Math. France 92 (1964), 363-84. [27J C. Schmoeger On isolated points of the spectrum of a bounded operator.
Proc. Amer. Math. Soc. 117 (1993), 715-19.
[28] C. Schmoeger (1995). Semi-Fredholm opemtors and local spectml theory., Demonstratio Math. 4, 997-1004. [29] M. 6 Searc6id , T. T. West Continuity of the genemlized kernel and mnge for
semi-Fredholm opemtors. Math. Proc. Camb. Phil. Soc. 105, (1989), 513-522. [30] P. Vrbova On local spectml properties of opemtors in Banach spaces. Czechoslovak Math. J. 23(98) (1973a), 483-92. [31]
T. T. West A Riesz-Schauder theorem for semi-Fredholm opemtors. Proc. Roy. Irish. Acad. 87 A, N.2, (1987), 137-146. DIPARTMENTO DI MATEMATICA ED ApPLICAZIONI, VIALE DELLE SCIENZE, UNIVERSITA DI
PALERMO, 90128 PAI,ERMO, ITALY E-mail address:paienafDmbox.unipa.it DEPARTAMENTO DE :tViATEMATICAS, FACULTAD DE CIENCIAS, UNIVERSIDAD UCLA DE BARQUISIMETO (VENEZUELA) E-mail
Contemporary Mathematics Volume 328, 2003
The Fejer-Riesz Inequality and the Index of the Shift John R. Akeroyd ABSTRACT. In this brief article we consider a result that can be characterized as the converse of the Fejer-Riesz inequality.
This result has bearing on theory concerning the index of the shift.
Let I-" be a finite, positive Borel measure with support in {z : Izl ::; I} (~ := {z : Izl < I}) and let p2(1-") denote the closure of the polynomials in L2(1-")' We assume throughout that p2(1-") is
irreducible (i.e., it contains no nontrivial characteristic functions). From this it follows that: a) I-"lalIJl« m (normalized Lebesgue measure on 8~), and b) for any w in ~, f f--+ f(w) defines a
bounded linear functional for polynomials f with respect to the L2(1-") norm, that is bounded independent of w in any compact subset of~; cf. [15], Theorem 5.8. In other words, ~ = abpe(p2(1-")) -
the collection of analytic bounded point evaluations for P2(1-")' In the case that 1-"(8~) > 0 and I-"llIJl is radially weighted area measure, there is much in the literature concerning which weights
have the property that p2(1-") is irreducible; for instance, cf. [9] and [10]. Returning to our general setting, notice that multiplication by the independent variable z is a bounded operator on p2
(1-"). We call this operator the shift (on p2 (1-")) and denote it by M z, suppressing reference to 1-". Let Lat(Mz ) denote the collection of closed invariant subspaces for the shift (on p2(1-"))'
If {O} "I- M E Lat(Mz), then, since 0 E abpe(p2(1-")), zM is a closed subspace of M and in fact dim(M e zM) 2: 1. In the case that 1-"(8~) = 0, C. Apostol, H. Bercovici, C. Foias and C. Pearcy have
shown that for any natural number n, and for n = 00, there exists Min Lat(Mz ) such that dim(MezM) = n; cf. [3], and for related work see [7]. This result is an indication of how very large Lat(Mz )
is in the case that 1-"(8~) = O. In fact, it is large enough to "model" the general invariant subspace problem for bounded operators on a Hilbert space; again, cf. [3] and [7]. A classical example
that falls under this heading (1-"(8~) = 0) is the Bergman space L~(~), which equals p2(1-") when I-" = A - area measure on ~. At the other extreme, if I-" = m, then P2(1-") represents the Hardy
space H2(~) and so, by Benrling's Theorem, dim(M e zM) = 1 for all nontrivial members M 1991 Mathematics Subject Classification. Primary 47 A53, 47B20, 47B38; Secondary 30ElO, 46E15. © 15
2003 American Mathematical Society
of Lat(Mz ). It has been conjectured that for any measure I-" with mass on the unit circle (i.e., 1-"(8lI))) > 0), the outcome mimics that of Hardy space case and dim(M8 zM) = 1 whenever {OJ =I- M E
Lat(Mz ); cf. [5]. There are a number of results in the literature that support this conjecture. The first of these is found in a paper ofR. Olin and J. Thomson (cf. [12]) who show that it holds
whenever I-" that has a so-called "outer hole" in its support. Subsequently (in [11]), L. Miller shows that it also holds in the case that I-" = A + mi.,!' where 'Y is some nontrivial sub arc of
8lI)). In [17], L. Yang extends this result of L. Miller to the case: I-" = A + mlE, where E is any compact subset of 8lI)) of positive Lebesgue measure that satisfies the Carleson condition 1
Ln m(In)log(m(In)) <
{In} are the intervals that are complementary to E in 8lI)). And then (in [16]) J. Thomson and L. Yang obtain this extension of L. Yang in the more general context of the shift on pt(I-"), for 1 < t
< 00. The conjecture has recently been established for any measure I-" for which there is a nontrivial subarc 'Y of 8lI)) such that
10g ( 1;; )dm >
with no special assumption made concerning 1-"10; cf. [2]. In [2], the author makes use ofresults in an earlier paper (cf. [1]) that are intricately related to the seminal work of R. Olin and J.
Thomson in [12]. Specifically, in [1] the author defines what it means for I-" to be strongly inscribed and shows that if I-" is such, then indeed dim(M 8 zM) = 1 for each nontrivial member M of Lat
(Mz ). To be explicit, I-" is said to be strongly inscribed if there is a Jordan subregion W of lI)) with rectifiable boundary (we let Ww denote harmonic measure on 8W for evaluation at some point in
W) with the properties: i) ww(8lI))) > 0, and ii) there is a nonnegative function h in LOO(ww) such that log(h) E L1 (ww)
law Ifl 2 w: ; f Ifl2dl-" hdw
for all polynomials
Since 8W is rectifiable, (i) is equivalent to: m((8W)n(8lI)))) > O. And this definition is not truly altered if we drop the requirement that W has rectifiable boundary, because if W were any simply
connected subregion of lI)) that satisfies (i) and (ii), then we could find a Jordan subregion V of W, where V has rectifiable boundary and V itself satisfies (i) and (ii); cf. [14], Proposition
6.23. It is still an open question as to whether or not our general assumptions concerning 1-", along with the hypothesis that 1-"(8lI))) > 0, together imply that I-" is strongly inscribed. In this
brief article we discuss what amounts to the converse of the Fejer-Riesz inequality and find that this converse has close ties to the definition of "strongly inscribed" . The Fejer-Riesz inequality,
whose statement follows, falls under the general heading of results concerning Carleson measures. For a proof, see [6], Theorem 3.13.
THE FEJER-RIESZ INEQUALITY AND THE INDEX OF THE SHIFT THEOREM 1 (Fejer-Riesz Inequality). If f E HP(][))) (0
< p < 00), then
If(tei'l')lPdt ::; 4121r If(eill)IPdO
for 0 ::; 'P < 27f. The constant
is best possible.
With z fixed in ][)), ( f-+ Pz (() := 1~=~J~2 is the Poisson kernel on 8][)) for evaluation at z. It is well-known that falf» Pz(()dm(() = 1 independent of z, and that if hE L1(m), then (by Fatou's
( Pz(()h(()dm(() ~ h(~) lalf» as z nontangentially approaches ~ for m-a.a. ~ in 8][)). One may consult [6] and [8] as good references for these results. We begin with a rather straightforward
observation whose proof appears in [4]; see the proof of Lemma 3.1 in this reference. LEMMA 2. Let rJ be a finite, positive Borel measure with support in ll} such that rJ(8][))) = O. Then lim r--+1-
for m-a.a.
1 - r2
If» 11-r~wI2
drJ(w) = 0
in 8][)).
Our next result can be viewed as the converse of the Fejer-Riesz inequality. THEOREM 3. Let v be a finite, positive Borel measure with support in ll} such that vlalf» « m and][)) = abpe(p2(v)). For c
> 0, let Bv(c) be the set of all ~ in 8][)) such that
11 If(t~Wdt J ::; c·
for all polynomials f. Then Bv (c) is a closed subset of 8][)) and :~ ;::: ~ (a. e. m) on Bv(c). Furthermore, if E is a Lebesgue measurable subset of Bv(c) and m(E) > 0, then XE ¢ p 2(v). Proof. That
Bv(c) is closed is an immediate consequence of the fact that any polynomial is uniformly continuous on ll}. Now if 0 < r < 1 and I~I = 1, then
g(w) := ----=1-r~w
is analytic in a region containing ll} and so, by Runge's Theorem, is the uniform limit (on ll}) of polynomials. Therefore we can apply our hypothesis to get that
11 Ig(t~)12dt J ::; c·
which yields:
1 - r2
- rt
)2 dt ::;c.
Igl 2dv,
1 - r2
1- r~wl
2 dv (W),
for 0 < r < 1 and any ~ in Bp.(c). Letting r --+ 1 and applying Lemma 2, we find that :~ ;::: ~ (a.e. m) on Bv(c). To finish the proof of this theorem, let E be a Lebesgue measurable subset of Bv(c)
such that m(E) > 0, and suppose
that XE E P2(v); we look for a contradiction. Now h := (1 - XE) E p 2(v) (since XE E p2(v)), and in fact, hg E P2(v) whenever 0 < r < 1 and ~ E E. Arguing as before, with hg now in the place of g, we
obtain: ~ ~ h· :::. = 0 a.e. rn on E clearly a contradiction.D It turns out that if f.L is strongly inscribed, then in fact there exists c > 0 such that rn(BJ.L(c)) > o. En route to this result
(Theorem 5, below), we make the following observation. PROPOSITION 4. Let f.L be a finite, positive Borel measure with support in iID such that p2 (f.L) is irreducible. Then the following are
equivalent. 1) f.L is strongly inscribed. 2) There is a Jordan subregion V ofH}, where 8V is rectifiable and rn((8V) n (8H})) > 0, and there is a positive constant M, such that
lav Ifl 2dw v
~ M·
for all polynomials f. Proof. We first assume (1), and so by definition there is a Jordan subregion W of H} and a nonnegative function h in LOO(ww) that satisfy certain requirements. One of these
requirements, namely that log( h) EL I (ww ), guarantees the existence of a bounded analytic function g in W such that g 0 'P is an outer function ('P is a conformal mapping from H) onto W) and Igl
has "boundary values" equal to h (a.e. ww). Applying Proposition 2.2 of [1], we can find a Jordan subregion V of W, where V has rectifiable boundary, wv(8H}) > 0 and Igl ~ e > 0 on V. So by the
subharmonicity of Ifl 21g1 in W, (2) holds, with M := ~. That (2) implies (1) is immediate, and our proof is complete.D THEOREM 5. Let f.L be a finite, positive Borel measure with support in iID such
that p2 (f.L) is irreducible. If f.L is strongly inscribed, then there exists c > 0 such that rn(BJ.L(c)) > o.
Proof. Assuming that f.L is strongly inscribed, Proposition 4 provides a Jordan subregion V of H} with the properties listed in (2). Let 'P be a conformal mapping from H} one-to-one and onto V, and
let 1/J = 'P -1. Since Wv (8H}) > 0, we can find (cf. [14], Theorem 6.8 and Theorem 3.7) a closed subset E of (8V) n (8H}), where each point in E is a point of tangency of 8V with 8H}, such that:
i) rn(E) > 0, ii) for any ~ in E and any Stolz angle ~ whose closure is contained in VU{O, there is a constant M > 1 such that
~ ~ 11/J'(z)l, 11/J(Z; =t~) I ~ M for all Z in ~, and iii) if ~ E E and 'Y is a smooth arc in V U {O having nontangential approach in V to ~, then 1/J("() is smooth and has nontangential approach in
H} to 1/J(~).
Now choose ~ in E. Since ~ is a point of tangency of 8V with 8H}, there exists s, o < s < 1, such that t~ E V whenever s ~ t < 1. Let'Y be the smooth curve in
THE FEJER-RIESZ INEQUALITY AND THE INDEX OF THE SHIFT ID>U{'l/J(~)}
defined by ')'(t) = 'l/J(t~), s::; t::; 1. Then, by (i) - (iii), there are positive (k = 1,2,3) independent of to in [s, 1) such that
lengthb([to, l])) 1 - b(to)1
It: W(t~)ldt
1 -1'l/J(to~)1 I~
C1 .
- to~1
1 - I'l/J(to~) I
I~ - to~1 C2· I'l/J(O - 'l/J(to~)1 < C3·
From this it follows that arclength measure on ')'([s, 1)) is a Carleson measure for H2(1D». And so there are positive constants Ck (k = 3,4) such that, for any polynomial f,
11 If(t~)12dt
llU 0 cp)(wWlcp'(w)ldlwl
·llU 0 cp)(wWdlwl
< C4· [ IU 0 cpWdm
C4· [
IfI 2ru.vv;
by Harnack's inequality, we may assume that cp(O) is the point in V of evaluation for wv. Once again recalling (2) (of Proposition 4), we can now find a positive constant C5 such that
11 If(t~Wdt
C5 •
for all polynomials f. Since ID> = abpe(p2(jj)), we may apply Lemma 2.6 of [13] and find another positive constant C6 such that, for all polynomials f,
1If(t~Wdt J s
C6 •
Ifl 2djj.
Consequently, ~ E B/-L(c) , for C := C5 + C6. Thus we have shown that E ~ U~=1 B/-L(n). Since m(E) > 0, we can assert that m(B/-L(n)) > 0 for some integer n, which completes the proof.D QUESTION 6.
Does the converse of Theorem 5 hold? That is, if jj is a finite, positive Borel measure with support in iij such that p 2 (jj) is irreducible, and if m(B/-L(c)) > 0 for some positive constant c, then
is jj strongly inscribed? Indeed, can we even assert that dim(M 8 zM) = 1 for each nontrivial, closed invariant subspace M for the shift on p2(jj)? We conclude this article with a rather anemic
response to Question 6 that supports an affirmative answer.
REMARK 7. There are other more general forms of the Fejer-Riesz inequality, where the integral on the left is taken over chords of the unit circle and not just over diameters. A converse to this
Fejer-Riesz inequality (for chords) can be established, and involves integrals over segments that have nontangential approach in lI} to certain points in alI}. Thus, analogues of B/-I(c) can be
defined, where the integral on the left is taken over segments in various Stolz angles. All of this leads to a counterpart of Theorem 5, whose converse appears to be manageable. To this author it
seems most likely that if m(B/-I(c)) > 0, then there is a sizeable subset E of B/-I(c) that is contained in these collections that are analogous to B/-I(c), and thus the converse of Theorem 5 is
likely a consequence of its counterpart in the context of the Fejer-Riesz inequality for chords. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
[10] [11] [12] [13] [14] [15] [16] [17]
J. Akeroyd, Another look at some index theorems for the shift, Indiana Univ. Math. J., 50 (2001),705-718. J. Akeroyd, A note concerning the index of the shift, Proc. Amer. Math. Soc., Vol. 130, No.
11 (2002), 3349-3354. C. Apostol, H. Bercovici, C. Foias, C. Pearcy, Invariant subspaces, dilation theory, and the structure of the predual of a dual algebra, I, J. Functional Analysis, 63
(1985),369-404. A. Aleman, S. Richter, C. Sundberg, The majorization function and the index of invariant subspaces in the Bergman spaces, J. Analyse Math., 86 (2002), 139-182. J. B. Conway, L. Yang,
Some open problems in the theory of subnormal operators, Holomorphic spaces, Cambridge University Press, 33 (1998), 201-209. P. L. Duren, Theory of HP Spaces, Academic Press, New York, 1970. H.
Hedenmalm, S. Richter, K. Seip, Interpolating sequences and invariant subspaces of given index in the Bergman spaces, J. Reine Angew. Math., 477 (1996), 13-30. K. Hoffman, Banach Spaces of Analytic
Functions, Prentice-Hall, Englewood Cliffs, N.J., 1962. S. Hruscev, The problem of simultaneous approximation and removal of singularities of Cauchy -type integrals, Trudy Mat. Inst. Steklov 130
(1978), 124-195; English transl., Proc. Steklov Inst. Math. 130 (1979), no. 4, 133-203. T. L. Kriete, B. D. MacCluer, Mean-square approximation by polynomials on the unit disk, Trans. Amer. Math.
Soc., vol. 322, no. 1 (1990), 1-34. T. L. Miller, Some subnormal operators not in A2, J. Functional Analysis, 82 (1989), 296-302. R. F. Olin, J. E. Thomson, Some index theorems for subnormal
operators, J. Operator Theory, 3 (1980), 115-142. R. F. Olin, L. Yang, A subnormal operator and its dual, Canad. J. Math., 48 (1996), 381-396. Ch. Pommerenke, Boundary Behaviour of Conformal Maps,
Springer-Verlag, BerlinHeidelberg, 1992. J. E. Thomson, Approximation in the mean by polynomials, Ann. Math., 133 (1991), 477-507. J. E. Thomson, L. Yang, Invariant subspaces with the codimension one
property in Lt(J.I), Indiana Univ. Math. J., vol. 44, no. 4 (1995),1163-1173. L. Yang, Invariant subspaces of the Bergman space and some subnormal operators in Al\A2, Mich. Math. J., 42 (1995),
301-310. DEPARTMENT OF MATHEMATICS, UNIVERSITY OF ARKANSAS, FAYETTEVILLE, ARKANSAS
E-mail address: jakeroydlDcomp.uark.edu
Contemporary Mathematics Volume 328, 2003
A Cauchy-Green Formula on the Unit Sphere in C 2 John T. Anderson and John Wermer In 1977 G. Henkin introduced an integral formula for solving where tL is a measure, on the boundary of a smooth
strictly convex domain. This result is closely related to a "Cauchy-Green" formula on the sphere (see Chen and Shaw [3]). We give a direct elementary proof of the Cauchy-Green Theorem on the unit
sphere and derive Henkin's solution of the 8b equation from this. We also give an application to an approximation result. ABSTRACT.
= tL
1. Introduction
Let n be a domain in the plane, with smooth boundary r. Cauchy-Green formula states that for any c/> E C 1 (0') and zEn, (1.1 )
c/>( z) = _1 27l'i
r c/>( () d( _ _ lnr o~ d( 1
lr ( - Z
The classical
d( Z
Note that the first term on the right of (1.1) is a holomorphic function 4> of Z in the domain n. In fact, 4> extends continuously to 0', and hence defines an element of the algebra A(O') consisting
of functions holomorphic in n and continuous on 0'. Of course, if c/> E A(O'), (1.1) reduces to the Cauchy integral formula and 4> = c/>. The representation (1.1) has many applications in complex
analysis. In the theory of approximation of continuous functions on a compact set K c C by rational functions with poles off K, one is led by considerations of duality to examine measures supported
on K. The Cauchy transform of such a measure J.l is defined by
p,(Z) =
dJ.l(() lK (-z The integral defining [J, converges absolutely for almost all z E C. Using (1.1), one can easily show that for any smooth compactly supported function C/>, (1.2)
c/>(z) dJ.l(z) = ~ °o~ p,(z) di 1\ dz lK 27l'zlc z That is, [J, satisfies the equation (1.3)
oi =-7l'J.l
1991 Mathematics Subject Classification. Primary 32A25, Secondary 32E30.
© 21
2003 American Mathematical Society
in the sense of distributions, and hence defines a holomorphic function on e \ K. The Cauchy transform is a key tool in rational approximation theory in the plane. We have been motivated by problems
of rational approximation for subsets of the boundary S of the unit ball in e 2 . It is posible to do a kind of function theory on S analogous to the theory of analytic functions in the plane. The
operator a / az is replaced by the tangential Cauchy-Riemann operator
= Z2-
x is well-defined 011 C l (S) and for any relatively open subset n of S, annihilates the restrictions to n of functions holomorphic in a neighborhood of n in e 2 • The solutions to X ¢ = 0 on n are
known as CR functions on n. A good general reference for the theory of CR functions is the book [2]. One would like an analogue of the Cauchy transform for measures on S. Given a measure JL on S, G.
Henkin in 1977 [4] constructed a function KJ1.' summable with respect to three-dimensional Hausdorff measure da on S, satisfying -
abKJ1. = -27r JL
in the sense of distributions, i.e.,
[ ¢(z) dJL(z) =
[ KJ1. X¢ da(z) 27r for all smooth ¢, provided that JL satisfies the necessary condition that P dJL = 0 for all polynomials P. Note that (1.7) implies that KJ1. is a CR function (in the sense of
distributions) off the support of JL. In attempting to use and understand Henkin's construction in the study of rational approximation on subsets of S, we were led to the analogue of the CauchyGreen
formula (1.1) that we present below. It plays the same role with respect to Henkin's formula (1.6) as the classical Cauchy-Green formula on the plane does to equation (1.4). The resulting formula,
which is contained in our Theorems 2.1 and 3.1 below, is not new. It is given in a more general setting in Chen and Shaw ([3], see the remarks following Corollary 11.3.5) as a consequence of the
theory of Henkin for solving the 8b equation on the boundary of a strictly convex domain in en. Our approach to establishing this Cauchy-Green formula on the sphere in e 2 is direct and elementary,
and leads immediately to the property (1.6) of Henkin's transform K J1.' Let A(B) denote the algebra of functions holomorphic in the open unit ball B of e 2 and continuous on its closure. We seek a
kernel H((, z), defined for ((, z) E S x S, such that for all ¢ E Cl(S), there exists E A(B) with
= (z) + c
H((, z) 8¢(() 1\ w(()
for all z E S, where w(() = d(l 1\ d(2, 8¢ = (a¢/azddz 1 + (a¢/az2)dz2, and c is a universal constant. We call (1.8) a "Cauchy-Green formula for S". We will demand that H have the following
A CAUCHY-GREEN FORMULA ON THE UNIT SPHERE IN C 2
a: H((, z) is continuous on S x S \ {z = 0; b: For all unitary transformations U of determinant 1, H (U (, U z) = H ((, z); c: jH((, el)j da(() < 00, where el = (1,0), and da is three-dimensional
Hausdorff measure 1 on S. Properties (b) and (c) together with the unitary invariance of da imply that H is uniformly summable with respect to da, Le., there exists a constant C so that
jH((, z)j da(() ::; C, Vz
They also imply that the integral
K(z) ==
H((, z) a¢(() /\ w(()
appearing in (1.8) is finite for all z E S, since a¢ /\ w is absolutely continuous with respect to da. A routine calculation gives
a¢ /\ w = 2(X¢) da
on S, where X is the operator in (1.5), for smooth ¢. We can say more about K: LEMMA
1.1. If H satisfies properties (a), (b) and (c), then K is continuous on
S. PROOF. Fix z E S. For f > 0, put S«z) = S \ {jz - (j ::; f} and S~ S n {jz - (j ::; fl. Let
H((, z) a¢(() /\ w(()
Then K< is continuous on S, by property (a) of H. For all z E S, by (1.11),
jK(z) - K,(z)j =
H((, z) a¢(() /\ W(()I ::; M
jH((, z)jda(()
ls~(z) Let el = (1,0) and choose a
where M is a constant independent of z and f. unitary transformation U of C 2 with Uel = Z; then U(S~(el)) = S~(z). Then using property (b),
ls~(z) Since
jH((, z)j da(()
jH(Ury, Uel)j da(Ury)
jH(ry, el)j da(ry)
Is jH(ry, edjda(ry) is finite by assumption (c), lim <-+0
It follows that K,
1rS~(e,) jH(ry, el)jda(ry) = 0
K uniformly on S, and so K is continuous, as claimed.
We say that a measure /-L on S is orthogonal to polynomials if
Pd/-L = 0, V holomorphic polynomials P
Given any measure /-L on S, define
K Il (() =
Ida is not normalized; oo(S) =
H((, z) d/-L(z), (E S
24 LEMMA
1.2. A kernel H((, z) satisfying (a), (b) and (c) satisfies (1.8) if and
if for each measure J.L on 8 orthogonal to polynomials
(1.14) for all ¢
¢ dJ.L = cis K,.. 8¢ A w
PROOF. Suppose first that H((, z) satisfies (a), (b), (c) and (1.8). Let J.L be a measure on 8 orthogonal to polynomials. Fix ¢ E C 1 (8). and let
Is (c Is Is (c Is Is
H((,z) 8¢(() AW(()) dJ.L(z) H((, Z)dJ.L(Z)) 8¢(() A w(()
K,..(() 8¢(() A w(()
so that (1.14) holds. The application of Fubini's theorem is justified by (1.9). Next, suppose that (1.14) holds, for H satisfying (a), (b) and (c). Choose a measure J.L on 8 orthogonal to
polynomials. Fix a function ¢ E C 1 (8), and define
H((, z) 8¢(() 1\ w(()
By Lemma 1.1,
Is Is
¢(Z)dJL(Z) - c
Is (Is
H((, Z)dJ.L(Z)) 8¢ A w(()
¢(z)dJ.L(z) - cis K,..(() 8¢(() A w(()
o by (1.14). Since this holds for all J.L orthogonal to polynomials,
( 1.15)
H(( z) =
(1 22 - (2 21
(, Z
1 2 ' where <, > denotes the Hermitian inner product < z, ( >=
Zl(l + Z2(2, and proved the formula (1.14) using this kernel. It is easy to check that H satisfies properties (a), (b) and (c) above. Formula (1.14) on 8 is actually very special case of a class of
general integral formulae on smooth convex domains established in [4]. In her thesis [5], H.P. Lee gave an elementary proof of Henkin's formula for 8j the paper [8] of Varopoulous also contains an
exposition of Henkin's results on the sphere. For applications of Henkin's formula to rational approximation, see the paper [6] of Lee and Wermer.
In this paper, we shall (1) give a direct proof of (1.8), using Henkin's kernel (1.15)j (2) give a formula for
A CAUCHY-GREEN FORMULA ON THE UNIT SPHERE IN C 2
1.1. Acknowledgment. The first author wishes to thank Joseph Cima for helpful conversations on the results in section 3.
2. A Cauchy-Green Formula using Henkin's Kernel With H as in (1.15) and ¢ E C 1 (S) as in section 1 put
K(z) = For a E int(6), put r = the z2-plane. LEMMA
2.1. Fix a
H((, z) 8¢(() 1\ w(()
Jl -lal 2 and denote by "fa the circle Z2 =
rT, ITI = 1 in
6. For n = 0,1,2, ... we have, putting z = (a, Z2),
(2.1) PROOF.
We denote the inner integral by J((). Multiplying both numerator and denominator of the integrand by T, we get
Let and Note that
T1 T2
1 -a1 -(
= 1. We have
Ir(21 2 -11 - a(11 2
= =
(1-laI 2)(1-1(112) -11 - a(11 2 1 - lal 2 - 1(11 2 + la1 211(112 - 1 - la1 21(112 + a(l -(laI 2 + 1(11 2 - a(l - a(t)
-Ia -
(11 2
+ a(l
(2.2) Let Se be the part of S lying over the region
{I(I -
f} n {I(II ::; I}
in the (I-plane. Let TE denote the boundary of SE' We claim that
la 1
K(Z)Z2 dZ 2 = -
!~~ [l,4>(()I(()W(()]
To establish the claim, note that
K(Z)Z2 dZ 2
lim [ 84> /\ w . I <-+0
lim [ d(4) wI) e-+O
since I is holomorphic on Se for equals
> O. By Stokes' Theorem, the latter integral - [ 4> wI
proving the claim. Note that T, is the torus (1 = a + fe ifJ , (2 = JI -1(II 2ei .p, 0::; 0, 'IjJ ::; 27l'.
A CAUCHY-GREEN FORMULA ON THE UNIT SPHERE IN C 2
On T< we have the following relations:
¢(() = ¢(a, rei1/J) + O(f);
d(1 = ife i9 dB, d(~ = -ife- i9 dB; d(2 = -(1 d(;. - (1 d(1 ei1/J + iJI -1(11 2ei 1/Jd'lj; = i re i1/Jd'lj; + O(f); 2JI-I(11 2 1 1 (1 - a fe i9 · Using this information together with (2.2) and (2.3)
we obtain
1K(z)z~ "fa
For fixed
= lim [-27ri [ ¢(()r2n+2 (
a(~1 )
+1 (( 1
1 -
a) d(11\. d(2]
we rewrite the expression in brackets as -27ri [ ¢(a, rei1/J)rnein1/JidB I\. irei'IjJd'lj; + O(f)
dZ2 =
This completes the proof of (2.1) and Lemma 2.1.
Next, we define an operator T on C 1 (8) as follows:
for z E 8, ¢ E C 1(8)
(T¢)(z) = 47r 2 ¢(z) - K(z),
Letting X denote the tangential Cauchy-Riemann operator on 8 as in section 1, using (1.11) we can write
T¢ = 47r 2 ¢ -
H((, z) (X¢)(()dcr(()
LEMMA 2.2. Fix ¢ E C 1 (8). Let L be a complex line in C2. Then the restriction ofT(¢) to L n 8 extends analytically to L n B. PROOF.
Lemma 2.1 gives us, for each a
int(h.), that
dZ 2 = 0, n = 0,1,2, ...
Note that "fa = La n 8, where La is the line {Z1 = a}. Then (2.5) implies that T¢ extends analytically to the disk La n B. Using the unitary invariance of H,cr, and X, it is not hard to check that
for all ¢ E C 1 (8),
Fix a complex line L. Let N denote the complex line passing through the origin which is orthogonal to L, and let zO denote the intersection point N n L. Write L = {zO + (t I t E C} for some unit
vector (. If U is a unitary transformation with Ue2 = (, where e2 = (0,1) then U maps the line {Z2 = O} to N, and maps some
point (a,O) to zOo Then U((a,O) + t(O, 1)) = zO + t(, for all t E C. So U maps the line La to L and maps the disk La n B to L n B. By (2.6), T¢ I Lns extends analytically to the disk L n B if and
only if (T¢) 0 U 1La ns extends to La n B. This last is true by (2.5), as we have noted earlier, and so the proof is complete. 0 By Lemma 1.1, since H satisfies properties (a), (b) and (c) of section
1, K and thus T¢ are continuous on S. By Lemma 2.2, T¢ has the "one-dimensional extension property" as defined by Stout in [7], p. 105. A theorem of Agranovskii and Val'skii [1] then gives that T¢
lies in the ball algebra A(B). Putting cP = T(¢), we have arrived at THEOREM 2.3. Let ¢ E C 1 (S). Then there exists cP E A(B) such that 47r 2 ¢(z) = cP(z)
H((, z) 8¢(() /\ w(()
where H is Henkin's kernel
3. The Cauchy-Green formula and the Cauchy transform In this section we identify the ball algebra function cP appearing in Theorem 2.3 as a certain principal value of the Cauchy transform of ¢. The
Cauchy kernel for B is 1 C(z,() = (1- < z,( »2 For z E S we set N. (z) = {( E S : I < (, z > I > 1 - f} and we denote the boundary of N. (z) by r f (z). THEOREM 3.1. Fix ¢ E C 1 (S). If cP is as in
Theorem 2.3, then for z E S,
cP(z) = 2 lim
¢(()C(z, () du(()
.-+0 JS\N.(z)
REMARK 3.2. Since C(z,·) Theorem 3.1 exists.
rt. Ll(du),
it is not immediate that the limit in
PROOF. As in sections 1 and 2, set
f H((, z) 8¢(() /\ w(() = JS
K(z) = For
f H((, z) 8¢(() /\ w(() JS\N, (z)
d[H((, z)¢(() /\ w(()]
> 0 fixed,
H((, z) 8¢(() /\ w(()
- f
[8H((, z)]/\ ¢(() /\ w(()
H((, z) 8¢(() /\ w(()
f JS\N.(z)
(XH)((,z) ¢(() du(()
A CAUCHY-GREEN FORMULA ON THE UNIT SPHERE IN C 2
by Stokes' theorem, if r e(z) is oriented as the boundary of S \ Ne (z). We have also used equation (1.11) from section 1. A computation shows (differentiation is in the ( variable) (XH)((, z) = -C
(z, () so that
K(z) = lim [ €-+O
H((, z) >(()
w(() - 2
C((, z) >(()dO"(()]
cI>(z) = 41r 2>(z) - K(z) by Theorem 2.3, the proof will be complete if we can show that lim
H((, z) >(()
1\ w(() =
To establish (3.1), choose a unitary map U with Ue1
H((, z) >(()
The torus r e(e1) parametrized by
1\ w(() =
= z. Then for fixed E > 0,
H(TJ, ed (> 0 U)(TJ)
1\ w(TJ)
= {TJ : ITJ11 = 1 - E}, oriented as the boundary of S \ Ne(ed, is
where Then on re(ed,
which gives
H((, z) >(() 1\ w(()
(3.2) where
r r 27r
IIel ::; C Jo Jo
11 - (1 -.: €)ei9112d(hd{;l2
for some C > o. An application of the Poisson integral formula shows that the first integral in (3.2) converges to 41r2(>oU)(e1) = 41r2>(z) as E ---+ 0, while lime-+o Ie = O. This completes the
proof. D
4. An Approximation Theorem Fix ¢ E C 1 (S). The quantity dist(¢, A(B)) = inf{ll¢ - gil : g E A(B)} where I . II is the uniform norm on S measures how closely ¢ can be approximated by polynomials on
S. THEOREM 4.1. There exists C > 0 so that/or all ¢ E C1(S),
dist(¢, A(B)) ~
PROOF. Let IIHl11 denote the L1 - drr norm of Henkin's kernel H(·, z) (which is independent of z E S). By the representation in Theorem 2.3, there exists E A(B) so that for z E S, 14rr2¢(z) - (z) I
H«(, z) 8¢«() Aw(OI
211s H«(, Z)(X¢)(Odrr(ol < 211 H lh11X¢11
from which the result follows.
References [1] M.L. Agranovskii and R.E. Val'skii, Maximality of Invariant Algebras of Functions, Siberian Math. J. 33 (1983),p. 227-250. [2] A. Boggess, CR Manifolds and the Tangential
Cauchy-Riemann Complex, CRC Press, 1991. [3] S.-C. Chen and M.-C. Shaw, Partial Differential Equations in Several Complex Variables, American Mathematical Society, 2001 [4] G. M. Henkin, The Lewy
Equation and Analysis on Pseudoconvex Manifolds, Russian Math. Surveys, 32:3 (1977); Uspehi Mat. Nauk 32:3 (1977),p. 57-118 [5] H. P. Lee, Orthogonal Measures for Subsets of the Boundary of the Ball
in C 2 , Thesis, Brown University, 1979. [6] H. P. Lee and J. Wermer, Orthogonal Measures for Subsets of the Boundary of the Ball in C2, in Recent Developments in Several Complex Variables, Princeton
University Press, 1981, pp. 277-289. [7] E.L. Stout, The Boundary Values of Holomorphic Functions of Several Complex Variables, Duke Math. J. 44, 1977,p. 105-108. [8] N. Th. Varopoulos, BMO functions
and the a-equation, Pac. J. Math. 71, no. 1 (1977). pp. 221273. DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE, COLLEGE OF THE HOLY CROSS, WORCESTER, MA 01610-2395 E-mail address:
andersonOradius.holycross .edu DEPARTMENT OF MATHEMATICS, BROWN UNIVERSITY, PROVIDENCE, RI 02912 E-mail address: wermerlDmath. brown. edu
Contemporary Mathematics Volume 328, 2003
On a-dual Algebras Hugo Arizmendi, Angel Carrillo, and Lourdes Palacios ABSTRACT. Let Ti(D) and TieD) be the algebras which consist of all holomorphic functions in the open unit disc D and in the
closed unit disc 75, respectively. These algebras, considered as algebras of sequences, are denoted by A and B. Let A" and B" be the a-dual spaces of A and B, respectively; we have that A" = Band B"
= A, and these sequence spaces with their normal topologies are topological algebras. A similar treatment can be applied to the algebra c of all entire functions and its a-dual space c" consisting of
all complex functions that are analytic in some neighborhood of the origin. Here we examine a more general situation. If A(ap,n) is a matrix algebra, we establish conditions under which the a-dual
space A" (ap,n) is a topological algebra relative to the normal topology. We also analyse some other important examples of topological algebras with such properties.
1. Introduction
A sequence space), is a vector space of complex sequences x = (Xi)~O' The vector space operations are the usual operations on the coordinates. ), can be considered as a linear subspace of the space w
of all complex sequences. To each sequence space ), we assign another sequence space N", its a-dual. ),01. is the set of all complex sequences Y = (Yi)~O for which the scalar product <Xl
yx =
I: XiYi
converges absolutely, for each x E ),.
The normal topology on ), is the topology determined by all the seminorms defined by <Xl
Ilxlly =
2: IXkYkl
k=O where Y runs over all ),01.. An important class of sequence spaces are the echelon and co-echelon spaces which have been studied by G. Kothe and O. Toeplitz. They are defined as follows: Let
(ap,n), p = 1,2, ... , n = 0,1, ... be an infinite matrix of non-negative numbers such that (1) 0:::; ap,n :::; ap+1,n (2) For every n, there exists p such that ap,n > O. 2000 Mathematics Subject
Classification. Primary 46; Secondary 30. Key words and phmses. Topological Algebras, Normal Topology, Matrix Algebra.
© 31
2003 American Mathematical Society
Let A = {(Xn) E
W :
E JXnJ ap,n < 00, p = 1,2, ... }.
A is called an echelon space
and N" a co-echelon space. We note that A = fu!!ll(ap,n)' p
In the following we denote the echelon space A by A(ap,n)' In [1] it is proved that if A(ap,n) is an algebra under the convolution product, then this product is jointly continuous and A(ap,n) is a
topological algebra. As a matter of fact, it is a metrizable locally convex complete algebra, i.e. aBo-algebra. It can be seen that a necessary and sufficient condition under which A(ap,n) is an
algebra is the following: (3) For each pEN, there exists q E N such that ap,n+m S aq,n aq,m; for n, m = 0, 1,2, .... If (ap,n) satisfies (1), (2) and (3), we call A(ap,n) a matrix algebra. Let 00
A = {(an)~=o: if
JzJ < 1,
then L anz n converges in C} n=O
and 00
B = {(bn)~=o: there exists a z
C, JzJ > 1, and L bnzn converges in C}. n=O
These sequence spaces A and B are called analytic sequence spaces and they are algebras under the usual linear operations and the convolution product. The transformation 00
(Xk)k=O ~ L Xk zk k=O identifies those sequence algebras A and B with the function algebras H(D) and H(D) consisting of all holomorphic functions in the open unit complex disc D and in the closed
unit complex disc D, respectively. Through this identification we shall 00
indistinctly write x = (Xk)k=O or x =
Xkzk to refer to an element x of A or B. k=O By the same fact we can consider in A the compact-open topology that is originally defined for H(D). We note that A ~ A(r;), n = 0,1, ... , p = 0,1, ... ,
where (rp) is an increasing sequence of positive numbers converging to 1, and the compact-open topology on A can be given by the sequence of seminorms 00
JJxJJp = L JXnJr;, n=O
where x
= LXnZnEA. n=O
In [5], O. Toeplitz studied the topological properties of the analytic sequences spaces A and B and proved that AO: = Band BO: = A. A similar treatment can be applied to the algebra c of all entire
functions and its a-dual space cO: consisting of all complex functions that are analytic in some neighborhood of the origin. In [2] it is proved that A and B are topological algebras when they are
endowed with the normal topology given by A and B, respectively. The same happens with the algebra of all entire functions c and its a-dual space cO:.
ON ",-DUAL ALGEBRAS
If ap n > 0, n ,
= 0,1,2, ... , p = 1,2, .... then A"'(ap n) = limlOO(-1-) as sets. ap,n ,
The inductive limit topology is stronger than the normal topology in A"'(ap,n)' Prom now on we are going to assume: ap,n > 0, n = 0,1,2, ... ,p = 1,2, .... If A(ap,n) is nuclear, then A"'(ap,n)
~:::'n~lOO(a:.J ~ l~l1(a:.J as topological p
vector spaces. Here we prove that l~l1(a:.J is an algebra if A(ap,n) satisfies: p
(*) for each p there exist q > p and Mp such that ap,n ap,m :::; Mpaq,n+m (or equivalentely _ 1 _ :::; Mp_1___1_) for all n,m. aq,n+m
a p . n ap,rn
And then the convolution product is jointly continuous. We also prove that if (ap,n) does not satisfy (*), then l~l1(a:.J is not an algebra. Therefore l~l1(a:.J p
is an algebra if, and only if, it is a topological algebra. Thus, if A(ap,n) is nuclear, then A"'(ap,n) is an algebra if, and only if, it satisfies (*). Therefore A(ap,n) and A"'(ap,n) are
topological algebras under the normal topology. This is a generalization of the properties of (A, B) and (c, c"'). We also study some other important examples of topological algebras with such
2. Definitions and Notation We recall some relevant definitions. Through this section we assume that X is a commutative complex topological algebra with unit element. X is called a locally convex
algebra if it is also a locally convex space. In this case its topology can be given by means of a family (11.11",)"'E;l of seminorms such that for each index Q E ~, there is an index f3 E ~ such
that (2.1) for all x, y EX. If relation (2.1) can be replaced by (2.2) for all x, y EX, then we say that X is locally multiplicatively convex (shortly m-convex) algebra. X is called a Bo-algebra if
it is a complete metrizable locally convex algebra. In this case its topology can be given by means of a sequence (1I.lIn)~=1 of seminorms satisfying
for n = 1,2, ... and for all x,y E X. Let (a-y,k), "I E r, k = 0,1, ... , be an infinite matrix of positive real numbers. Assume that for each "I E r there is a "I' E r such that
(2.3) for all k, l = 0, 1, ....
a-y,k+l :::; a-y',k a-y',l
The matrix algebra A(a/"k) associated with the matrix (a/"k) is the algebra of 00
all formal complex power series x =
Xkzk such that, for each "I E
L a/"k IXkl <
IIxli/' =
k=O By (2.3), A(a/"k) is a complete locally convex algebra under the usual linear operations and the convolution product. If r = N, then A(a/"k) is a Bo-algebra. If "I = "I' in (2.3), then A(a/"k) is
an m-convex algebra. We shall be mainly interested in a Bo matrix algebra A(ap,n)' We note that 00
= {(zn)~=o
L IZnYnl <
V Y = (Yn)~=o in AO(ap,n)}'
n=O We have the following: REMARK
2.1. AOO(ap,n) = A(ap,n)'
For each p, the row (ap,n) is an element of AO(ap,n)' Therefore, if
Z E AOO(ap,n), then
IZnap,nl <
and hence Z E A(ap,n)'
3. A Matrix Algebra and its a-dual 3.1. AO(apn ) = limloo(-I-) , -; ap,n
Let Y
= {(Yn);:"=o
liml oo
1 13 p: sup IYn< oo} as sets. n Qp,n
then sup IYn
for some p.
= sup IYn-I-I, we have that IYn 1 :s M ap,n for each n. Therefore, for n ap,n 00
:s M L
Ixnap,nl = M Ilxlip and hence Y E AO(ap,n)' n=O n=O Conversely, let us suppose that there exists Y E AO(ap,n) such that oo (_I_); that is, yrf.loo(_I_) for each p. Since for each p, sup IYn yrf.liml
= 00, -+ ap,n Qp,n n ap,n
x E A(ap,n)'
then there exists an increasing sequence (np) such that IYn p ap,np define the sequence x
= (xn);:"=o as follows: Xn =
_I_~ ap,np
> p2. Let us
ifn#np =n .
if n
We claim that x E A(ap,n): For arbitrary q, IIxllq q "
aq,np 1
L.J a
p=O p,n"
+ "L.J
aq,np 1
p=q+l ",n"
" aq,np..!.. = n=O Ixnlaq,n = p=O 1~~laq,np = p=O L.J a p,np p2 v,np P
as claimed •
Nevertheless n=O IXnYnl = n=O I~~ Yn,,1 = 00 since I~~pl p,n" P p,np Yn,,1 > 1 for every n. This is a contradiction to the assumption that Y E AO(ap,n)' 0 PROPOSITION 3.2. The inductive limit
topology is stronger than the normal topology in AO(ap,n).
PROOF. Let us recall that 11.11 is a continuous seminorm in limloo(-I-) --+ av,n p
and only if,
each p,
there exists a
~ Cp lIylll"" = Cps~p IYna:.n I for every Y E loo(a:.J·
-1 =
1 Let Y E 100 (_1_), then sup IYnap,n n ap,n
AI <
Cp such that
1and therefore IYnlap,n 00
for each n.
Hence, for x E A(ap,n) we have that IIYllx
~ AI
IXnYnl ~ <
n~o AI IXnl ap,n = AI n~o IXnl aq,n = AI IIxllp = s~p IYna:.n Illxll p = Ilxllp Ilyllloo'
In [4] it is proved that an echelon space A(ap,n) is nuclear if, and only if, for each p there exist q > p and a sequence u = (un);:::'=o E II such that ap,n = Un aq,n, n = 0,1, ... PROPOSITION 3.3.
If A(ap,n) is nuclear, then AO(ap,n), endowed with the normal topology, is such that AO(ap,n) ~ limloo(-I-) ~ limll(-I-). -+ ap,n -+ ap,n p
PROOF. Due to the nuclearity of A(ap,n) it is easy to see that limll(-I-) ~ limloo(-I-). By the previous Proposition we know that the inductive --+ ap,n -+ ap,n p
limit topology is stronger than the normal topology in AO(ap,n). Let 11·11 be a seminorm in limll(-I-). Let (e n );:::'=1 be the sequence of the -+ ap,n p
canonical vectors and note that (e n );:::'=1 E 11(_1_) for each p. Therefore there ap,n exists Cp > 0 such that lIenll ~ Cpllenill oo = CPa :. n for each p. (Xn);:::'=1 = (1Ien ll);:::'=I' We will
prove that (Xn);:::'=1 EA(ap,n)' 00
Indeed, IIxnllp
We put
I: Cqllenllqunaq,n
1L~1 ynenll ~ n~IIYnlllenll
lIenllunaq,n <
I: Un < 00
Therefore, for any Y E loo(a:.J, IlylI
IYnllxnl = IIYllx <
SO II . II is continuous with respect to the normal topol-
D Let A(ap,n) be a matrix algebra. We define the following condition:
DEFINITION 3.4. (*) for each p there exist q ap,n ap,m ~ AIp aq,n+m for all n,m.
P and AIp such that
Let us recall that the absolutely convex hull of a subset V of a vector space is the set r(V) =
{I: .Aiai, I finite, ai EV, iEI
EC and
I: l.Ail ~ I}.
It is a straight for-
ward matter to check that if V and Ware two subsets of an algebra, then the property (x, Y E V ~ xy E W) implies the property (x, Y E r(V) ~ xy E r(W)). PROPOSITION 3.5. If the matrix algebra A(ap,n)
satisfies (*), then l~ll(a:.J p
is a topological algebra under the convolution product.
PROOF. Let x, Y E 11(_1_). From condition (*), let q > p and Mp be such that
Mp.: ••":_,, The:'il:YII, ~ "~O I(xy)" I .;," ~ I
.;," <;
C~O IXnl a:,n) C~O IYnl a:. n) =
n=O k=O
IXkYn-kl a:q,n
:s n=O f (t Mp IXka:-k IIYn-k k=O ".
l_k p.n
Mp IIxlip IIYllp <
Ikt IXk!ln-kll·;"
So far, limll( a:-) is an algebra. From the previous proof it is easy to see that p.rt.
--+ p
1 if x E II and Y E II then xy E II and IIxYllq ::; MpllxllrllYIIs, where q and Mp are any two numbers that satisfy (*) for p = max(r, s). Now let n be a neighborhood of the origin in limll(-I-);
then --+ ap,n
n= r
CQl V(O, II·IIp' ep)), where ep > 0 for all p ~ 1.
For each p
1, let qp and Mp be two numbers that satisfy (*) for p and let us
take 0 < t5p < min ~-). If x E V(O, II.IIr,er) and Y E V(O, II·IIs,es), then IIxYllqp MpllxllrilYlls, where p = max(r,s); therefore IIxyllq" < eq", thus xyEV(O, II.IIqp,eq,,)' From the result stated
before this proposition, it follows that,
x, Y E r
CQl V(O, II·IIp' t5p)) ::::} xy r CQl V(O, II·IIp' ep)) ~ n. E
This shows that
the multiplication is continuous.
COROLLARY 3.6. If the matrix algebra A(ap,n) is nuclear and satisfies (*), then AQ(ap,n) is a topological algebra under the normal topology. PROPOSITION 3.7. If the matrix algebra A(ap,n) does not
satisfy (*), then limll(-I-) is not an algebra. -+ ap,n p
PROOF. The hypothesis gives a Po with the property that for each q > Po and Mq there exist m and n such that apo,n apo,m > !l-lq aq,n+m, or, equivalently, _ 1 _ > M _1_ _ _ 1_. q
Fix an i that
a pQ •n
Po and choose Mi = 22i. There are positive integers mi, ni such > Mi _1_ _ _1_.
Let x
= L:
for i ~ 1 and an
anz n and Y
= L:
bnz n , with ani
= bn = 0 otherwise.
IIxli po =
2ill)illpo' bmi
~ 2illz~illpo IIznili po
= 1 and
.=1 00
IIyllpo = ~ .=1
2'lIz;'ql" IIzm; IIpo
= 1; but, since xy=
= L: anibm;zni+m; .=1
n+m=j 00
we have IIxYllq ~ . L: 2i+illzni lI~o IIzmj IIp() II zn;+mj IIi = 00 .=q+l Therefore the space limll(-I-) can not be an algebra (see also Proposition -+ ap,n p
ON ",,-DUAL ALGEBRAS COROLLARY
3.8. A(a pn ) satisfies (*) if, and only if, limll(-I-) is a topological , -+ ap,n p
algebra under the convolution product. COROLLARY 3.9. If A(ap,n) is nuclear, then it satisfies (*) if, and only if, A""(ap,n) is a topological algebra under the convolution product. EXAMPLES
(1) Let us consider the matrix algebra A(ap,n) where ap,n
In this case, A(ap n) I
= (n + l)P, n = 0, 1,2, ....
E w: lim (n n-+oo
+ l)P IXnl = 0,
= 1,2, ... }.
A(ap,n) and A""(ap,n) are nuclear algebras and one is the a-dual of the other. A(ap,n) is nuclear since for each p we can take q = 3p and
(un)~=o =
p) ~=o =
(( (n;l)p )
2) ~=o'
which clearly is in ll,
and satisfies ap,n = unaq,n for all n. A""(ap,n) is an algebra due to Corollary 3.6 since for each p we can take q = 2p and Mp = 1 to satisfy ap,nap,m ::; Mpaq,n+m for all n, m. (2) Let us consider
the matrix algebra A(ap,n) where 1
a p,n - n 2 -P
In this case, A(ap,n) satisfies (*). For, if p is given, let q = p + 1 and 2- p - 1 Mp=2 . Note that for arbitrary n , m 'nnt.~ < n+m = .1+..!.. < 2. Then ap,n ap,m = m nm n m aq,n+m 1
;:;:z=P ~ _ (n+m) ;
_ (n+m) -
_ (n+m) -
n2 m 2
< 2TP -
1 -
M p.
However, the algebra A(ap,n) is not nuclear. For, if p < q are arbitrary, _1_ 2- P 2- q 2q-2P ~ = ~ = (.1) = (.1) ~ E II only if ~ > 1, which is aq,n 2-q n n 2P q impossible. (3) If w is the linear
space of all complex sequences, we note that w = A(ap,n) where ap,n
I { 0
if n > p . Then A (ap,n)
= I!m(N). Both of them are
nuclear topological algebras. Here Corollary 3.9 does not apply because it is not true that ap,n > 0 for all indices p and n.
References [IJ R. Arizmendi. "Matrix Algebras and m-convexity". Demostratio Mathematica, Vol. XVII, no. 3, 1984. [2J R. Arizmendi; A. Carrillo; L. Palacios. "On continuous multiplicative Mappings on
Ananlytic Sequence Spaces". Commentatione Mathematicae, XXXIX (1999). [3] G. Kothe. Topological Vector Spaces l. A series of Comprehensive Studies in Mathematics, Springer-Verlag, 1966. [4] A.
Pietsch. Nuclear Locally Convex Spaces. Springer-Verlag, 1969. [5] O.Toeplitz. Die linearen volkommenen Riiume der Funktionentheorie. Comment. Math. Relv. 23 (1949), 222-242.
(Hugo Arizmendi, Angel Carrillo) INSTITUTO DE MATEMATICAS. UNIVERSIDAD NACIONAL AUTON OM A DE MEXICO, APDO POSTAL 14455, MEXICO, D.F.
E-mail address:hugolDservidor.unam.mx (Lourdes Palacios) DEPARTAMENTO DE MATEMATICAS, UNIVERSIDAD AlTTONOMA METROPOLITANA. Av. SAN RAFAEL ATLIXCO 186, COL. VICENTINA, 07340 MEXICO, D.F.
E-mail address:pafalDxanum.uam.mx
Contemporary Mathematics Volume 328, 2003
A Connected metric space that is not separably connected Richard M. Aron and Manuel Maestre ABSTRACT. We construct a subset of the unit ball of foo that is connected but not separably connected.
A topological space X is said to be separably connected if for every two points X, Y E X there exists a connected and separable subset C{x,y} such that x,y E C{x,y}' The reader may be interested,
perhaps even amazed, at the fact that this concept arises naturally in economics. In fact, in their work on utility theory, J. Candeal, C. Herves, and E. Induntin [B] ask whether there is a connected
metric space that is not separably connected. They observe that there are examples of connected topological spaces which are not separably connected. However, no example of a connected metric space
which is not separably connected is given and, in fact, this issue is explicitly raised in [A]. The purpose of this note is to give a negative answer to this question. In fact, this problem was
solved by R. Pol over 25 years ago, who produced a somewhat different construction in [C]. The construction presented here seems natural, in view of its similarity to the standard construction of a
non-measurable subset of JR. Our example will be a subset of the Banach space (foo, 11·1100) of all sequences (Xn)~=l in lK = JR or C, satisfying II(xn)~=11l := sup{lxnl : n = 1, 2... }
< 00,
Given U E Roo and T/ > 0, we denote by B(u, T/) := {v E Roo : IIv - ull ::; T/}, i.e., B( u, T/) is the closed ball with center u and radius T/. We will single out the unit vector e := (1,0, ..... )
in Roo and the associated linear functional c.p : Roo ---+ lK, c.p( (xn)) = x}, for all (xn) E f oo . Clearly, Ker c.p = ((xn) E Roo : c.p( (xn)) = Xl = o}. Let .N denote the set consisting of all
non-empty subsets S C {2, 3, .. , n, ... }, and for each such S, let Us = (Xn)~=l E foo : Xn = 1 if n E Sand Xn = 0 if n ~ S, n = 1,2 .... The set £ == {us : S E .N} is uncountable since .N is, any
element of £ has norm one and Ilu - vii = 1 for all u,v E £, u I- v. Recall now 2000 Mathematics Subject Classification. Primary 54D05, 46B26. Key words and phrases. Metric spaces non separably
connected, Banach spaces. The first author was partially supported by the Ministerio de Educacion y Cultura of Spain (SAB1999-0214). The second author was partially supported by MCYT and FEDER
Project BFM2002-01423. © 2003 American Mathematical Society 39
the following equivalence relation on [0, 1], which is used to prove the existence of a non-Lebesgue measurable subset of [0,1]: x '" y if and only if x - y E Q. We denote the equivalence classes of
[0,1] defined by this relation by {x : x E [0, I]}. By applying the Axiom of Choice, we can choose one element from each class, in particular taking E O. Let us denote the set so obtained by A.
LEMMA 1.1. The following properties hold: (1) Card (A) = Card([O, 1]). (2) Given x, yEA, x =1= y and r, SEQ, x + r =1= y + s. (3) If a E [0,1]' then there exist a unique x E A and a unique r E Q with
a = x+r.
(4) Given a < b E JR and x E JR, the family {x +r : r E Q} n (a, b) is a dense subset of [a, b].
The proof of the above properties is immediate. Let us remark that (4) can be obtained by using the fact that for each x E JR, the map ¢ : JR --+ JR defined by ¢(t) := x + t is a homeomorphism and
the fact that Q n (a - x, b - x) is always a dense subset of [a - x, b - x]. Also it is very easy to construct a bijection R : A --+ £, R(x) = ex. We define our set C as C = C 1 U C2 , where
C1 =
re+ [O,eo]
(x + r)e + (0, ex].
For an intuitive, geometric idea of what C is, consider a type of "comb" set in the plane given by {(s, t) : 0::; s ::; 1, s ~ Q, t ::; I} U {(s, t) : 0::; s ::; 1, s E
°: ;
We endow C with the metric induced by the norm of loc. Our goal in this note is to prove the following: THEOREM 1.2. C is a connected metric space that is not separably connected. PROOF. By Lemma
1.1.(2), given x, YEA, x =1= y, r, SEQ and any u, v E Ker rp we have that rp((x + r)e + u) = x + r =1= y + s = rp((y + s)e + v). Hence for x E A \ {O} and r E Q with x + r E [0,1]' we will have that
+ r)) U rp-l((x + r, +00)). Consider a connected subset DeC such that 0, e + eo E D. If there exist x E A \ {O} and r E Q with x + r E (0,1) and D n ((x + r)e + (0, ex]) = 0, then, from rp(O) = < x +
r < 1 = rp(e + eo) and (1.1), we obtain that D is not connected. Hence for all x E A \ {O} and r E Q with x + r E (0,1) we have that there exists C \ {(x
+ r)e + (0, ex]} c
rp-l(( -00, X
Ax ::; 1 with
(x + r)e + Axex E D. Thus, {ex}xEA\{O} C span{e, D}. However, the family {ex}xEA\{O} is uncountable, and Ile x - ey II = 1 for all x, yEA \ {O}, x =1= y. As a consequence span {e, D} is not
separable, which implies that D is non-separable. Thus, it only remains to show that C is connected. If C were not connected, then there would exist a separation of C by subsets U and V; that is,
there would exist open (and closed) sets U and V such that U =I-
0 =I- V, U U V
= C,
UnV =
To simplify the notation, given Z E [0,1] we denote by z the only x E A such that = X, and we denote by Iz = ze + (0, e z] if z fj. Q and I z = ze + [0, eo] if Z E Q. Since Iz is connected we have
that given Z E [0,1] if I z n U =I- 0 then Iz C U (and the same with respect to V). As eo E U U V, we will assume that eo E U and hence 10 C U. Consider Z E Q n [0,1] such that I z C U. Since U is
open, there exists TJ = TJ(z) > a such that the closed ball B(ze,2TJ) n C c U and moreover such that (z - TJ, z + TJ) C [0,1] if Z E (0,1) n Q. Now it is immediate that for every u E f(X), Ilull = 1,
every y E (z-TJ, z+TJ)n[O, 1] and every A, 0:$ A:$ TJ, lIye+Au-zell :$ 2TJ. By Lemma 1.1.(3), there exist unique Xl E A and rl E Q with y = Xl +rl. If Xl = 0, then ye + [0, TJeo] = rle + [0, TJeo] C
B(ze, 2TJ) n C l c U and hence rle + [0, eo] C U. Alternatively, if Xl E A\ {a}, then ye+(O, TJe X1 ] = (Xl +rl)e+(O, TJe X1 ] C B(ze, 2TJ)n C2 c U, and again it follows that (Xl + rde + (0, eX!] C
U. In either case, we see that Iy C U for all y E (z - TJ,Z +TJ) n [0,1]. Let A:= {z E Qn [0,1] : Iz C U} and B:= {z E Qn [0,1] : Iz C V}. Let K := UzEA(Z - TJ, Z + TJ) n [0,1] and L := UzEB(Z - TJ,
z + TJ) n [0,1] where TJ = TJ(z) is given above. K and L are open sets in [0,1] and Q n [0,1] c L U K. If t is in k, the closure of K, then given n E N we can find Zn E A such that (Zn - TJn, Zn +
TJn) n (t - ~, t + ~) n [0, 1] =I- 0, where 1Jn = TJ(zn). Hence there exists a < b such that (a, b) C (zn - TJn, Zn + TJn) n (t - ~, t + ~) n [0, 1]. By Lemma 1.1.(4) there exists r n E Q such that t
+ r n E (a, b). Thus, the preceding paragraph shows that I Hrn C U and as t + rn = f, we have (t + rn)e + ef E U. Since Irnl < ~ for all nand U is closed in C we have that te + er E U, thus It cU.
The same argument applied to L implies that if S E L, then Is C V. Hence k and L are two closed disjoint sets. But as Q n [0,1] c K U L c [0,1] we have [0,1] = k U L. Since [0,1] is a connected set
and a E A c K we have that k = [0,1] and hence C = U, a contradiction. Therefore C is connected. 0
REMARK 1.3. This result extends to any non-separable Banach space. To see this, consider a non-separable Banach space F. As above we can take e E F with lIell = 1 and a continuous, linear functional
a and an uncountable set c Ker
C l :=
U rEQn[O,l]
re + [0, eo] and C2 :=
(X + r)e + (0, ex].
x+rE[O,l] xEA\{O}, rEQ
If C is endowed with the metric induced by the norm of F, then, by a proof similar to the one given in Theorem 1.2, C is a connected metric space that is not separably connected.
ACKNOWLEDGEMENT. This note evolved during visits the first author made to the Departamento de Amilisis Matematico of the Universidad de Valencia, and while the second author was a Visiting Professor
in the Department of Mathematical Sciences at Kent State University for the 1998-99 academic year. The authors express their thanks to the Departments concerned for their hospitality. In addition,
the authors are very grateful to the Referee, whose careful reading of the original version of this paper led to the present considerably simplified proof of Theorem 1.2.
References [A] Balbas, A., Estevez, M., Herves, C., and Verdejo, A. Espacios separablemente conexos, Rev. R.Acas. Cien. Exact. Fis. Nat. (Spain), 92, 1, (1998), 35-40. [B] J. Candeal, C. Herves and
E. Indurain, Some results on representation and extension of preferences, Journal of Mathematical Economics 29 (1998), 75-81. [C] R. Pol, Two examples of non-separable metrizable spaces, ColI. Math.
33, (1975),209-211. DEPARTMENT OF MATHEl\IATICS, KENT STATE UNIVERSITY, KENT. OH 44242 USA E-mail address: aronClmcs. kent. edu DEPARTAMENTO DE ANALISIS MATEMATlCO, FACULTAD DE MATEMATICAS,
UNIVERSIDAD DE VALENCIA, 46100 BURJASOT (VALENCIA). SPAIN E-mail address: manuel.maestreCluv. es
Contemporary Mathematics
Volume 328, 2003
Weighted Chebyshev Centres and Intersection Properties of Balls in Banach Spaces Pradipta Bandyopadhyay and S Dutta ABSTRACT. Vesely has studied Banach spaces that admit weighted Chebyshev centres
for finite sets. Subsequently, Bandyopadhyay and Rae had shown, inter alia, that Lt-preduals have this property. In this work, we investigate why and to what extent are these results true and thereby
explore when a more general family of sets admit weighted Chebyshev centres. We extend and improve upon some earlier results in this general set-up and relate them with a modified notion of minimal
points. Special cases when we consider the family of all finite, or more interestingly, compact subsets lead to characterizations of Lt-preduals. We also consider some stability results.
1. Introduction
Let X be a Banach space. We will denote by Bx[x,r] the closed ball ofradius r > 0 around x EX. We will identify any element x E X with its canonical image in X**. Our notations are otherwise
standard. Any unexplained terminology can be found in either [6] or [10). In this paper we continue the study of Banach spaces that admit weighted Chebyshev centres that began with [3). DEFINITION
1.1. Let Y be a subspace of a Banach space X. For A ---? ~+, define
Y and
¢A,p(X) = sup{p(a)llx -
all : a E A}
A point Xo E X is called a weighted Chebyshev centre of A in X for the weight p if ¢A,p attains its minimum at Xo. When A is finite, Vesely [18) has shown that if X is a dual space, A admits weighted
Chebyshev centres in X for any weight p, that the infimum of ¢ A,p over X and X** are the same, and THEOREM 1.2. [18, Theorem 2.7) For a Banach space X and aI, a2, ... ,an E X, the following are
equivalent : 2000 Mathematics Subject Classification. Primary 4IA65, 46B20; Secondary 41A28, 46B25, 46E15, 46E30. Key words and phrases. Weighted Chebyshev centres, minimal points, central subspaces,
I-complemented subspace, I PI,oo, Ll_preduals. © 43
2003 American Mathematical Society
BANDYOPADHYAY AND DUTTA (a) Ifrl, r2, ... , rn > 0 and ni=IBx " [ai, riJ =10, then ni=IBx[ai, riJ =10. (b) {al,a2, ... ,an } admits weighted Chebyshev centres for all weights rl,r2, ... ,rn > O. (c)
{aI, a2,' .. , an} admits f-centres for every continuous monotone coercive f : IR+. -4 IR (see [18J for the definitions).
In this work, we investigate why and to what extent are these results true and thereby explore when a more general family of sets admit weighted Chebyshev centres. Extending the notion of central
subspaces introduced in [3], we define an A-C-subspace Y of a Banach space X with the centres of the balls coming from a given family A of subsets of Y, the typical examples being those of finite,
compact, bounded or arbitrary sets. The first gives us the central subspace a la [3J and the last one is related to the Finite Infinite Intersection Property (I Pj,oo) [8J. We extend and improve upon
some results of [3, 18J in this general set-up and relate them with a modified notion of minimal points. We also improve upon one of the main results of [4J on the structure of the set of minimal
points of a compact set. As in [3], special cases when we consider the family of all finite, or more interestingly, compact subsets lead to characterizations of Ll-preduals. We also consider some
stability results.
2. General Results We first extend VeselY's result in [18J on dual spaces from finite sets to all the way upto bounded sets and also strengthens its conclusions. We need the following notions.
DEFINITION 2.1. Let X be a Banach space and A ~ X. (a) We define a partial ordering on X as follows : for Xl, X2 EX, we say that Xl :SA X2 if IlxI - all IIx2 - all for all a E A. We will denote by mx
(A) the set of points of X that are minimal with respect to the ordering :SA and often refer to them as :SA-minimal points of X. Note that :SA defines a partial order on any Banach space containing A
and we will use the same notation in all such cases. f(x2) (b) A function f : X -4 IR+ is said to be A-monotone if f(xd whenever Xl :SA X2· (c) Let Y be a subspace of X and A ~ Y. Following [9], we
say X E X is a minimal point of A with respect to Y if for any y E Y, Y :SA X implies y=x. We denote the set of all minimal points of A with respect to Y in X by Ay,x, Note Ay,x ;2 A. For A ~ X, the
set Ax,x will be called minimal points of A in X, and will be denoted simply by min A. (d) For A ~ X bounded, the Chebyshev radius of A in X is defined by
r(A) = inf sup Ilx xEX aEA
THEOREM 2.2. (a) If A ~ X is bounded and X ~ A + r(A)B(X), then there exists y E X such that y :SA x. (b) If X = Z· is a dual space and A is bounded, then every A-monotone and w*-lower semicontinuous
(henceforth, lsc) f : X -4 IR+ attains its minimum. In particular, for every p, ¢ A,p attains its minimum.
(c) If X = Z* is a dual space, for every Xo E X, there is a Xl E mx(A) such that Xl ::;A Xo. In particular, the minimum in (b) is attained at a point ofmx(A). PROOF. (a). Let X ~ A + r(A)B(X). Then,
there exists c > 0 such that IIx - all > r(A) + c for all a E A. By definition of r(A), there exists y E X such that sUPaEA Ily - all < r(A) + c. Clearly, y ::;A x. (b). By (a), if X ~ A + r(A)B(X),
there exists y E X such that y ::;A X, and hence, f(y) ::; f(x). Thus, the infimum of f over X equals the infimum over A + r(A)B(X). Moreover, since X is a dual space and f is w*-lsc, it attains its
minimum over any w*-compact set. Thus f actually attains its minimum over X as well. Since the norm on X is w*-lsc, so is cPA,p for every p. (c). Consider {x EX: x ::;A xo}. Let {Xi} be a totally
ordered subset. Let z be a w*-limit point of Xi. Since the norm is w*-lsc, we have liz -
all ::; lim inf Ilxi - all =
inf Ilxi
- all
for all a E A.
Thus the family {Xi} is ::;A-bounded below by z. By Zorn's lemma, there is a Xl E mx(A) such that Xl ::;A Xo. Now let Xo be a minimum for f. There is a Xl E mx(A) such that Clearly, f attains its
minimum also at Xl.
::;A Xo. 0
REMARK 2.3. (a) It follows that for any bounded set A, minA C A + r(A)B(X). This improves the estimates in [9] or [18]. (b) Apart from cPA,p, there are many examples of A-monotone and w*lsc f : X =
Z* ---- lR+. One particular example that has been treated extensively in [4] is the function cPf.,l defined by cPf.,l(x) = fA Ilx - aI1 2 dJ.L(a), where J.L is a probability measure on a compact set
A ~ X. (c) Observe that though minimal points of A are ::;A-minimal, there is some distinction between the two notions. The two notions coincide if X is strictly convex. See Proposition 3.1 below.
Now, if A is a bounded subset of a Banach space X, then by Theorem 2.2, A has a weighted Chebyshev centre in X**. But what about a weighted Chebyshev centre in X? When A is finite, Vesely [18] has
shown that the infimum of cPA,p over X and X** are the same, and A admits weighted Chebyshev centres in X for any weight p if and only if X satisfies Theorem 1.2(a). We now show that both of these
are special cases of more general results. We need the following definition. DEFINITION 2.4. Let Y be a subspace of a Banach space X. Let A be a family of subsets of Y. (a) We say that Y is an almost
A-C-subspace of X if for every A E A, X E X and c > 0, there exists y E Y such that
Ily - all::; IIx - all + c forall a E A. (b) We say that Y is an A-C-subspace of X if we can takec = 0 in (a). (c) If A is a family of subsets of X, we say that X has the (almost) A-IP if X is an
(almost) A-C-subspace of X**.
Some of the special families that we would like to give names to are : (i) F = the family of all finite sets, (ii) K = the family of all compact sets, (iii) B = the family of all bounded sets, (iv)
'P = the power set. Since these families depend on the space in which they are considered, we will use the notation F(X) etc. whenever there is a scope of confusion. REMARK 2.5. (a) Note that
F-C-subspaces were called central (C) subspaces in [3], 'P-C-subspaces were called almost constrained (AC) subspaces in [1, 2]. Also if X has the F-IP, it was said to belong to the class (GC) in [18,
3], and the 'P-IP was called the Finite Infinite Intersection Property (IPj,oo) in [7, 2]. (b) The definition of almost A-C-subspace is adapted from the definition of almost central subspace defined
in [17]. The exact analogue of the definition in [17] would have, in place of condition (1), sup Ily aEA
all::; sup Ilx - all + c. aEA
Clearly, our condition is stronger. We observe below (see Proposition 2.7) that this definition is more natural in our context. (c) By the Principle of Local Reflexivity (henceforth, PLR), any Banach
space has the almost F-IP. More generally, if Y is a!l ideal in X (see definition below), then Y is an almost F-C-subspace of X. DEFINITION 2.6. A subspace Y of a Banach space X is said to be an
ideal in X if there is a norm 1 projection P on X* with ker(P) = y.l. PROPOSITION 2.7. Let Y be a subspace of a Banach space X. Let A be a family of bounded subsets of Y. Then the following are
equivalent: (a) Y is an almost A-C-subspace of X (b) for all A E A and p : A ---t lR.+, if naEABx [a, p(a)] ¥- 0, then for every e > 0, naEABy[a, p(a) + c] ¥- 0. (c) for every bounded p, the infimum
of ¢A,p over X and Yare equal. PROOF. Equivalence of (a) and (b) is immediate and does not need A to be bounded. (a) ::::} (c). Let Y be an almost A-C-subspace of X, A E A and p : A ---t lR.+ be
bounded. Let M = supp(A). Let e > O. By definition, for x E X, there exists y E Y such that lIy - all::; Ilx - all + e for all a E A. It follows that
p(a)lly - all ::; p(a)llx -
all + p(a)e ::; p(a)llx - all + Me for all a E A.
and hence,
¢A,p(Y) ::; ¢A,p(X)
+ Me.
Therefore, inf ¢A,p(Y) ::; inf ¢A,p(X)
+ Me.
As e is arbitrary, the infimum of ¢A,p over X and Yare equal.
(c) =} (a). Let A E A, x E X and e > O. We need to show that there exists y E Y such that Ily - all IIx - all + e for all a E A. If x E Y, nothing to prove. Let x E X \ Y. Let N = sUPaEA IIx - all.
Let p(a) = l/llx - all. Since x r/: Y and A ~ Y, p is bounded. Then ¢A.p(X) = 1, and therefore, inf ¢A,p(X) S 1. By assumption, inf ¢A,p(Y) = inf ¢A,p(X) S 1, and so, there exists y E Y, such that
¢A,p(y) 1 + e/N. This implies Ily - all S Ilx - all + ell x - all/N S Ilx - all + e for all a E A. 0
As noted before, by PLR, any Banach space has the almost F-IP. And therefore, the result of [18J follows. PROPOSITION 2.8. Let A and Al be two families of subsets of Y such that for every A E A and e
> 0, there exists Al E Al such that A ~ Al + cB(Y). If Y is an almost AI-C-subspace of X, then Y is an almost A-C-subspace of X as well. Consequently, any ideal is an almost K-C -subspace and any
Banach space has the almost K-IP. In particular, if A is a compact subset of X and p : A ---+ lR+ is bounded, then the infimum of ¢A,p over X and X** are the same. PROOF. Let A E A and e > O. By
hypothesis, there exist Al E Al such that Al + eB(Y). Let x E X. Since Y is an almost AI-C-subspace of X, there exists y E Y such that A
lIy - alii S Ilx - alII
+ e/3 for all al
E AI.
Now fix a E A. Then there exists al E Al such that lIa - alii < e/3. Then Ily - alii + lIa - alii S Ilx - alii + 2e/3 Ilx - all + Iia - alii + 2e/3 S IIx - all + e. Therefore, Y is an almost
A-C-subspace of X as well. Since any Banach space has the almost F-IP, by the above, it has the almost K-IP too. The rest of the result follows from Proposition 2.7. 0 Ily - all
S S
EXAMPLE 2.9. Vesely [18J has shown that if A is infinite, the infimum of ¢A,p over X and X** may not be the same. His example is X = co, A = {en: n ;::: I} is the canonical unit vector basis of Co
and p == 1. Then inf ¢A,p(X) = 1 and inf ¢A,p(X**) = 1/2. The example clearly also excludes countable, bounded, or, taking Au {O}, even weakly compact sets. Thus Co fails the almost B-IP, almost P-IP
and if A is the family of countable or weakly compact sets, then Co fails the almost A-IP too. Stronger conclusions are possible for A-IP. LEMMA 2.10. Let Y be a subspace of a Banach space X. For A ~
Y, the following are equivalent : (a) For every A-monotone f : A ---+ lR+ and x E X, there exists y E Y such that f(y) S f(x). ( b) For every p : A ---+ lR+ and x EX, there exists y E Y such that
¢A,p(Y) S ¢A,p(X). (c) For every continuous p : A ---+ lR+ and x EX, there exists y E Y such that ¢A,p(Y) S ¢A,p(X). (d) For every bounded p : A ---+ lR+ and x EX, there exists y E Y such that ¢A,p
(Y) S ¢A,p(X).
(e) Any family of closed balls centred at points of A that intersects in X also intersects in Y. (f) for any x EX, ther·e exists y E Y such that y :SA x. It follows that whenever any of the above
conditions is satisfied, for every Amonotone f : A --+ lR+, the infimum of f over X and Yare equal and if A has a weighted Chebyshev centre in X, it has a weighted Chebyshev centre in Y.
PROOF. (a) => (b) => (c), (b) => (d) and (e) ¢} (f) => (a) are obvious. (c) or (d) => (f). As in the proof of Proposition 2.7, let p(a) = l/llx-all. Then p is continuous and bounded and
We now conclude the discussion so far by obtaining the extension of Theorem 1.2. THEOREM 2.11. For a Banach space X and a family A of bounded subsets of X, the following are equivalent: (a) X has the
A-IP. (b) For every A E A and every f : X** --+ lR+ that is A-monotone and w*-lsc, the infimum of f over X** and X are equal and is attained at a point of (c) For every A E A and every p, the infimum
We now study different aspects of A-C-subspaces. DEFINITION 2.12. Let Y be a subspace of a Banach space X. Let A x E X and X* E B(X*), define U(x,A,x*)
= inf{x*(y) + IIx - yll : yEA}
L(x, A, x*)
Y. For
sup{x*(y) -lix - yll : yEA}
The following lemma is in [1]. We include the proof for completeness. LEMMA 2.13. Let Y be a subspace of a Banach space X and A ~ Y. For Xl, X2 EX, X2 :SA Xl if and only if for all x* E B(X*), U(X2,
A, x*) U(XI, A, x*).
PROOF. If X2 :SA Xl, then for all x* E B(X*), x*(y) + IIx2 - yll x*(y) + IIxI -YII. And therefore, U(x2,A,x*):S U(xI,A,x*). Conversely, suppose IIx2 - Yo II > IIxI - Yo II for some Yo E A. Then there
exists c > 0 such that IIx2 - Yo II - c ~ IIxI - Yoli. Choose x* E B(X*) such that IIxI - Yo II IIx2 - yolI- c < X*(X2 - Yo) - c/2. Thus U(XI, A, x*) x*(yo) + IIxI Yoll < X*(X2) - c/2 < U(X2, A, x*).
REMARK 2.14. Instead of B(X*), it suffices to consider the unit ball of any norming subspace of X*. We compile in the following propositions several interesting facts about A-Csubspaces and the A-IP.
PROPOSITION 2.15. Let Y be a subspace of a Banach space X. For a family A of subsets of Y, the following are equivalent: (a) Y is an A-C-subspace of X
(b) for every x E X and A E A, there exists y E Y such that U(y, A, x*) ::; U(x,A,x*) for every x* E B(X*). (c) for any A E A, Ay,x ~ Y.
PROOF. This follows from Lemma 2.13 and the definition of Ay,x.
COROLLARY 2.16. X has the P-IP if and only if for every x** E X**, there exists x E X such that x is dominated on B(X*) by the 1~pper envelop of x** considered as a function on B(X*) equipped with
the w*-topology. PROOF. Observe that for any x E X, U(x,X,·) == x on B(X*) and for x** E X**, U(x**, X, x*) is the upper envelop of x** considered as a function on B(X*) equipped with the w*-topology
(see [8]). 0 PROPOSITION 2.17. (a) Let X be a Banach space and let Y be a subspace of X. Let A be a family of subsets of Y and let Al be a subfamily of A. If Y is a A-C -subspace of X, then Y is a Al
-C -subspace of X as well. In particular, P-IP implies B-IP implies /C-IP implies F-IP. (b) 1-complemented subspaces are A-C-subspaces for any A. (c) Let Z ~ Y ~ X and let A be a family of subsets of
Z. If Z is an A-C-subspace of X, then Z is an A-C-subspace of Y. And, if Y is an A-C-subspace of X, then the converse also holds. PROOF. The proof follows the same line of argument as in [3,
Proposition 2.2]. We omit the details. 0 PROPOSITION 2.18. For a family A of subsets of a Banach space X, the following are equivalent : (a) X has the A-IP (b) X is a A-C-subspace of some dual space.
(c) for all A E A and p : A ---+ lR+, ni=l Bx [ai, p(ai) + c] =f. 0 for all finite subset {aI, a2, ... ,an} ~ A and for all c > 0 implies naEABx [a, p(a)] =f. 0. In particular, any dual space has the
A-IP for any A. Let S be any of the families F, /C, B or P. The S-IP is inherited by S-C-subspaces, in particular, by 1-complemented subspaces. PROOF. Clearly, (a) =} (b), while (c) =} (a) follows
from the PLR. (b) =} (c). Let X be an A-C-subspace of Z*. Consider the family {Bz. [a, p(a)+ c] : a E A, c > O} in Z*. Then, by the hypothesis, any finite subfamily intersects. Hence, by
w*-compactness, naEABz • [a, p(a)] =f. 0. Since X is an A-C-subspace of Z*, we have naEABx[a,p(a)] =f. 0. 0 The following result significantly improves [3, Proposition 2.8] and provides yet another
characterization of the A-IP. PROPOSITION 2.19. Let Y be an almost F-C subspace of a Banach space X. Let A be a family of subsets of Y. If Y has the A-IP, then Y is an A-C -subspace of X. In
particular, the conclusion holds when Y is an ideal in X. PROOF. Let x EX, A E A. Since Y be an almost F-C subspace of X, for all finite subset {aba2, ... ,an} ~ A and for all c > 0, nb:IBy[ai,
Ilx.,... aill + c] =f. 0. Since Y has the A-IP, by Proposition 2.18(c), naEABy[a, Ilx - alll =f. 0. 0 Since X is always an ideal in X**, the following corollary is immediate.
COROLLARY 2.20. For a Banach space X and a family A of subsets of X, the following are equivalent : . (a) X has A-IP. (b) X is an A-C-subspace of every superspace Z in which X embeds as an almost F
-C subspace. (c) X is an A-C-subspace of every superspace Z in which X embeds as an ideal.
3. Strict convexity and minimal points PROPOSITION 3.1. If a Banach space X is strictly convex, then for every A X, min A = mx(A).
PROOF. As we have already observed, min A ~ mx(A). Let Xo E mx(A) and Xo ~ minA. Then there is an x E X such that x f= Xo and x :S::A Xo. Since Xo E mx(A), we must have Ilx - all = Ilxo - all for all
a E A. Since X is strictly convex, II (x + xo)/2 - all < IIxo - all for all a. This contradicts that Xo E mx(A). Hence Xo EminA. D REMARK 3.2. If X is strictly convex, by a similar argument, for
every Xo EX, there is at most one Xl E mx(A) such that Xl :S::A Xo. Thus for a strictly convex dual space, for every Xo E X*, there is a unique xi E mx· (A) such that xi :S::A Xo' PROPOSITION 3.3.
Let X be strictly convex. Let A be a compact subset of X. For each continuous p, A admits at most one weighted Chebyshev centre. PROOF. Suppose A admits two distinct weighted Chebyshev centres Xo, Xl
E X. Then
3.4. Let X be a Banach space such that (i) X has the F-IP; and (ii) for every compact set A ~ X, mx(A) is weakly compact. Then X has the K-IP. Moreover, if X** is strictly convex, then the converse
also holds. THEOREM
Let X have the F-IP and for every compact set A ~ X, let mx(A) be weakly compact. Observe that for any B ~ A, we have mx(B) ~ mx(A). Let A ~ X be compact and let X** E X**. By Lemma 2.10, it suffices
to show that there is a Zo E X such that Ilzo - all :s:: Ilx** - all for all a E A. Let {an} be a norm dense sequence in A. Take a sequence Ck --+ O. By compactness of A, for each k, there is a nk
such that A ~ U~k BX[an,ck]. Since X has the F-IP, there exists Zk E nlk Bx [an' Ilx** - anll] and Zk E mx( {al, a2 ... ank }) ~ mx(A). Then IIzk-all :s:: Ilx**-all+2ck for all a EA. Now, by weak
compactness of m x (A), we have, by passing to a subsequence if necessary, Zk --+ Zo weakly for some Zo EX. Since the norm is weakly Isc, we have Ilzo -all :s:: lim inf IIZk -all :s:: Ilx** -all for
all a E A. Conversely, let X have the K-IP and X** be strictly convex. Let A ~ X be compact. It is enough to show that any sequence {xn} ~ mx(A) has a weakly convergent subsequence. Without loss of
generality, we may assume that {xn} are PROOF.
all distinct. By Remark 2.3 (a), mx(A) ~ A + r(A)B(X) is bounded. Let x** be a w*-cluster point of {x n } in Xu. It suffices to show that x** EX. Suppose x** E X** \ X. Since X has the K-IP, there
exists Xo E mx(A) such that Ilxo - all ::; Ilx** - all for all a E A. Since X** is strictly convex, lI(x** +xo)/2 - all < IIx** - all for all a E A. Since (x** +xo)/2 E X** \X, by K-IP again, there
exists Zo E mx(A) such that Ilzo -all::; II(x** +xo)/2-all < IIx** -all for all a E A. Since A is compact, there exists c > 0 such that Ilzo - all < IIx** - all - c for all a E A. Observe that
Ilzo - all < Ilx** - all- c ::; liminf IIx n n
all- c for all a E A.
Therefore, for every a E A, there exists N(a) EN such that for all n
IIzo - all < Ilxn - all- c. By compactness, there exists N E N such that IIzo - all < IIx n - all - c/4 for all n ~ N and a E A. Thus, Zo ::;A Xn for all n ~ N. Since Xn E mx(A) and X is strictly
convex, Zo = Xn for all n completes the proof.
N. This contradiction
REMARK 3.5. In proving sufficiency, one only needs that {Zk} has a subsequence convergent in a topology in which the norm is lsc. The weakest such topology is the ball topology, bx . So it follows
that if X has the F-IP and for every compact set A ~ X, mx(A) is bx-compact, then X has the K-IP. Is the converse true? COROLLARY 3.6. [4, Corollary 1] Let X be a reflexive and strictly convex Banach
space. Let A ~ X be a compact set. Then min(A) is weakly compact. REMARK 3.7. Clearly, our proof is simpler than the original proof of [4]. If Z is a non-reflexive Banach space with Z*** strictly
convex, then X = Z· is a non-reflexive Banach space with K-IP such that X** is strictly convex. Thus, our result is also stronger than [4, Corollary 1].
4. L l -preduals and PI-spaces Our next theorem extends [3, Theorem 7], exhibits a large class of Banach spaces with the K-IP and produces a family of examples where the notions of FC-subspaces and
K-C-subspaces are equivalent. DEFINITION 4.1. (a) [12] A Banach space X is called an Ll-predual if X* is isometrically isomorphic to £1(J-t) for some positive measure J-t. (b) [11] A family {Bx
[xi,ri]} of closed balls is said to have the weak intersection property iffor all x* E B(X*) the family {Ba[x*(xd,ri]} has nonempty intersection in JR. THEOREM 4.2. For a Banach space X, the
following are equivalent: ( a) X is a K -C -subspace of every superspace (b) X is a K-C -subspace of every dual superspace ( c) X is a F -C -subspace of every superspace (d) X is an almost F -C
-subspace of every superspace (e) X is a F-C-subspace of every dual superspace (f) X is an almost F-C-subspace of every dual superspace (g) X is an Ll-predual.
PROOF. Observe that if X ~ Y ~ y** and X is a A-C-subspace of Y**, then X is a A-C-subspace of Y. Thus (a) {::} (b) and (c) {::} (e). And clearly, (a) ::::} (c) ::::} (d) ::::} (f). (f) ::::} (g).
Since the definition of almost central subspaces in [17J is weaker than our definition of almost F-C-subspaces, this follows from [17, Theorem 1, 2::::} 3J (g) ::::} (a). Suppose X is an LI-predual,
and let X ~ Y. Let A ~ X be compact with at least three points. Let Yo E Y. Then the family of balls {Bx[a, Ilyo - allJ : a E A} have the weak intersection property. Since X is an £l-predual and
since the centres of the balls are in a compact set, by [14, Proposition 4.4J, naEABx [a, IlyoallJ ¥- 0. If A has two points, observe that two balls intersect if and only if the distance between the
centres is less than or equal to the sum of the radii, it is independent of the ambient space. 0 COROLLARY 4.3. Every LI-predual has the /C-IP and hence also the F-IP.
PROPOSITION 4.4. Suppose X is an LI -predual space. Then for a subspace X, the following are equivalent : (a) Y is an ideal in X (b) Y is a /C-C-subspace of X (c) Y is a F-C-subspace of X (d) Y is an
almost F-C-subspace of X (e) Y itself is an LI -predual
PROOF. (e) ::::} (b) follows from Theorem 4.2 and (e) ::::} (a) follows from [16, Proposition IJ. And clearly, (b) ::::} (c) ::::} (d) and (a) ::::} (d). (d) ::::} (e). This again is an easy
adaptation of the proof of [17, Theorem 1, 2 ::::} 3J. We omit the details. 0 The analog of Theorem 4.2 for 'P-C-subspaces involves 'PI-spaces. DEFINITION 4.5. Recall that a Banach space is a
'PI-space if it is 1complemented in every superspace. THEOREM 4.6. For a Banach space X, the following are equivalent: (a) X is a 'PI -space ( b) X is i-complemented in every dual space that contains
it ( c) X is a 'P -C -subspace of every superspace (d) X is a 'P -C -subspace of every dual space that contains it ( e) X is isometric to C (K) for some extremally disconnected compact Hausdorff
space K. PROOF. (a) {::} (b) and (c) {::} (d) follow as in the first paragraph of Theorem 4.2. And clearly, (a) ::::} (c). (d) ::::} (a). By Proposition 2.18 and Theorem 4.2, (d) implies X is an LI_
predual with 'P-IP. Recall that [12, Theorem 3.8J a Banach space X is a 'PI-space if and only if every pairwise intersecting family of closed balls in X intersects. And that X is a LI-predual if and
only if X** is a 'PI-space. Now given a pairwise intersecting family of closed balls in X, since X** is a 'PI-space, they intersect in X**. And since X has 'P-IP, they intersect in X too. (a) {::}
(e) is also observed in [12, Section 11J. 0
PROPOSITION 4.7. Let A be a family of subsets of X such that F ~ A. Then, the following are equivalent : (a) X is an Ll -predual with A-IP (b) X is an A-C-subspace of every superspace (c) for every A
E A, every pairwise intersecting family of closed balls in X with centres in A intersects. PROOF. (a) =? (b). Since X has the A-IP, it is an A-C-subspace of every superspace in which it is an ideal
(Proposition 2.20) and since X is an L1-predual, it is an ideal in every superspace [16, Proposition 1]. Thus (b) follows. (b) =? (a). Since F ~ A, this is immediate. (a) =? (c). This is similar too
the proof of Theorem 4.6 (d) =? (a). (c) =? (a). If every finite family of pairwise intersecting closed balls in X intersects, then X is an L1-predual. And that X has the A-IP follows from Propo0
sition 2.18 (c). Let C(T, X) be the space of all X-valued bounded continuous functions on a topological space T equipped with the sup norm. We now characterize when C(T, X) is a real L1-predual.
First we need the following lemma. LEMMA 4.8. Suppose Y is a subspace of a Banach space X and Y is a real L1-predual. Let A ~ Y be a compact set and r : A -7 lR,+ be such that naEABx[a,r(a)] =I- 0.
Let y E naEABy[a,r(a) + c] for some c > O. Then there exists z E naEABy [a, r(a)] such that Ily - zll S c. PROOF. Since naEABx [a, r(a)] =I- 0, and intersection of intervals is an interval, for any
y* E B(Y*), naEABJR[y*(a), r(a)] =I- 0 and is a closed interval. As y*(y) E naEABJR[y*(a),r(a) + c] for any y* E B(Y*), the family {By[y,c].By[a,r(a)] : a E A} is a weakly intersecting family of
balls in Y. Since Y is a L1-predual,
By[y,C]nnaEABy[a,r(a)] =1-0.
PROPOSITION 4.9. A Banach space X is a real Ll -predual if and only if for each paracompact space T, C(T, X) is a real L1-predual. PROOF. Since X is I-complemented in C(T, X), hence a K-C-subspace,
by Proposition 4.4, if C(T, X) is an L1-predual, then so is X. Conversely, suppose X is a real L1-predual. Let Z = C(T, X), {!I, 12,···, fn} ~ Z and rl, r2,···, rn > 0 be such that the family {Bz[fi,
ri] : i = 1, ... , n} intersects weakly. Then for each t E T, the family {B x [fi (t), r i] : i = 1, ... , n} intersects weakly, and since X is a real L I-predual, they intersect in X. Consider the
multi-valued map F: T -7 X given by F(t) = nf=lBx[Ji(t),ri]. Note for each t, F(t) is a nonempty closed convex subset of X. CLAIM : F is lower semicontinuous, that is, for each U open in X, the set V
= {t E T: F(t) n U =I- 0} is open in T. Let to E V. Let Xo E F(to) n U. Let c > 0 be such that lI:r - xoll < c implies x E U. Let W be an open subset of to such that t E W implies 111;(t)- fi(to) II
< c/2 for all i = 1, ... , n. We will show that W ~ V. Let t E W. Then for any i = 1, ... , n, Ilxu - fi(t)11 S Ilxo - f.i(tO) II + 111;(to) fi(t)11 Sri + c/2. Therefore, Xo E nf=lBx[fi(t),rj + c/2].
By Lemma 4.8, there exists z E F(t) = nf=lBx[fi(t), ri] such that Ilxo-zll S c/2 < c. Then z E F(t)nU, and hence, t E V. This completes the proof of the claim.
Now since T is paracompact, by l\Iichael's selection theorem, there exists 9 E Z such that g(t) E F(t) for all t E T. It follows that 9 E ni'=l Bz[f;, ril, 0 REMARK 4.10. For T compact Hausdorff,
this result follows from [13, Corollary 2, p 43]. But our proof is simpler.
5. Stability Results In this section we consider some stability results. With a proof similar to [3, Proposition 14], we first observl' that PROPOSITION 5.1. K-IP is a separ-ably determined property.
i.e .. if every sepamble subspace of a Banach space X have K-IP. then X also has K-IP. DEFINITION 5.2. [10] A subspace Y of a Banach space X is called a semi-Lsummand if there exists a (nonlinear)
projection P : X ----> Y such that P(>..x
+ Py) Ilxll
+ Py, and IIPxl1 + Ilx - VI'II
for all x, y EX. >.. scalar. In [3]. it was shown that semi-L snmmands are .1'-C-subspaces. Basically the same proof actually shows that PROPOSITION 5.3. A semi-L-summand is an A-C-subspace for any
A. Our next result concerns proximinal subspaces. DEFINITION 5.4. A subspace Z of a Banach space X is called proximinal if for every :r E X, there exists Zo E Z such that Ilx - zoll = d(J;, Z) =
infzEz Ilx - zll· The map PZ(J:) = {zo E Z : Ilx - zoll = infzEz lI:r - zll} is called the metric projection. PROPOSITION 5.5. Let Z ~ Y ~ X, Z proximinal in X. (a) Let A be a family of subsets of Y
/ Z. Let A' be a family of subsets of Y s'ltch that for any :1; E X and A E A. there exists A' E A' such that for any a + Z E A. {a + Pz(x - a)} n A' i=- 0. S-uppose Y is a A'-C -subspace of X. Then
Y/Z 'is a A-C-subspace of X/Z. Let S be any of the families .1', l3 or P. (b) IfY is a S(Y)-C-subspace of X. then Y/Z is a S(Y/Z)-C-8'ubspace of X/Z. (c) Suppose the metric projection has a
continuous selection. Then. if Y is a K(Y)-C-subspace of X, Y/Z is a K(Y/Z)-C-subspace of X/Z. (d) Let Z ~ Y ~ X*, Z w*-closed in X*. 11' Y is a S(Y)-C-subspace of X*, then Y/Z is a S(Y/Z)-C-subspace
of X* /Z. and hence. has the S(Y/Z)-IP. (e) Let X have the S(X)-IP. Let l'd ~ X be a reflexive subspace. Then X/M has the S(X/l'd)-IP. PROOF. (a). Let A E A and x + Z E X/Z. Choose A' as above. Then,
for a + Z E A, there exists z E Pz(x - a) (depending on .1: and a) such that a + zEA'. Since Y is a A' -C-subspace of X, there exists Yo E Y snch that Ilyo - a - zll ~ Ilx - a - zll for all a + Z E
A. Clearly then Ilyo - a + ZII ~ Ilyo - a - zll ~ 11:1; - a - zll = IIx - a + ZII·
If S is the family under consideration in (b) and (c) above and A = S(Y/Z), then for any choice of A' as above, S(Y) ~ A'. Hence, (b) and (c) follows from (a). For (d), we simply observe that any
w*-closed subspace of a dual space is proximinal. And (e) follows from (d). 0 As in [3, Corollary 4.6], we observe PROPOSITION 5.6. Let Z ~ Y ~ X, Z proximinal in Y and Y is a semi-Lsummand in X.
Then Y/Z is a P-C-subspace of X/Z. Let us now consider the Co or fp sums. THEOREM 5.7. Let r be an index set. For all 0: E r, let Ya: be a subspace of Xa:. Let X and Y denote resp. the Co or fp (1
:::; p :::; 00) sum of Xa: 's and Ya: 'so (a) For each 0: E r, let Aa: be a family of subsets ofYa: such that {O} E Aa: and for any A E Aa:. there exists B E Aa: such that Au {O} ~ B. Let A be a
family of subsets of Y such that for any 0: E r, the 0:section of any A E A belongs to Aa:. Then Y is an A-C-subspace of X if and only if for each 0: E r, Ya: is an Aa:-C-subspace of Xa:. Let S be
any of the families F, 1C, B or P. (b) Y is a S(Y)-C-subspace of X if and only if for any 0: E r, Ya: is a S(Ya:)-C-subspace of Xa:. (c) The S-IP is stable under fp-sums (1 :::; p:::; 00). PROOF.
(a). The proof is very similar that of to [3, Theorem 4.7]. We omit the details. (c). Xa: has S-IP if and only if Xa: is a S-C-subspace of some dual space Y';. Now the fp-sum (1 :::; P :::; 00) of
Y';'s is a dual space. 0 REMARK 5.8. The result for F-IP has already been noted by [18] with a much different proof. The stability of the P-IP under f1-sums is noted in [15] again with a different
proof. [18] also notes that F-IP is stable under co-sum. And Corollary 4.3 shows that Co has the IC-IP. However, we do not know if the IC-IP is stable under Cosums. As for the B-IP or P-IP, we now
show that co-sum of any infinite family of Banach spaces lacks the B-IP, and therefore, also the P-IP. This is quite similar to Example 2.9. PROPOSITION 5.9. Let r be an infinite index set. For any
family of Banach spaces Xa:, 0: E r, X = fficoXa: lacks the B-IP. PROOF. For each
0: E
r, let Xa:
be an unit vector in Xa: and define ea: E X by if f3 = 0: otherwise
Then the set A = {ea: : 0: E r} is bounded and the balls Bx--[ea:, 1/2] intersect at the point (1/2xa:) E X**, but the balls Bx[ea:, 1/2] cannot intersect in X. 0 REMARK 5.10. As before, taking AU
{O}, it follows that X lacks the A-IP even for A = weakly compact sets. Coming to function spaces, we note the following general result.
PRDPOSITION 5.11. Let Y be a subspace of a Banach space X and A be a family of subsets of Y. (a) For any topological space T, if C(T, Y) is a A-C-subspace of C(T, X), then Y is a A-C -subspace of X.
Moreover, if C(T, X) has A-IP, X has A-IP. (b) Let (n, L-, fl) be a probability space. If for some 1 ~ P < 00, LP(fl, Y) is aA-C-subspace of LP(fl, X), then Y is aA-C-subspace ofX. Moreover, if LP
(fl, X), has A-IP, then X has A-IP. PROOF. For (a) and (b), let F(X) denote the corresponding space of functions and identify X with the constant functions. In (a), point evaluation and in (b),
integral over n gives us a norm 1 projection from F(X) onto X. Thus X inherits A-IP from F(X). Now suppose F(Y) is an A-C-subspace of F(X). Let P : F(Y) -+ Y be the above norm 1 projection. Let x E X
and A E A. Then, there exists g E F(Y) such that IIg-ali ~ IIx-all for all a E A. Let y = Pg. Then, lIy-all ~ IIg-all ~ IIx-all for all a E A. 0 The following Proposition was proved in [3].
PROPOSITION 5.12. (a) Let X has Radon Nikodym Property and is 1complemented in Z* for some Banach space Z. Then for 1 < p < 00, LP(IL,X) is 1-complemented in U(fl, Y)* (l/p+ l/q = 1), and hence has
the 'P-IP. (b) Suppose X is separable and 1-complemented in X** by a projection P that is w*-w universally measurable. Then for 1 ~ P < 00 LP(fl, X) is 1-complemented in Lq(fl,X*)* (l/p+ l/q = 1),
and hence has the 'P-IP. Since the B-IP or 'P-IP is inherited by I-complemented subspaces and the B-IP, the next result follows essentially from the arguments of [17].
PROPOSITION 5.13. (a) Let X be a Banach space containing Co and let Y be any infinite dimensional Banach space. Then X 181" Y fails the B-IP and 'P-IP. (b) If C(K, X) has the B-IP, then either K is
finite or X is finite dimensional. C(K,X) has the 'P-IP if and only if either (i) K is finite and X has the 'P-IP or (ii) X is finite dimensional and K is extremally disconnected. (c) For· any
nonatomic measure space (n,L-,/L) and a Banach space X containing co, £1 (fl, X) fails the B-IP. In the next Proposition, we prove a partial converse of Proposition 5.11(a) when Y is finite
dimensional and K is compact and extremally disconnected. PROPOSITION 5.14. LetS be any ofthefamiliesF, IC, B or'P. LetY be a finite dimensional a S(Y)-C-subspace of a Banach space X. Then for any
extremally disconnected compact space K, C(K, Y) is a S(C(K, Y))-C-subspace of C(K, X). PROOF. We argue similar to the proof of [3, Proposition 4.11]. Let K be homeomorphically embedded in the
Stone-Cech compactification .B(r) of a discrete set r and let ¢ : .B(r) -+ K be a continuous retract. Let A E S(C(K, Y)) and g E C(K, X). Note that since Y is finite dimensional, by the defining
property of .B(r), any Y-valued bounded function on r has a norm preserving extension
in C(,B(r), Y). Thus C(,B(r), Y) can be identified with EBeoc(r) Y. Lift A to this space. In view of Theorem 5.7, this space is S(Y)-C-subspace of EBooX. This latter space contains C(,B(r) , X). Thus
by composing the functions with 4>, we get a f E C(K, Y) such that Ilf - hll ~ Ilg - hll for all h E A. Hence the result. 0 And now for a partial converse of Proposition 5.11(b). THEOREM 5.15. Let Y
be a separable subspace of x. IfY is a P-C-subspace of X, then for any standard Borel space 0 and any a-finite measure jl, Lp(jl, Y) is a P-C-subspace of Lp(jl,X). PROOF. Let f E Lp(jl, X). Since Y
is a P-C-subspace of X, for each x E X, nyEyBy[y, IIx - yll] =1= 0. Define a multi-valued map F : 0 ----+ Y, by
F(t) = {
By[y, Ilf(t)
- ylll
f(t) EX \ Y
f(t) E Y
Let G = {(t, z) : z E F(t)} be the graph of F. Claim: G is a measurable subset of 0 x Y. To establish the claim, we show that GC is measurable. Since Y is separable, let {Yn} be a countable dense set
in Y. Observe that z ~ F(t) if and only if either f(t) E Y and z =1= f(t) or f(t) E X \ Y and there exists Yn such that liz - Ynll > IIf(t) - Ynll· And hence,
G C = {f(t) E Y and z
f(t)} u
U {f(t) EX \ Y and liz - Ynll > Ilf(t) -
is a measurable set. By von Neumann selection theorem, there is a measurable function 9 : 0 ----+ Y such that (t,g(t)) E G for almost all tEO. Observe that Ilg(t)11 ~ IIf(t)1I for almost all t. Hence
9 E Lp(jl, Y). Also for any h E Lp(jl, Y) we have Ilg(t) - h(t)11 ~ Ilf(t) - h(t)1I for almost all t. Thus, Ilg - hll p ~ IIf - hll p for all h E Lp(jl, Y). 0 QUESTION 5.16. Suppose Y is a separable
lC-C-subspace of X. Let (0, ~,jl) be a probability space. Is LP(jl, Y) a lC-C -subspace of LP(jl, X)? REMARK 5.17. This question was answered in positive in [3] for F-C-subspaces and we did it for
P-C-subspaces. Both the proofs are applications of von Neumann selection Theorem. The problem here is for a compact set A in LP(jl, Y) and wE 0 the set {f(w) : f E A} need not be compact in Y.
ACKNOWLEDGEMENTS. Partially supported by a DST-NSF grant no. RP041/2000. The first-named author availed this grant to visit Southern Illinois University at Edwardsville, USA in May-June 2002 and
attended the Fourth Conference on Function Spaces, where he presented a talk based on this work. He would like to thank Professor K. Jarosz for the warm hospitality and a wonderful conference. We
also thank the referee for suggestions that improved the paper.
References [1] P. Bandyopadhayay, S. Basu, S. Dutta and B. L. Lin Very nonconstmined subspaces of Banach spaces, Preprint 2002. [2] P. Bandyopadhayay and S. Dutta, Almost constmined subspaces of
Banach spaces, Preprint 2002. [3] Pradipta Bandyopadhyay and T. S. S. R. K. Rao, Centml subspaces of Banach spaces, J. Approx. Theory, 103 (2000),206-222. [4] B. Beuzamy and B. Maurey, Points
minimaux et ensembles optimaux dans les espaces de Banach, J. Functional Analysis, 24 (1977), 107-139. [5] J. Diestel, Geometry of Banach Spaces, selected topics, Lecture notes in Mathematics,
Vo1.485, Springer-Verlag (1975). [6] J. Diestel and J. J. Uhl, Jr., Vector' measures, Mathematical Surveys, No. 15, Amer. Math. Soc., Providence, R. 1. (1977). [7] G. Godefroy, Existence and
uniqueness of isometric preduals: a survey, Banach space theory (Iowa City, lA, 1987), 131-193, Contemp. Math., 85, Amer. Math. Soc., Providence, RI, 1989. [8] G. Godefroy and N. J. Kalton, The ball
topology and its applications, Banach space theory (Iowa City, lA, 1987), 195-237, Contemp. Math., 85, Amer. Math. Soc., Providence, RI, 1989. [9] G. Godini On minimal points, Comment. Math. Univ.
Carotin., 21 (1980), 407-419. [10] P. Harmand D. Werner and W. Werner, M-ideals in Banach spaces and Banach algebras, Lecture Notes in Mathematics, 1547, Springer-Verlag, Berlin, 1993. [11] O.
Hustad, Intersection properties of balls in complex Banach spaces whose duals are L1 spaces, Acta Math., 132 (1974), 283-313. [12] H. E. Lacey, Isometric theory of classical Banach spaces, Die
Grundlehren der mathematischen Wissenschaften, Band 208, Springer-Verlag, New York-Heidelberg, 1974. [13] J. Lindenstrauss, Extension of compact opemtors, Mem. Amer. Math. Soc., No. 48, 1964. [14] A.
Lima, Complex Banach spaces whose duals are L1-spaces, Israel J. Math., 24 (1976), 59-72. [15] T. S. S. R. K. Rao, Intersection properties of balls in tensor products of some Banach spaces II, Indian
J. Pure Appl. Math., 21 (1990),275-284. [16] T. S. S. R. K. Rao, On ideals in Banach spaces, Rocky Mountain J. Math., 31 (2001), 595-609. [17] T. S. S. R. K. Rao Chebyshev centers and centmble sets,
Proc. Amer. Math. Soc., 130 (2002), 2593-2598. [18] L. Vesely, Genemlized centers of finite sets in Banach spaces, Acta Math. Univ. Comen., 66 (1997),83-115. (Pradipta Bandyopadhyay) STAT-MATH
DIVISION, INDIAN STATISTICAL INSTITUTE, 203, B. T. ROAD, KOLKATA 700 108, INDIA, E-mail: pradiptaClisical.ac.in (S Dutta) STAT-MATH DIVISION, INDIAN STATISTICAL INSTITUTE, 203, B. T. ROAD, KOLKATA
700 108, INDIA, E-mail: sudipta.rGlisical.ac . in
Contemporary Mathematics Volume 328, 2003
The boundary of the unit ball in HI-type spaces Paul Beneker and Jan Wiegerinck ABSTRACT. This is a survey on the extreme, exposed and strongly exposed boundary points of the unit baH in various
Hl-type spaces.
1. Introduction
Theorems like the Krein-Milman theorem and Phelps' theorem assert the existence of many extreme points in certain convex sets. However, to decide whether a particular point in the boundary of a
convex set is extreme, exposed or strongly exposed is quite a different question. In this paper we survey what is known in the very special situation of the unit ball of certain Hardy spaces. Of
course we refer to the literature for the majority of the proofs. But we have included some, and sketched others, hoping that this will clarify the exposition. After introducing extreme, exposed and
strongly exposed points, we end the introduction with definitions of the Hardy-spaces in which we study our problem. These are: the standard Hl(]I))) of the disc ]I)) in e, HI(0.) of a domain of
finite connectivity 0. c e, the Bergman space Al(]I))) of the disc and Hl(lB n ) of the unit ball in en. In Section 2 we will study the boundary points of the unit ball of HI (]I))). Here good
function theoretic descriptions of extreme and strongly exposed points are possible. Exposed points seem to escape such a description. The ball of HI (0.) is the topic of Section 3. We will see that
results for the disc in general do not extend to this case. In Section 4 we will turn to the Bergman space. Now all boundary points of the unit ball are exposed and many well-behaved functions, like
polynomials, are strongly exposed. However, there are boundary points that are not strongly exposed. The final section is devoted to the little we know about the higher dimensional situation. The
paper borrows heavily from the expository parts of the PhD-thesis of the first author, [3]. 1.1. Generalities. Let X be a Banach space with (only in this section) real dual space X* and let A be a
bounded closed subset of X. We recall some notions that are connected with convexity properties at a boundary point of A. Research of the first author was supported by the Dutch research organization
© 59
2003 American Mathematical Society
DEFINITION 1.1. A point x E A is called extreme if x = tp + (1 - t)q for some p, q E A and 0 < t < 1 implies that p = q = x. A point x E A is called exposed if there exists L E X* such that Lx = 1
while Ly < 1 for every yEA \ {x}. The functional L is called an exposing functional for x.
For L E X* we set
M(L, A) = sup{Lx : x E A}
(1.1) and (1.2) If IILII
1\1(A) = sup{llxll : x
= 1 and a > 0 we
define a slice of A as
S(L,a,A) = {x E A: Lx> 1\1(L, A) - a}.
Let B be the closed unit ball of X. DEFINITION 1.2. A subset A of X is called dentable if for every c > 0 there is a point x E A such that x is not in the norm-closed convex hull of A \ (x + cB).
Equivalently, A is dent able if for every c > 0 there is a slice S(L, a, A) of diameter less than c. DEFINITION 1.3. A point x E A c X is called a denting point if for every c > 0 there exists a
slice S(L, a, A) of diameter less than c that contains x. A point :1: E A c X is called a strongly exposed point if there exists L E X* such that for every c > 0 there exists a > 0 such that the
slice S(L, a, A) contains x and has diameter less than c. In other words, there exists L E X* such that Lx = 1\1(L, A) and if LX n ---+ 1\1(L, A) for {x n } C A, then Xn ---+ x in the norm of X.
With these definitions at hand we make some observations. A strongly exposed point in A is exposed. Normalizing the functional L in the previous definition will yield an exposing functional for x. An
exposed point of A is extreme and an extreme point of A belongs to the boundary of A. A theorem of Phelps, [38], states that if every bounded subset of X is dentable, then every bounded closed convex
set is the convex hull of its strongly exposed points. It is known that separable dual spaces have the property that every bounded set is dentable, cf. [10], Chapter 6. However, neither Phelps'
theorem nor its proof gives us any information about, say, which of the exposed points of a convex set C are strongly exposed points. In the rest of the paper it will be more convenient to use an
equivalent, more analytic definition of strongly exposed point: An exposed point x of the set A C X with exposing functional L is strongly exposed if and only if every sequence (xn)in A with the
property that LX n ---+ 1 converges strongly to x. Moreover, we will work over C, so that the definition of exposed point becomes slightly adapted. A point x E A is an exposed point of A, if there
exists a (complex) functional L on X with the following property: Lx = 1 and for all yEA, Y =1= x ~ Re L(y) < 1. (For A E C, Re A denotes the real part of A.) Again we say that the functional L is an
exposing fUIlctional for x, or simply that L exposes the point x.
1.2. Hardy spaces. Next we briefly recall the definition and some properties of Hardy spaces. Excellent introductions are [12, 16, 24], and for domains of finite connectivity [13]. We will denote the
space of holomorphic functions on a domain D c en by H(D). Let !Dl be the unit disc in e and 'JI' its boundary. DEFINITION 1.4. Let 0 < p < 00. The Hardy space HP = HP(!Dl) consists of all
holomorphic functions I on !Dl for which 11111it-v:= sup
dO II(re i9 W - <
The Hardy space HOC(!Dl) consists of the bounded functions in H(!Dl). It is a closed subspace of LOC(!Dl) under the sup-norm 11.lIoc. If IE HP(!Dl), then
(1.4) exists for almost allO. The function 1* belongs to U('JI'). For alII::; p::; 00, II.IIHP is a norm, IIIIIHP = 111*IILp and HP(!Dl) becomes a Banach space that is a subspace of U('JI'). We will
simply write 1IIIIp instead of 11111Hv. Alternatively, the Hardy space HP(!Dl) (1::; P < 00) may be described by
HP(!Dl) = {f E H(!Dl) : IIIP admits a harmonic majorant} = closure of holomorphic polynomials in £P ('JI'). The value at 0 of the least harmonic majorant of IIIP,
uf(z) = inf{h(z) : h harmonic and h> IIIP} equals the norm, 1IIIIp = u(O?/p. Equivalent norms are given by (uf(z)?/P, (z E
!Dl) . These descriptions are useful for defining HP-type spaces on a domain n in
= {f E H(n) : IIIP admits a harmonic majorant}.
Norms, all equivalent, can be defined via the least harmonic majorant uf by (Uf(Z))l/P, (z En). If n has smooth boundary an then HP(n) is again a closed subspace of U(an). Similarly on the unit ball
~n of (n ~ 2):
= closure of holomorphic polynomials in
We end this section by defining the Bergman space AP(!Dl). Let 1 ::; p < IE H(!Dl) is such that
1111I~p:= i'D[11(zW dxdy < 00, W then we say that I belongs to the Bergman space AP(!Dl). In other words, AP(!Dl) = H(!Dl) n U(!Dl). It is a closed subspace of U(!Dl). One motivation for looking at
the Bergman space Al(!Dl) is that
Al(!Dl) ~ {f E Hl(~2): I(Zl,Z2) = I(Zl'O)}. Therefore B(Al(!Dl)) may serve as a testing ground for B(H1(!Dl)).
2. The boundary of unit ball in H1(]]))) 2.1. Extreme points. The subject started with a famous theorem of deLeeuw and Rudin, [26], that characterizes the extreme points of B(H 1 (]])))). It will be
presented below. Recall the inner-outer decomposition for f in H1(]]))), cf. [16]:
f(z) = I(z)U(z). Here I E H(]]))) is an inner function, that is, I E HOC(]]))) with 1I*(ei9 )1 = 1 a.e. and U E H(]]))) is an outer function, i.e. log IUI(z) = P[log IUI](z), where P is the Poisson
Pu z = [ ]()
1 - Izl2 1 - 2Re (ze- i9 )
+ Izl2 u (e
d() -. ) 27r
The inner factor thus contains all the zeros of f. In fact, every inner factor is of the form
I(z) = B(z)S(z),
with B a Blaschke product and S is the singular factor (2.2) where J.L is a positive, singular measure on 'JI' and - denotes harmonic conjugation. EXAMPLES 2.1. Outer functions are zero free, and
e.g. a holomorphic function on]])) that is continuous and away from 0 on iij is outer. Taking a root and applying a limit argument shows then that f E HI (]]))) that assumes values only in C \
(-00,0) is outer too. EXAMPLE 2.2. If I is an inner function then (1
+ 1)2
is outer.
DeLeeuw and Rudin proved THEOREM 2.3 ([26]). A function f is an extreme point of the unit ball of HI (]]))) if and only if f is an outer function and IIfl11 = 1. PROOF. Suppose f is an outer function
of unit norm and suppose g in H1 is such that IIf + gill = Ilf - gill = 1. Let dJ.L be the probability measure If I on the unit circle. Then, with k = gl f on 'JI', the relations Ilf + gill = Ilf -
gill = 1 imply that J;11" 11 + kl + 11 - kldJ.L = 2. Because for all z E C: 11 + zl + 11 - zl 2 2, with equality if and only if z E [-1,1], we conclude that for almost dJ.L-every ~ E 'JI' : g(~)1 f(O
E [-1,1]. Observe that this inclusion also holds for almost d()-every ~ E 'JI', because ~ = If I i:- 0 almost d()-everywhere on 'JI'. In particular, Igl :::; If I d()- a.e. on 'JI'. By the fact that
f is outer, then also for all zED: Ig(z)1 :::; If(z)l. Hence gI f E HOC has real boundary values on 'JI'. The Poisson integral representation of glf yields that glf is constant. Finally, because Ilf
+ gill = 1, this constant is zero, i.e., g == 0, so f is extreme. In the other direction, suppose f = I· F is of unit norm, but I is a non-trivial inner function. Because for every ei9 E 'JI' \ {±1}:
±(1 ± ei9 )2 e
---'-----:-.,,::--' --- = 2 ± 2Re (e t9 ) = 2 ± 2 cos( ()) > 0, t17
the functions I and (1 ± I)2 have the same argument a.e. on'lI'. Let 9 = We have III ± gill =
l±t .F.
h[2'" 1F1(1 ± Re (I)) 2w = 1± Re (h[2'" IFI' I ~ ).
Now if we replace I by AI throughout the preceding, where A E 'lI' is arbitrary, then we obtain: 2", dB II/±glll=I±Re(A IFI·I-). o 2w
Therefore, we can choose A in such a way that Re (A J:'" IFI . I quently, III ± gill = 1, but 9 "¥= 0, so I is not extreme.
EXAMPLE 2.4. Normalized polynomials without zeros in j[)) are extreme points. 2.2. Exposed points in B(Hl(j[)))), function theoretic methods. As far as the authors know, there is no clearcut function
theoretic characterization of exposed points of B«Hl(j[)))). The necessary and sufficient condition of Helson, [21] given below in Theorem 2.8 probably comes closest. The following lemma identifies
the exposing functionals. LEMMA 2.5.
II I
is exposed in B(Hl(j[)))), then Lf:
J7 9
dB UT2w
is the unique exposing functional. PROOF. Suppose I is an exposed point of B(HI(j[)))) with exposing functional L. By the Hahn-Banach theorem there exists a function cp E V"', Ilcplloo=l, which
represents the action of L: [21< dB L(g) = gcp 2w'
Because L(f) = 11/111 = 1, and because the function I has mass (almost) everywhere on 'lI', there is only one cp that has the desired properties, namely cp = 711/1 = exp( -i arg(f)). In particular we
see that the exposing functional for I is unique and of the above form. D DEFINITION 2.6. A function I E HI (j[))) is called rigid if 9 E HI (j[))) and arg 9 = arg I implies that 9 = cl for some c >
O. An HI-function is rigid if its argument on 'lI' determines the function up to a multiplicative constant. By the previous lemma exposed points are precisely the rigid functions of norm 1. Not all
outer functions are rigid as the following example demonstrates. EXAMPLE 2.7. By the DeLeeuw-Rudin theorem (Theorem 2.3), no HI-function with non-trivial inner factor is rigid. Inspection of the
proof of the theorem explicitly gives another HI-function with the same argument: if I = I· F with F outer, then 9 = (1 + 1)2. F is an outer function, cf. Example 2.2 that has the same argument as f.
This argument can be reversed: suppose I "¥= 0 is divisible in HI by the outer function (1 + I)2, where I is any non-trivial inner function, then I
and I I / (1 + I) 2 have the same argument, so non-trivial inner functions I we have (2.3)
If I is rigid, then 1/(1
is not rigid. In other words, for all
+ 1)2 rt HI
Nakazi [28, 29] conjectured that an outer function with fI(l + cz)2 rt HI for every c E 'JI' must be rigid. E. Hayashi [17] gave a counter example. Later Sarason [44, 45] conjectured that (2.3)
characterizes rigid functions. Inoue [22] subsequently gave an example of a non-rigid function I E HI with the property that 1/(1 + 1)2 rt HI for all non-trivial inner functions I. Helson showed that
a variant of (2.3) indeed characterizes rigid functions; THEOREM 2.8 ([21]). Let I be an outer lunction in HI. Then I is rigid il and only il lor all inner lunctions p and q, not both constants, I/(p
+ q)2 rt HI. The drawback of Helson's theorem is that his condition is difficult to check in practice. More user friendly are the following results of Yabuta. THEOREM 2.9. (1) II I E HI is invertible
in HI, then I is rigid. [51] (2) Suppose ther'e exists h E Hoo \ {O} such that Re(h!) 2: 0 a.e. on 'JI'. Then I is rigid. [52] The proof uses a result of Neuwirth and Newman [36], that is interesting
in its own right. LEMMA 2.10 ([20],[36]). Every lunction I E H 1 / 2 that is positive a.e. on'JI' is constant. Nakazi [29] gives another proof of Yabuta's results. EXAMPLE 2.11 ([23]). Normalized
polynomials P without zeros on II)) and at most simple zeros on 'JI' are exposed. Indeed, the argument of such a function makes jumps of height 7r at the zeros in 'JI', hence by adding a smooth
+iv> yields a function with positive real part and the second of Yabuta's criteria applies. 2.3. Strongly exposed points. Whereas it seems impossible to give a characterization of exposed points that
allows to check if a given function in aB(HI (II)))) is an exposed point, it is possible to characterize the strongly exposed points. This is the content of work of Nakazi and, independently, the
first author. Let us describe strongly exposed points in terms of properties of the exposing functional. THEOREM 2.12 ([2,4]). II alunction IE 8B(H 1 (1I)))) is strongly exposed, then d( Tfr, HOO) <
1. (d = Loo-distance). PROOF. Suppose that the LOO-distance of
The space HOO + G('JI') that appears in the next theorem, has been studied extensively, see [42] and cf. [16]. It is a closed subalgebra of LOO('JI').
2.13 ([47]). Let IE B(Hl(]])))) be exposed. Ild(
&!. Hoo+C(1I')) < 1,
then I is strongly exposed. We will explain how the proof goes if 'P := 111/1 E Hoo + C(1I'). The exposing 7r functional L for I is defined by L(g) = g'Pg!. Suppose that (fn)f is a sequence in the
unit ball of Hl such that L(fn) --) 1 as n --) 00. Clearly this implies that II In I 1 --> 1. We claim that there is a subsequence (fnk) that converges to I in HI. By weak* compactness of the unit
ball of (C(1I'))* the sequence of measures Ing! has a weak* convergent subsequence Ink g! with limit dJ.L({}) , and IIJ.LII :::; 1. For all r 27r em . (J n = 1,2,3, ... , we have Jo dJ.L({}) = O. Now
the F. & M. Riesz theorem [16] precisely states that J.L is of the form dJ.L( (}) = F g! for some F in the unit ball of HI . Write 'P = 'Pl + 'P2 with 'Pl E H[J = zHoo and 'P2 E C. Because H[J
annihilates HI, the weak* convergence of Ink g! implies that
We conclude that F = I because I is exposed. Again, by weak* convergence Ink converges pointwise to I. The following result of D.J. Newman ensures that the convergence is in Hl with limit I. The
argument shows that every subsequence of (fn) has a subsubsequence that converges in HI to I, and therefore In --) I in HI. 2.14 ([35]). Let (fn)i be a sequence 01 functions in the unit ball Suppose
In converyes pointwise to I E Hl on ]])) and I In I 1 --> 11/111 as Then In converges to I in HI-norm.
01 Hl. n
--> 00.
The method for the general case in [47] is to work in the maximal ideal space of LOO(1I') and use the Helson-Lowdenschlager generalization of the F. & M. Riesz theorem. Cf. [30] for a related
approach. 2.4. Strong exposedness and Helson-Szego weights. In this section we will explicitly describe the strongly exposed points in the unit ball of HI: they are the outer functions induced by
so-called Helson-Szego weights on the unit circle. We will discuss three different (albeit related) roads to this result. The first one quickly establishes that strongly exposed points are induced by
Helson-Szego weights and uses little background on exposed points and Hardy space theory. The second one uses function theory to show that Helson-Szego weights give rise to strongly exposed points
and is essentially due to T. Nakazi [31]. The third approach (to be discussed in Section 2.5) uses operator theory and the relation between exposed points and Toeplitz operators [2] to prove that
strongly exposed points and HelsonSzego weights are "the same". These last two proofs use the characterization of strong exposedness obtained in the previous section (Theorem 2.13). To put the
results in perspective, we will mention some surprising properties of strongly exposed points. Let J.L be a finite (positive) measure on 1I'. Let P be the collection of polynomials in z and let Q be
the space of polynomials in z vanishing at the origin. On the unit
circle P and Q consist of the trigonometric polynomials of the form Ln>o ane inlJ and Ln
p:= sup I iPqd/-Ll, where the supremum is taken over all pEP and q E Q restricted by Ilpll£2(/J) ~ 1 and Ilqll£2(/J) ~ 1. By Cauchy-Schwarz, 0 ~ P ~ 1. We see that the spaces P and Q are orthogonal
if and only if P = O. On the other hand, when the L 2 (/-L)-closures P of P and Q of Q have a non-trivial intersection, then clearly p = 1. The size of the number p is related to the question when
the sum P + Q is closed in L 2 (/-L). Namely, when p < 1, then for all p E P, q E Q in L 2 (/-L)-norm, lip + ql12 ~ IIpl12
+ IIqll2 -
2p· Ilpll· Ilqll ~ (1- p)(llpl12
+ IlqIl2),
which implies that P + Q is closed in When p < 1 we say that P and Q are at the positive angle cos-I(p) > O. When the spaces P and Q are at positive angle, the projection
L 2 (/-L).
P+(Lane inlJ ) = LaneinlJ , -N
L 2 (/-L),
which is densely defined on extends to a bounded operator on L 2 (/-L). Conversely, if the intersection of P and Q in L 2 (/-L) is trivial, then the definition of P+ acting on trigonometric
polynomials is well-defined and extends to a bounded operator on L 2 (/-L) if and only if P and Q are at positive angle. DEFINITION 2.15. We say that a function w ~ 0 on 'll' is a Helson-Szego weight
(on 'll') ifthere exist real valued u,v E Loo('ll') with Ilvll oo < I such that w = eU+V. (Here v is the boundary function of the harmonic conjugate of the Poisson integral of v to JIll.) The
following theorem of H. Helson & G. Szego elegantly describes all measures /-L for which P and Q are at positive angle. THEOREM 2.16 ([19]). The subspaces P and Q are at positive angle in L 2 (/-L)
if and only if the measure /-L is of the form d/-L = wdO, for some Helson-Szego weight won'll'. COROLLARY 2.17. If the function f is strongly exposed in the unit ball of HI, then If I is a
Helson-Szego weight on'll'. PROOF. Assume f is an exposed point, such that If I is not a Helson-Szego weight. We will show that f is not strongly exposed. Let /-L be the probability By the theorem of
Helson and Szego the spaces P and Q are at measure If I zero angle (p = 1). Thus we can find sequences (Pn) and (qn) in the L 2(/-L)-unit balls of P and Q respectively, such that
Pnqn d/-L
as n --+ 00. For n = 1,2, ... , let fn be the HI-function Pnqnf. These functions are contained in the unit ball of HI:
THE BOUNDARY OF THE UNIT BALL IN Hl_TYPE SPACES
If we set
L(g) =
dO g
Now (2.4) expresses that L(fn) ~ 1 as n ~ 00, yet the functions fn do not converge to f in HI because fn(O) = qn(O) = O. We conclude that f is not strongly exposed. D Already we can mention a
surprising property of strongly exposed points. COROLLARY 2.18 ([2]). If the function f is strongly exposed in the unit ball of HI, then for all small c > 0, the functions f and 1/ f are contained in
H1+ e . The corollary together with Theorem 2.13 show that if f E B(HI(][})) satisfies the distance condition in Theorem 2.13, then f is strongly exposed if and only if 1/ f E HI. (It is worth noting
that there exist exposed points f E HI such that 1/ f is in no HP, p > 0, [37, 44].) Since f and l/f are outer functions, the corollary is immediate from the following lemma (which is actually an
exercise in [16]). LEMMA 2.19. If the function w is a Helson-Szego weight on 'JI', then for all small c > 0, we have w E £1+ e ('JI') and l/w E L1+e('JI'). LEMMA 2.20. (Cf. [16], Lemma IV. 3. 3, p.
148.) If'lf; is a measurable real function, then the LOO-distance of e-i.p to Hoo is less than 1 if and only if there exist c > 0 and h E H oo such that 7r (2.5) Ihl 2:: c and I'lf; + arghl $ "2 - c
(mod 27r)
almost everywhere on 'JI'. THEOREM 2.21 ([2, 31]). Let f be an extreme point of the unit ball of HI. Then f is strongly exposed if and only if If I is a Helson-Szego weight. PROOF. ([31]) By
Corollary 2.17 we need only to prove the backward implication. Let us assume that the outer function f is such that If I is a Helson-Szego weight on 'JI', say, If I = exp(u + v), where Ilvll oo < l
We remark that the function is exposed by Theorem 1 and Lemma 2.19. Next, again using the fact that f is outer, we notice that
f(z) = eu(z)+iu(z) . ev(z)-iv(z),
because the right hand side is an outer function with the appropriate absolute values on 'JI'. We set
O. Recall Corollary 2.18: if f is strongly exposed then for all small c > 0, the functions p+e and 1/ f1+ e are contained in HI. This result can be sharpened somewhat using Theorem 2.21:
PROPOSITION 2.23 ([2]). Let f be a strongly exposed point in the unit ball of HI. Then for all sufficiently small c > 0, the (normalized) functions ce f1+E: and del f1+E: of unit norm are again
strongly exposed in the unit ball of HI. The Proposition explains the following examples from [47]. EXAMPLES 2.24. A polynomial is rigid in HI if and only if its zeros on the unit circle are single
zeros (and it is zero-free on llJ), obviously). Any normalized polynomial P with at least one single zero on '][' is not strongly exposed however, because 1/ P ¢ HI Let f#(z) be the extreme point c(1
+ z) log2(1 + z) in the unit ball of HI. Because f# is outer and because 1/lf#1 E LI, we conclude that f# is exposed (Theorem 2.9. 1). However, 1/lf#1 ¢ L1+E:, so f# is not strongly exposed.
2.5. Toeplitz operators and De Branges-Rovnyak spaces. Let P+ be the orthogonal projection of L2('][') onto H2: 00 00 P+(Lane inll ) = Lane inll , -00 0 and let P_ be the orthogonal projection of L2
onto (H2).l. 00 -1 P_(Lane inll ) = Laneinll . -00 -00
DEFINITION 2.25. Given a bounded function '¢ E LOC the Toeplitz operator T", is the bounded map T", : H2 _ H2 given by T",(f) = P+('¢f).
We say that '¢ is the defining function of the Toeplitz operator T",. We see that the norm of the Toeplitz operator T", is at most 1I,¢1I00. It is not difficult to show that the norm of T", is in
fact equal to 1I'¢1I00, but we will not need this result. Also, it is a routine exercise to verify that the adjoint of T", is the Toeplitz operator Ttji. Clearly, if'¢ E Hoo, then T",(f) = '¢f.
Combining these two observations we have the following result: LEMMA 2.26. If cp or'¢ is contained in H oo , then T-qiT", = TVi",. Given a function '¢ E Loo, the Hankel operator H", (with defining
function '¢) is the bounded operator H",(f)
= '¢f - T",(f) = (I - P+)(,¢f) = P_('¢f)
from H2 into (H2).l.. By the same reasoning the norm of H", is at most 1I,¢1I00. If two functions cp and '¢ in Loo differ by an element of H OO , then the associated Hankel operators coincide. Hence
the operator norm of H", is at most Loo-dist('¢,HOO). The basic fact about Hankel operators, due to Z. Nehari, is that equality holds: THEOREM 2.27 ([34]). The operator norm of H", equals the Loo
-distance of '¢ to HOO.
The following result (and its corollaries) will be of great importance. We only have as reference Sarason's notes [43]. THEOREM 2.28. (Devinatz-Rabindranathan) If'IjJ is unimodular, then T1fJ is
left-invertible if and only if Loo-dist('IjJ,Hoo) < 1.
A bounded linear operator L : X -+ Y is said to be left-Fredholm if the range of X under L is closed and of finite codimension in Y. For left-Fredholm operators one has the following result.
COROLLARY 2.29 ([11]). (Douglas-Sarason) If'IjJ is unimodular, then T1fJ is left-Fredholm if and only if the Loo -distance of'IjJ to HOO + C is less than 1. See [43], p. 119 ff. also for a proof of
the next result COROLLARY 2.30 ([9, 49]). (Devinatz-Widom) If'IjJ is unimodular, then T1fJ is invertible if and only if'IjJ can be written as ei(uH), where u and v are real functions in Loo such that
IIvll oc < 'IT". We will now explain the relation between Toeplitz operators and rigidity (exposedness) of functions in HI. LEMMA 2.31. Let f be an outer function in HI. Then f is rigid if and only
if the Toeplitz operator with defining function
{af, ag)M := (f, g)2.
This makes M(a) a Hilbert space, and the Toeplitz operator Ta : H2 -+ M(a) unitary. Clearly, when lIall oo ::; 1, the inclusion i : M(a) -+ H2 is a contraction.
Let b be a non-constant function in the unit ball of Hoo. We will construct the space 1t(b), the so-called complementary space to M(b), in a similar fashion; the narile reflects the fact that M(b) +
1t(b) = H2 (although the intersection M(a) n 1t(b) is trivial only when b is an inner function). Observe that I - TbTj) is a positive contraction on H2, hence the operator (I - nTj»)l/2 is
well-defined on H2 and contractive. As a linear space 1t(b) consists of all H 2-functions in the range of the operator (I - TbTj»)l/2 on H2. We also use this map to give 1t(b) a Hilbert space
structure. Namely, if lor 9 is orthogonal in H2 to ker(I -TbTj»)l/2 = ker(I -TbTj»), we set
((I - TbTj»)l/2 I, (I - TbTj»)l/2g)b:= (/,g)2. As a consequence (I -TbTj)?/2 is a co-isometry from H2 onto 1t(b) and the inclusion map i : 1t(b) --+ H2 is another contraction. Moreover, we see that
if Ilbll oo < 1, then 1t(b) is all of H2 with an equivalent norm. Given an outer function I E Hl (not a constant) Sarason constructs three auxiliary holomorphic functions: 27T
(2.7) (2.8)
G(z) = {
ifJ e, + z I/(eifJ)1 dO, e,fJ - z 211'
( ) _ 2v'f(Z)
a z - G(z)
+ l'
G(z) - 1 b( z) = G ( z) + 1 '
where we may take any branch of -IT Observe that we can recover the function I from a and b: 1= F2, with F = a/(1 - b) E H2. The function G is reasonably well-behaved on JI): because ReG(z) ~ I/(z)1
~ 0, G is contained in HP for all o < p < 1. The real part of G is positive and majorizes hence a and bare contained in the unit ball of Hoo. The fact that Re G = III a.e. on 'II' gives us that lal 2
+ IW = 1 a.e. on 'II'. By a theorem in [26] the functions a and b are not extreme in the unit ball of Hoo. It is known in such cases that the polynomials are contained and dense in 1t(b) (cf. [46],
We shall also encounter Toeplitz operators on H2 with unbounded defining functions. For a function 't/J E L2, the Toeplitz operator T,p (with defining function 't/J) acts as follows. Take a function
9 E H2, then 't/Jg is an L l ('II')-function with Fourier series E~oo cn['t/Jg]einfJ and we set 00
T,p(g) =
L en ['t/Jg]zn
For bounded 't/J this coincides with the Toeplitz operators as defined earlier. For unbounded 't/J, the map T,p is a densely defined operator from H2 to H2 and is continuous from H2 into the space of
holomorphic functions on JI) with the topology of uniform convergence on compact subsets. In this light it is perhaps surprising that the product of the operators Tl-b and Tp (the former is ordinary
multiplication by 1 - b) is bounded on H2. This is a consequence of the following theorem of Sarason. The proof uses that G = ~ is the Herglotz integral of an absolutely continuous measure (cf.
(2.6)); it may be found in [46], IV-13, p. 30.
THEOREM 2.32. The operator T1-bTF is unitary from H2 onto 1i(b). Next one can show that TI-bTFTF/F = TI-bTFF/F = TI-bTF = Ta
(compare with Lemma 2.26). Therefore T1-bTF maps the range ofTF/F in H2 onto M(a) C 1i(b) (Theorem 2.32). Consequently, M(a) is dense in 1i(b) if and only if the range ofTF/F on H2 is dense in H2, or
equivalently ifTF/F = T;/F is injective on H2. Combining this with Lemma 2.31 we find Sarason's result:
THEOREM 2.33 ([44]). Let I be an extreme point is exposed il and only il M(a) is dense in 1i(b).
01 the
unit ball in HI. Then
This fundamental result has several striking consequences, one of them being that if I is exposed, then so are the squares of the H 2-functions FA = a/(1 - )"b) ().. E 1l'), cf. [44, 46] As a next
step one may even replace).. by any inner function u. Sarason proves that the square of Fu = a/(I- ub) is exposed in HI. We now formulate a variant on Theorem 2.33 that deals with strongly exposed
THEOREM 2.34 ([2]). Let I be an extreme point is strongly exposed il and only il M(a) = 1i(b).
01 the
unit ball
01 HI.
PROOF. First observe that for any FE H2 \ {O}, the Toeplitz operator TF/F is injective. Sarason's reasoning preceding Theorem 2.33 shows that (with 1= F2 as before) (2.9)
M(a) = 1i(b) if and only if TF/F maps H2 onto H2.
Hence, (2.10)
M(a) = 1i(b)
TF/F is invertible
TF/F is invertible.
Suppose I is strongly exposed. By Theorem 2.13, the LOO-distance of F / F to H OO + C is less than 1. The Douglas-Sarason theorem (Corollary 2.29) implies that the operator TF/F has closed range in
H2. As we have seen its adjoint TF/F is injective, hence the range of TF/F is dense in H2 too. Thus TF/F : H2 -+ H2 is surjective. The operator is also injective by the rigidity of I = F2 (Lemma
2.31). By (2.10), M(a) = 1i(b). To establish the other half of the theorem suppose M(a) = 1i(b). We observe that I is exposed by Theorem 2.33. By (2.10), the operator TF/F is invertible, in
particular, it is left-invertible. Theorem 2.28 implies that the LOO-distance of 711/1 to HOC is less than 1. We conclude that I is strongly exposed by Theorem 2.13. 0 There is an alternative proof
of the implication
"I strongly exposed =* 1/1 E
HI" that lies on the surface: if I is strongly exposed, then 1 E 1i(b) = M(a), because 1i(b) contains all polynomials. Thus l/a E H2 and 1/1= (l-b)2/ a2 E HI. lt should come as no surprise that l/a
is actually in HP for slightly larger p > 2. Indeed, equality of the spaces M (a) and 1i(b) occurs if and only if the operator Ta/a
is invertible and the non-extreme pair (a, b) satisfies the so-called Corona condition: inf
la(zW + Ib(zW > 0,
see [44] and [46], IX-5, p. 66. We see in particular that for strongly exposed f, a2 Illall~ is also strongly exposed (cf. (2.10)), and that lla E H2+£, 1/(I-b) E Hi+£ for sufficiently small E > O.
Let us go back to the distance condition used in Theorem 2.13 and assume f is an extreme point such that the LOO-distance of 7/1fl to HOO + C is less than 1. Starting with the Douglas-Sarason theorem
(Corollary 2.29) and exploiting the full strength of the statement that TF/F is (left-)Fredholm it follows that the operator TF/F has closed range in H2 of finite codimension, say, N. Hence M(a) is
closed and of finite co dimension N in 1i(b). Using [20], Theorem 6 or [46], X-18, p. 77, we conclude that f is of the form f = p2g, where 9 is strongly exposed in the unit ball of Hi and p is a
polynomial of degree N that has all its zeros on 11'. Conversely, if f is of the above form, then
distallfl, H oo
+ C) = dist(z2N . gllgl, H oo + C) = dist(gllgl, H oo + C) < 1, by the algebra structure of Hoo + C, but dist(7/lfl,H OO ) = 1 when N > O. This calculation serves to illustrate that if for a given
extreme point f the distance of 7Ilfl to H oo + C is less than 1, the ("only") thing that can prevent f from being exposed (and hence strongly exposed) is the divisibility of f in Hi by functions of
the form (1 - U)2 with u(z) = >.z (>. E 11') a particularly simple inner function. Functions in Hi that lack this divisibility property are called strong outer functions. We see that for any strong
outer function that is not exposed, like Inoue's example, [22], it is true that the distance of 7/1fl to H OO +C is 1. Alternatively, the strongly exposed points are the (normalized) strong outer
functions f that satisfy distallfl, H oo + C) < 1.
3. The boundary of B(Hi(O)) Having studied the sets of exposed and strongly exposed points in the unit ball of the classical Hardy space Hi of the unit disc, we will investigate how these results
hold up for the Hardy space Hi of a domain of finite connectivity ("finite domain") 0 c C. Such domains are e.g. conformally equivalent to domains with a smooth boundary consisting of finitely many
components or to a disc from which a finite number of disjoint slits are deleted, cf. [33]. We defined Hi (0) in Section 1.2. There are two important differences with the classical Hardy space that
make the analysis very different. On domains of finite connectivity, Hi-functions may not allow a classical factorization using Blaschke products, (singular) inner functions and outer functions.
Indeed, the argument principle prevents the existence of a holomorphic function f on the annulus A = {I < Izl < 2}, such that If I = 1 a.e. on 8A and f has exactly one zero on A. Secondly, and in a
way related, there now exist extreme points in the unit ball with (finitely many) zeros (Section 3.1). While such zero sets are somewhat generic for extreme points, their location plays a
surprisingly crucial role in its being a (strongly) exposed point (Section 3.2). As in Section 2.1 we call a function I E HOO(O) an inner function if 11*1 = 1 almost everywhere with respect to arc
length (dO') on 80. Consequently, II(z)1 ::; 1 for all z E O. Let w(·, z) denote harmonic measure on 80 at z. We call a function F E Hi(O) an outer function if for every z in 0 (equivalently, for at
least one
THE BOUNDARY OF THE UNIT BALL IN Hl_TYPE SPACES
zEn): log IF(z)1 = (
log IF*(~)I dw(~, z).
An outer function is zero free on n. Green's function G(z; zo) and the Blaschke factor Bzo(z) = t:z:oz for are related by G(z; zo) = log IBzo(z)l. This suggests the following definition.
DEFINITION 3.1. Let G(z; zo) be Green's function on a finite domain n with pole at Zo and let G(z; zo) denote its (multiple valued) harmonic conjugate. Suppose that Zl, Z2, . .. satisfy the Blaschke
condition En G(z; Zn) < 00. Then the Blaschke product with zeros at Zl, Z2, ... is the multiple valued function
B(z) = exp( - L G(z; zn)"- iL G(z; Zn». n
DEFINITION 3.2. A multiple valued function F on n is modulus automorphic if • IF(z)1 is well-defined; • locally on n, IFI coincides with the absolute value of a holomorphic function. If in addition,
IF(z)IP admits a harmonic majorant on n (0 < p < oo), then we say that F E MHP(n}. If IFI is bounded, we say that F E MHOO(n}. I E MHOO(n} is called innerif 11*1 = 1 a.e on r. I is singular inner if
I is in addition zero-free. Blaschke products are modulus holomorphic inner functions. One can show that if FE MHP(n}, then IFI has non-tangential limits (denoted IF*!) a.e. on an, IF*I E p(an,da},
and unless F == 0, 10glF*1 E LI(an,da) and (3.2)
log IF*(z)l:::; (
log IF*(~}ldw(~, z),
for all zEn.
If equality holds in (3.2) at all points zEn (equivalently, for at least one zEn), we call F an outer function in MHI(n). While a single valued inner-outer decomposition is impossible in HP(n), a
decomposition in modulus automorphic inner and outer factors is possible.
THEOREM 3.3 ([48]). Let f E HP(n) be not identically zero. Then there exist a Blaschke product BE MHOO(n), a singular inner function 8 E MHOO(n}and an outer function F E MHP (n) such that for all zEn
If(z}1 = IB(z)I·18(z}I·IF(z}l· This factorization is unique in the following sense: if B 1 , 8 1 and FI are a Blaschke product, a singular inner function and an outer function on n for which If(z)1 =
IB1 (z}I·181 (z}I·IFI(z)l, then IBI = IBII, 181 = 181 1 and IFII = IFI· 3.1. Extreme points in HI(n}. The question arises which functions are extreme in the unit ball of Hl(n). After the
deLeeuw-Rudin theorem 2.3, the following result is elementary: LEMMA 3.4. Iff E HI(n} is an outer function of unit norm, then f is extreme in the unit ball of HI(n}. Any attempt to copy the proof of
the deLeeuw-Rudin theorem in the other direction will break down. For suppose f = I· F where I is a non-trivial inner function (in MHOO(n)}. Then deLeeuw and Rudin look at the function 9 = (1+12)F
and show that IIf ± gill = 1. However, unless I is a single valued inner function, 1 + 12 is not a well-defined modulus automorphic function. There is no remedy for this problem, because as we have
already mentioned, when m ~ 2 there exist extreme points with a non-trivial inner part. The following theorem of F. Forelli is crucial in understanding which inner functions can appear in the
inner-outer factorizations of an extreme points. THEOREM 3.5 ([14]). Let f be an extreme point of the unit ball of H1(n). Then the codimension of the H 1-closure of f . Hoo in H1 is at most T' when n
is bounded by m + 1 closed smooth curves. All functions in the H 1-closure of f . HOO(n) inherit the zeros of f, hence Forelli's theorem implies that an extreme point can only have a limited number
of zeros. In fact one has: COROLLARY 3.6 ([14]). If f is an extreme point of the unit ball of H1 (n), then zeros. the inner part of f is a finite Blaschke product with at most
Inspection of the proof of the deLeeuw-Rudin theorem gives us the following criterion for extremity (where we identify H 1-functions with their boundary values): LEMMA 3.7. Let f E H1(n) be of unit
norm. Then f is not extreme if and only if there exists a non-constant real function k E LOO(an) for which kf E H1(n). Suppose f = I . F is of unit norm where I is a finite Blaschke product, and F
outer. Let us suppose that f is not an extreme point of the unit ball of H1(n). Then let k be as in the lemma, and let 9 E H1(n) have boundary values kf. Because F is an outer function, for all zEn:
Ig(z)1 $ IIkll oo ·1F(z)l. Hence, because also III = 1 everywhere on an, the meromorphic function h = g/ f is bounded near an, real-valued (a.e.) on an, and has its poles in the zeros of f (with
corresponding mUltiplicities). Conversely, if h is a meromorphic function on n with these three properties, then with k := h on an and 9 := hf E H 1(n), we have 9 = kf on an, so by the previous
lemma, f is not an extreme point. We come to the following definition. DEFINITION 3.8. Let I be a finite Blaschke product. We say that I is an extremal Blaschke product if there exists no meromorphic
function h on n that is bounded near an, real-valued on an and has its poles in the zeros of I, with no greater multiplicity than the zeros of I. The conclusions of the previous paragraphs may thus
be summarized as follows: PROPOSITION 3.9. Let the norm of f E H1(n) be 1. Then f is an extreme point of the unit ball of H1(n) if and only if the inner part of f is an extremal Blaschke product. We
wish to stress that Forelli's theorem also gives us an upper bound for the number of zeros of an extremal Blaschke product on n. Also, by the previous proposition we see that it is only the location
of the zeros of a function in H1 (n) and not so much the outer factor that decides whether or not the function is extreme in the unit ball (after normalization). The problem of determining the
extreme points of H1 (n) has thus been reduced to a problem on meromorphic functions on n with pre-described poles, that is: a problem concerning meromorphic divisors on n.
THE BOUNDARY OF THE UNIT BALL IN Hl_TYPE SPACES
Let I be a finite Blaschke product with zeros Zl, Z2, ... , Zn repeated according to multiplicity. Thus I has n zeros on O. Let 8 := 1'ZI +1'Z2+' -+1,zn be the divisor on 0 associated with I. If 8' =
L:zEf! d'(z) . Z is another divisor on 0 we say that 8' ~ 8 if at every Z E 0: 8'(z) ~ 8(z). The space of all meromorphic differentials w on 0 that are real-valued on 00 and for which the associated
divisor (w) satisfies (w) ~ 8 is a real linear space of dimension MD(8). Using a theorem of H.L. Royden [39], based on the Riemann-Roch theorem, T.W. Gamelin & M. Voichick proved the following
result: THEOREM 3.10 ([15]). The Blaschke product I with zeros Zl, Z2, ... , Zn and associated divisor 8 is extremal if and only if MD(8) + 2n = m. In particular, using only the fact that the inner
factor of an extreme point is a finite Blaschke product (as shown by Forelli), Gamelin & Voichick also arrived at Forelli's upper bound (W-) for the number of zeros of an extremal Blaschke product.
They proved that this upper bound is also sharp. THEOREM 3.11 ([15]). The HI-closure of the set of extreme points in the unit ball of HI(O) is the collection of all functions in HI(O) that have unit
norm and no more than W- zeros. There is a special type of finite domains where Gamelin and Voichick described the zero sets of extreme points explicitly, namely the so called real slit domains.
These are usually defined as the extended complex plane with a finite number of intervals deleted. In our situation we prefer a (conformly equivalent) definition. DEFINITION 3.12. We will call any
domain n of the form JDl \ (It u ... U 1m), where It, 12', • .• ,Im are disjoint, bounded and closed intervals in (-1, 1) a real slit domain.
THEOREM 3.13 ([15]). Let be a real slit domain and let Zl, Z2, ... , Zn be points of (not necessarily distinct). Then the Blaschke product with zero set Zl, Z2, ... ,Zn is extremal if and only if:
• n::; W- and • for all i, j: Zi =I- Zj. (In particular, none of the Zi is real.)
We omit most of the proof, and only observe that because the meromorphic function Z~Zi + z~z; + l-~iz + l-kz is bounded and real-valued on an c JR., no zeros of an extremal Blaschke product are
3.2. Strong exposedness and the location of zeros. In this section we investigate exposed and strongly exposed points in the unit ball of HI(O). We give several examples and criteria for (strongly)
exposed points. Also we show that nontrivial properties of strongly exposed points in HI (JDl) (for example: Ll-invertibility on the boundary) have no analogue for finite domains. Finally, we again
look at the zero sets of extreme points and the question of divisibility of extreme functions by functions of the form (1 + U)2, where u is a non-constant inner function. The Hahn-Banach theorem
again gives that for f an exposed point of the unit ball of HI(O) the exposing functional L for f is unique and given by;
L : 9 E HI
fan 9 I~I da.
Hence, like H1(1I))), a function I in the boundary of the unit ball of H1(n) is exposed if and only if it is rigid: apart from (positive) constant multiples of I there is no H 1-function with the
same argument a.e. (dO") on an. The following criteria for rigidity of H 1 (1I)))-functions carryover to finite domains word for word: • If IE H1 and 1/1 E Hl, then I is rigid (Theorem 2.9. 1). • If
there is agE Hoo such that Re(fg) > 0 a.e. on an, then I is rigid (Theorem 2.9. 2). • If u is a non-constant inner function such that 1/(1 + u)2 is in Hl, then I is not rigid (or I == 0). A priori
the first two conditions can only be used to demonstrate rigidity of outer functions. In both cases III cannot be too small near the boundary of 0.: if I satisfies the second condition, then 1/1 E H1
-"'(an) for all E: > 0, so 1/1 is "nearly" in H1. The first condition can be modified to allow for exposed points with zeros on n. PROPOSITION
3.14 ([4]). II I is extreme in H1(n) and 1/1/1 E L1(an) then I
is exposed.
Similar to Theorem 2.13 is: THEOREM 3.15. Let I be a lunction in H1(n). Then I is strongly exposed in the unit ball 01 H1 il and only il I is exposed and L oo-dist(7 /1/1, Hoo+C(an)) < 1. Throughout
the remainder of this section the domain R will be a real slit domain with m slits that contains the origin. Note that on 11'\ {i} the function (z+i)2 has the same argument as iz so (z+i)2 is not
rigid in H1 (II))). LEMMA 3.16 ([4]). For all m ~ 2, the normalized lunction I(z) = c(z + i)2 is strongly exposed in the unit ball 01 H 1(R). The proof is an amusing exercise in elementary function
theory. One supposes that 9 E H1 has the same argument as I a.e. on oR and sets h = g/ f. Schwarz's reflection principle eventually shows that h extends to a rational function which turns out to have
no poles at all. Hence I is exposed and by Theorem 3.15 strongly exposed. REMARK 3.17. Similarly, if m > k + 1, then the normalized function hk(Z) = c(z + i)2k is strongly exposed in the unit ball of
H 1 (R). In particular we see that there exist strongly exposed points in the unit ball of H1 (R) that are "small" on the boundary: 1/lhkl rt L 1 / 2k (OR). We recall that for I = I . F to be an
extreme point the only requirement is that the inner part I of I is an extremal Blaschke product - a generic zero set; "most" of the properties of the function I then follow from its outer factor,
i.e., the size of Ilion the boundary an. It is reasonable to ask whether exposedness is also essentially a property of the outer factor. We make this question precise in the following sense: if I E
H1 is a rigid outer function, I is an extremal Blaschke product on 0., and 9 is invertible in MHOO(n) and such that Ig E HOO(n) (a single valued function), is the extreme point Ig· 1/IIIg . 1111 also
exposed? (Compare
THE BOUNDARY OF THE UNIT BALL IN Hl_TYPE SPACES
with the first example in Section 2.2.) Proposition 3.14 tells us the answer is yes if 11f E Ll(80). Theorem 3.18 below shows that in general the answer is no. We mentioned in Section 2.2 Helson's
criterium for exposedness of outer functions, 2.8. It fails for finite domains: THEOREM
3.18 ([4)). For m = 3, there exists ~
R such that the function
c(z - ~)(z + i)2
is extreme in the unit ball of Hl(R), but not exposed.
We will only describe a non-trivial HI (R)-function 9 with the same argument
f a.e. on an, for suitable ~ E R \ lR and refer to [4J for details.
Suppose the three slits are the intervals [Xl,YlJ, [X2,Y2J, [X3,Y3J. Let k(z) be a rational function with poles of order 2 in ~,~, 1/~, 1/~, and poles of order 4 in ±i l , and double zeros at ±1,
while k(oo) = -1. and zeros at the 12 points Next, let q be the (well-defined) square root of k on R with q(oo) = -i. Now with a suitable choice of ~ and an appropriate polynomial p of degree 8 that
is positive on lR U '][', one can show that the function
xtl, yt
p(z) + q(z) (z - ~)(z - i)2
1 (1- ~z)(l- ~z) belongs to HOO(R) and has the same argument as f on 8R \ {-i}, which implies that f is not rigid. Because the inner part of f is the extremal Blaschke product with zero at ~, f is
extreme, however. By Lemma 3.16, (z + i)2 is rigid. We conclude that if Helson's criterion were valid, for any two inner functions u, v not both constant on R, (z + i)2/(u(z) + v(z))2 (j. Hl(R),
hence also (z-~)(z+i)2/(u(z)+v(z))2 (j. Hl(R), a contradiction! =
4. The Bergman space AI(JD)) Recall from Section 1.2 the definition of the Bergman space
AP(JD)) = H(JD)) n LP(JD)) In the Bergman space extreme and exposed points are extremely simple. Indeed, every f E 8B(Al(JD))) is exposed, and (hence) extreme. The exposing functional is
Lf :
If Lfg = 1 for some 9 E 8B(Al(JD))), the argument of 9 would be equal to that of f, (a.e) and this clearly implies f = g. However, to study strongly exposed points we need some machinery.
4.1. Bergman projection and Bloch space. An important tool for the study of strongly exposed points will be the Bergman projection; this is the orthogonal projection It is given by an integral
operator (4.1)
= =
(f, kz)
iD (1 _ zw)2 dA(w)
fUn+ 1)· 1m f(w)'W'dA(w))zn. n=O
Elementary properties are, see e.g. [18], • If cp has compact support in]l)), then Pcp is holomorphic on a neighborhood of Jij. • If cp is Coo on Jij, then Pcp is Coo on Jij. The effect of the
Bergman projection on Loo(]I))) and on C(Jij) will be very important for us. We need the Bloch spaces. DEFINITION 4.1. The Bloch space B consists of all holomorphic functions f on ]I)) with the
property that (1-lzI2)lf'(z)1 is bounded on]l)). Equipped with the norm
IlfllB := If(OI
+ sup (1 -lzI 2)1f'(z)l, zED
B becomes a Banach space. The set of all functions f in B for which the expression (1-lzI2)1f'(z)1 Izl --> 1 is a closed subspace of B, called the little Bloch space Bo.
0 as
Let Co denote the continuous functions on Jij that are zero on T. We have the following theorem of R. Coifman, R. Rochberg and G. Weiss: THEOREM 4.2 ([8]). The Bergman projection P maps Loo(.]I)))
boundedlyonto B. Furthermore, P maps both C(D) and Co boundedlyonto Bo. The result proved in [8] is much more general than Theorem 4.2. As we state it, the theorem may be found in [18], Theorem 1.12,
with an elementary proof. There we also find the following results THEOREM 4.3. (1) The dual space of Al is the Bloch space B under the following pairing:
g E B : f E Al !--+lim ( fr(z)g(z)dA(z). rTl
(2) The dual space of the little Bloch space Bo is the Bergman space A l under the pairing: f
Al : g
lim ( f(z)gr(z)dA(z). rTl
REMARK 4.4. When one identifies (A1)* with the Bloch space in Theorem 1 B, the dual norm on B yields a norm that is equivalent with, but not equal to the norm 11.1113 that we have previously defined
on B. Hence, there exists a norm 11.11. on the Bergman space A I that is equivalent to 11.111 and is such that the dual norm of 9 E B = (AI)* equals IIgIiB. The strongly exposed points in the unit
ball of Al with the norm 11.11. have been described by C. Nara ([32]), who also showed that up to isometrical isomorphisms, Al with the 11.11. norm is the unique pre-dual of B. 4.2. Strongly exposed
points of AI(.]I))). The following theorem is a consequence of Theorems 5.3 and 5.2, the fact that Al(]I))) may be identified with the subspace of HI (lI£2) consisting of functions that depend only
on one variable, and the fact that these functions are exposed in B(HI(lI£2). We set
THE BOUNDARY OF THE UNIT BALL IN Hl-TYPE SPACES
THEOREM 4.5. Let I E Al be 01 unit norm. Then I is strongly exposed in Ball(AI) il and only il the LOO-distance 01 filII to the space (AI)1. + C(D) is less than one. Henceforth we will simply write
instead of (AI)1.
+ C{D).
The question now is: how can we estimate the distance in Loo of cp = 1/111 to (AI)1. +C, where I is a given function in AI? (Clearly the distance cannot exceed one. ) Let us first look at polynomials
of a particularly simple form: I (z) = c( z - 0:) n , where c is normalizing. We will assume n 2: 1 because the constant functions are strongly exposed by Theorem 4.5. We distinguish three cases in
order of increasing difficulty: 10:1 > 1, 10:1 < 1 and 10:1 = 1. The case where 10:1 > 1 is very easy: 1/111 is continuous on D, so I is strongly exposed. In fact, we may even take non-integer powers
n and products of such functions and we always obtain strongly exposed points after normalization. When 10:1 < 1, let us write cp = 1/111 = 'l/Jo + 'l/JI, where 'l/JI is compactly supported in 1D>
and cp == 'l/JI on a neighborhood of 0:, while 'l/Jo is smooth on D. From (4.1) we see that P'l/JI is holomorphic across the unit circle because 'l/JI is compactly supported in 1D>. Next, because 'l/
Jo is smooth on D, also P'l/Jo is smooth on D. Hence Pcp is continuous on D. Now cp - Pcp is bounded, so cp = (cp - Pcp) + Pcp is contained in (AI)1. + C. By Theorem 4.5 I is strongly exposed. Again,
our reasoning readily shows that the normalized product I of functions (z - O:i)n;, for all ni and all O:i ft 'll', is strongly exposed. Now suppose 10:1 = 1; we may take 0: = 1. Let us write In{z) =
cn (1- z)n and CPn = In/l/nl· Introducing polar coordinates and applying Cauchy's theorem one finds that the exposing functional L for h is given by
L(g) = [ g(z) ~1 -
z~: dA{z) = Z
[ Ig{z) (1 -
JD -
z) dA{z) =
+ CIg'{O).
But then there exists a polynomial P2 such that L(g) = fD gP2 dA. Therefore CP2 - P2 is contained in (A I) 1., hence CP2 E (A I) 1. + C so h is strongly exposed. Similarly, for all even n, CPn is
contained in (A I) 1. + C and In is strongly exposed in AI. Again, we may introduce non-integer exponents. Let 1f3 = cf3(1 - z)f3, where (3 > -2 to ensure that I f3 E A I; the constant Cf3 > 0 is
normalizing. Set
cpf3 = 1f3/I/f3I· PROPOSITION 4.6 ([5]). For all (3 > -1, the Loo-distance 01 CPf3 to (AI)1. + C is at most Isin( f327i") I· In particular, lor all (3 > -1, (3 =I- 1,3,5, ... , the lunction I f3
is strongly exposed in the unit ball 01 A I. The proof consists of observing that if 1(3 - 2nl < 1, then IIcpf3 - CP2nli00
IsinCf)1 < 1. We will come back to odd exponents in Section 4.4 and end this section with and example of a boundary points of B (A I (1D>)) that is not strongly exposed. 2
EXAMPLE 4.7 ([5]). The normalized function I(z) = (I-z)2~~g2(I--cz) is not strongly exposed in the unit ball of A I. The functions I f3 tend pointwise to O. However, limf3!-2 fD /{3"(j5 dA = 1.
·4.3. The space (AI).1 + C. We have observed that (AI).1 + C plays the same role in Theorem 4.5 with respect to the Bergman space as (HI).1 + C('1I') = Hoo +C('1I') with respect to the Hardy space HI
(JI))) (Theorem 2.13). We mentioned already in Section 2.3 that Hoo + C is closed in L oo . From this then it followed relatively easily that Hoo + C('1I') is in fact an algebra, cf. [41], Theorem
6.5.5. How far do these results extend to the space (A I ).1 + C? From [5] we quote (1) Al (JI))).1 + c(il~) is a closed subspace of LOO(JI))). (It equals P-I(80 )!) (2) AI(JI))).1 + C(ii)) is a C
(ii))- module. (3) A I (JI))).1 + C(ii)) is not an algebra. (4) The space (A 1).1 + C is invariant under composition with holomorphic automorphisms of JI)). EXAMPLE 4.8. [5] As for (3), let J{3 = (1-
z){3 and let 'P{3 = J{3/IJ{31 for {3 E R Then 'P2 and 'P-4 E (AI).1, but 'P-2 is not contained in (AI).1 + C. The space (A I ).1 + C is not an algebra. The properties of (A 1).1 proposition.
lead in a straightforward way to the following
PROPOSITION 4.9. Let J be a strongly exposed point in AI. Then (a) iJu is an automorphism oJJI)), then the normalized Junction FI = CI(fou) is strongly exposed; (b) iJ v E A(JI))) is zero-Jree on the
circle, then the normalized Junction F2 = C2!v is strongly exposed. Furthermore, the functions J IIJI, FdlFII and F2/1F21 have the same Loo-distance to (AI).1 + C. 4.4. Strong exposedness of (1 - z)
{3. We saw in Section 4.2 that the functions J{3 = c{3(1 - z){3 are strongly exposed in the unit ball of Al for all {3 > -1 except possibly when {3 = 1,3,5, .... This was deduced from rather
straightforward estimates of the L 00 -distances of the functions 'P = ff3 I IJ{31 to the space (A 1).1 + C (Proposition 4.6). In [5] a much sharper result is proved. THEOREM 4.10. For all {3 :::::
0, the Bloch distance oj the Junction P'P{3 to 8 0 4I sin({h)1
equals 1r
{3+T .
SKETCH OF PROOF. It is convenient to rewrite 'P{3 as 'P{3(w) = (1- w){3/2/(1w){3/2. Using the series expansions for the Bergman kernel 1/(1- zw)2 (see (4.2)), as well as for (1 - W){3/2, and 1/(1 -
w){3/2, we evaluate the Bergman projection P'P{3' One obtains P'P{3 = E~=o c{3,nzn, where
c{3,n =
-(n + 1) sin(~) ~ r(m + ~)r(m + n - ~) 271" ~ m!(m + n + I)! .
It is proved in [5] that for fixed (3 > 0:
~r(m+~)r(m+n-~) =:0 m!(m + n + I)!
where the o(I)-term tends to zero as n
= n2{3({3 + 2) (1 + 0(1)),
-+ 00.
This implies that
c{3,n = 7I"({3 + 2)n (1
+ 0(1)),
THE BOUNDARY OF THE UNIT BALL IN where the o{I)-term vanishes as n
--> 00.
But then,
so the Bloch distance of PCP(3 to Bo is at least large N, I{ ~ c
On the other hand, for
zn)'1 < ~ nlc 1.l z ln-1 < 21 sin{~)ll + 0(1) - n~ (3,n - 7r(,B + 2) 1 - Izl '
where the o{I)-term tends to zero as N increases. Using the fact that the polynomials are contained in Bo it follows that the Bloch distance of PCP(3 to Bo is at most 4I sin (¥)1 o ... ((3+2) .
COROLLARY 4.11. Let d(cp(3, (Al)1. (Al)1. + C. Then for all,B 2: 0, (4. 7)
+ C)
denote the LOO-distance of CP(3 to
~ 1sin( !?f) 1 < d( (A 1) 1. + C) < ~ 1sin( ~) 1 < ~ 2 ,B + 2 CP(3, - 7r ,B + 2 - 7r'
In particular, all f(3 are strongly exposed for ,B 2: O. SKETCH OF PROOF. Let q: B --> B/Bo be the quotient map. By Theorem 4.2, the map q 0 P : L OO --> B/Bo is continuous and surjective. The kernel
of the map q 0 P is the space (Al)1. + C, cpo Property (I) from Section 4.3. Hence the derived map
P* : L oo / ((Al)1.
+ C)
B /Bo
is bijective and bounded (by ~ as follows from the proof in [18] of Theorem 4.2). This gives the lower bound for d(cp(3, (Al)1.
+ C), because
IIP*cp(311 =
~ ISi~~y)l.
By the closed graph theorem, the inverse P* -1 of P* is also bounded. Actually, one can show directly that II P* -111 ::; 1, which in turn yields the upper bound for d{cp(3, (Al)1.+C). Supposing that
FE B/Bo has norm 1, we will show that P*-I(F) has norm at most 1 in L oo /((Al )1.+C). For any c > 0, we can find a representative fEB of the coset F such that IIfl18 < 1 + c. In the proof of Theorem
4.2 in [18] one finds that
'¢(w) = (I -IWJ2) . f'(w) ~ f'(O)
L oo
w satisfies f(z) - P'¢{z) = f(O) + j'(O)z E Bo. Thus '¢ is a representative of P*-I(F) in LOO. As a consequence IIp*-l{F)IILOO/((AI)J.+C) ::; d(,¢, (Al)1.
+ C)
::; lim esssuPr
l 2 lim sup 1(1 -lwI )f'(w)1 ::; IIfll8 < 1 + C. Iwl-->l
o The module structure of (Al)1.
and the previous corollary lead to
COROLLARY 4.12. Suppose that g E A(JI1» vanishes nowhere on'll'. Let Zl, Z2, E 'll' be distinct and let {31, {32, ... ,(3n be real numbers greater than - 2. Then the normalized Junction J(z) = cg(z)
n~=l (1- ZZi),8i is strongly exposed in the unit ball oj A I iJ and only iJ all Junctions J,8i = C,8i (1 - z ),8i are strongly exposed. In particular, all choices oj {3i > -1 yield strongly exposed
points and all normalized
•.. , Zn.
polynomials are strongly exposed in the unit ball oj A I. We end this section with a conjecture on the functions J,8 for -2 < {3 < 0, for which strong exposedness is already implied by Proposition
4.6 when -1 < {3 < O. In [5] it is shown that
-2< (.J
with equality in the limit as {3 ! -2. This leads us to surmise the following: CONJECTURE 4.13. For all-2 < (3 < 0, d(cp,8,(AI)1. +C) = particular, the Junctions J,8 are strongly exposed Jor all said
~lsi~~y)J. In
5. Scattered results for B(HI (lRn )) The geometry of B(HI(lR n )) is still quite out of reach. At one time it was conjectured that all boundary points would be extreme, [41], but this conjecture was
refuted as soon as the existence of inner functions on lR n was proved, [1, 27]. Just as in Theorem 2.3 one finds that inner functions, or more generally products with an inner factor cannot be
extreme. However, it is well known that there is no inner-outer decomposition possible for B(HI(lR n )), because the zero sets of functions in various HP classes are essentially different in size.
Rudin already observed in [41] that a function J E 8B(H 1 (lR n )) which is continuous on an open set in 8lRn must be extreme. The proof goes by considering the intersection of lR n with appropriate
complex lines and applying I-dimensional theory. The same idea can be used to show that such functions are in fact exposed, [50]. EXAMPLE 5.1. Not all extreme points are exposed: c(1 has the same
argument as I.
+ 1)2
is extreme but
Example 4.7 also is an example of an exposed point in B(HI(lR2)) that is not strongly exposed. Let us write S = 8lRn , HI = HI(lR n ) and HJ for the functions in HI that vanish at o. We define
(HI)1. =
E Loo(S) :
{JjdA = 0 for allJ E HI}
Analogously we define (HJ)1.. Let d denote the Loo-distance on S. Theorems 2.13 and 2.12 have a full analogue in B(H 1 ). THEOREM 5.2 ([50]). Let J be an exposed point in B(H 1 ). Then exposed iJ and
only il d(J 11/1, C(S) + (HI)1.) < 1.
is strongly
The proof again uses methods from uniform algebra, but is more involved. Noticing that (HJ)1.) is a fairly small subspace of (HI)1. +C(S), this theorem can be strengthened in one direction. THEOREM
5.3. Suppose that d(J IIJI, (HJ)1.) < 1.
is a strongly exposed point in B(H 1 ). Then
THE BOUNDARY OF THE UNIT BALL IN Hl_TYPE SPACES
EXAMPLES 5.4. Homogeneous polynomials in 8B(H 1 ) are strongly exposed, see [50J. Corollary 4.12 shows that polynomials of one variable in 8B(H 1 (B 2 )) are strongly exposed.
It seems natural to try to generalize ideas of Section 4 to HI. The Bergman projection should be replaced by the Szego projection P and the Bloch spaces 8 and 8 0 by BMOA, the space of analytic
functions of bounded mean oscillation and VMOA, the space of functions of vanishing mean oscillation (with respect to the non-isotropic balls in S). It is known that P maps Loo(S) surjectively to
BMOA and C(S) surjectively to VMOA, see [25J for an overview of these kind of results.
So we again have a map P*: Loo(S)/«H1)1. +C)
BMOA/VMOA. However,
we have no idea how to estimate (p*)-l, nor how to estimate the norm of P*cp for cp = p/ipi with p for example a polynomial. Answers to these questions would be very interesting. References [1] A.B.
Aleksandrov, A. B. The existence of inner functions in a ball, (Russian) Mat. Sb. (N.S.) 118(160) (1982), no. 2, 147-163,287. (32A40 46J15) [2] P. Beneker, Strongly exposed points, Helson-Szego
weights and Toeplitz operotors, J. Int. Eq. Oper. Th. 31 (1998), 299-306. [3] P. Beneker, Strongly exposed points in unit balls of Banach spaces of holomorphic functions, Academisch Proefschrift,
Universiteit van Amsterdam, 2002. Electronically available at http://remote.science.uva.nl/ ~beneker /thesis.ps [4] P. Beneker, J. Wiegerinck, Exposedne'ss in Hardy spaces of domains of finite
connectivity, Indag. Math., N.S., 11 (4),2000,487-497. [5] P. Beneker, J. Wiegerinck, Strongly exposed points in the ball of the Bergman space, http://arXiv.org/abs/math.CV /0208234 to appear. [6] L.
de Branges, J. Rovnyak, Square Summable Power Series, Holt, Rinehart and Winston, New York, 1966. [7] L. de Branges, J. Rovnyak, Appendix on square summable power series, in "Perturbation Theory and
its Applications in Quantum Mechanics", John Wiley and Sons, New York (1966), 347-392. [8] R.R. Coifman, R. Rochberg, G. Weiss, Factorization theorems for Hardy spaces in severol variables, Ann. of
Math. (2), 103 (1976), 611-635. [9] A. Devinatz, Toeplitz operotors on H2 spaces, Trans. Amer. Math. Soc. 112 (1964), 304-317. [10] J. Diestel, Geometry of Banach paces - Selected topics LNM 485,
Springer, Berlin etc. 1975. [11] R.G. Douglas, D.E. Sarason, Fredholm Toeplitz operotors, Proc. Amer. Math. Soc. 26 (1970), 117-120. [12] P.L. Duren, Theory of HP-spaces, Academic Press, 1970. [13]
S.D. Fisher, Function Theory On Planar Domains, John Wiley & Sons, 1983. [14] F. Forelli, Extreme points in Hl(R), Can. J. Math. 19 (1967), 312-320. [15] T.W. Gamelin, M. Voichick, Extreme points in
spaces of analytic functions, Can. J. Math. 20 (1968), 919-928. [16] J. Garnett, Bounded holomorphic functions, Pure & Applied Mathematics, 96, Academic Press, Inc., New York-London, 1981. [17] E.
Hayashi, The solution of extremal problems in Hl, Proc. Amer. Math. Soc. 93 (1985), 690-696. [18] H. Hedenmalm, B. Korenblum, K. Zhu, Theory of Bergman Spaces, Springer Graduate Texts in Mathematics
199, 2000. [19] H. Helson, G. Szego, A problem in prediction theory, Ann. Mat. Pura Appl. 51 (1960), 107-138. [20] H. Helson, D. Sarason, Past and future, Math. Scand. 21 (1967), 5-16. [21] H.
Helson, Large analytic functions II, Analysis and partial differential equations, Lecture Notes in Pure and Appl. Math., 122, Dekker, New York (1990), 217-220.
[22] J. Inoue, An example of a non-exposed extreme function in the unit ball of Hl, Proc. Edinburgh Math. Soc. 37 (1993), 47-51. [23] J. Inoue; T. Nakazi, Polynomials of an inner function which are
exposed points in Hl. Proc. Amer. Math. Soc. 100 (1987), no. 3, 454-456. [24] P. Koosis, Introduction to HP spaces, Cambridge University Press, 1980. [25] S.G. Krantz, Geometric Analysis and FUnction
Spaces CBMS Regional Conference Series in Mathematics, 81. Amer. Math. Soc., Providence, RI, 1993. [26] K. de Leeuw, W. Rudin, Extreme points and extremum problems in Hl, Pacific J. Math. 8 (1958),
467-485. [27] E. Ll1lw,A construction of inner functions on the unit ball in CP. Invent. Math. 67 (1982), no. 2, 223-229. [28] T. Nakazi, Exposed points and extremal problems in Hl, J. Funet. Anal.
53 (1983), 224-230. [29] T. Nakazi, Exposed points and extremal problems in Hl, II, T6hoku Math. J. 37 (1985), 265-269. [30] T. Nakazi, Existence of solutions of extremal problems in Hl , Proc.
Edinburgh Math. Soc. (2) 34 (1991), no. 2, 99-112. [31] T. Nakazi, personal communication. [32] C. Nara, Uniqueness of the predual of the Bloch space and its strongly exposed points, Illinois J.
Math, 34 (1990), no. 1,98-107. [33] Z. Nehari, Conformal Mapping, McGraw-Hill, 1952. [34] Z. Nehari, On bounded bilinear forms, Ann. of Math. 65 (1957), 153-162. [35] D.J. Newman, Pseudo-uniform
convexity in HI, Proc. Amer. Math. Soc. 14 (1963), 676-679. [36] J. Neuwirth, D.J. Newman, Positive Hl/2 functions are constant, Proc. Amer. Math. Soc. 18 (1987),958. [37] A. Nicolau, The
coefficients of Nevanlinna's pammetrization are not in HP, Proc. Amer. Math. Soc 106 (1989), 115-117. [38] R.R. Phelps, Dentability and extreme points in Banach spaces, J. Funet. Anal. 17 (1974),
78-90. [39] H.L. Royden, The boundary values of analytic and harmonic functions, Math.Z. 78, (1962), 1-24. [40] W. Rudin, Analytic functions of class H p , Trans. Amer. Math. Soc. 78 (1955), 46-66.
[41] W. Rudin, FUnction Theory in the Unit Ball of en, Grundlehren der Mathematischen Wissenschaften, 241, Springer-Verlag, New York-Berlin, 1980. [42] D.E. Sarason, Algebras of functions on the unit
circle, Bull. Amer. Math. Soc. 79 (1973), 286-299. [43] D.E. Sarason, Function Theory on the Unit Circle, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, 1979. [44] D.E.
Sarason, Exposed points in Hl, I, Operator Theory: Advances and Applications 41 (1989), Birkhiiuser Verlag Basel, 485-496. [45] D.E. Sarason, Exposed points in Hl, II, Operator Theory: Advances and
Applications 48 (1990), Birkhiiuser Verlag Basel, 333-347. [46] D.E. Sarason, Sub-Hardy Hilbert Spaces in the unit disc, Univ. of Arkansas Leeture Notes in the Math. Sc. 10, Wiley-Interscience, 1995.
[47] D. Temme, J. Wiegerinck, Extremal properties of the unit ball of Hl, Indag. Mathern., N.S., 3 (1) (1992), 119-127. [48] M. Voichick, L. Zalcman, Inner and outer functions on Riemann surfaces,
Proc. Amer. Math. Soc. 16 (1965), 1200-1204. [49] H. Widom, Inversion of Toeplitz matrices III, Notices Amer. Math. Soc. 7 (1960), 63. [50] J. Wiegerinck, A chamcterization of strongly exposed points
of the unit ball of Hl, Indag. Mathern., N.S., 4 (4) (1993), 509-519. [51] K. Yabuta, Unicity of the extremum problems in Hl(un), Proc. Amer. Math. Soc. 28 (1971), 181-184. [52] K. Yabuta, Some
uniqueness theorems for HP(un) functions, T6hoku Math. J. 24 (1972), 353-357. FACULTY OF MATHEMATICS, UNIVERSITY OF AMSTERDAM, PLA:'ITAGE I\[UIDERGRACHT
Contemporary Mathematics
Volume 328, 2003
Complete isometries - an illustration of noncommutative functional analysis David P. Blecher and Damon M. Hay ABSTRACT. This article, addressed to a general audience of functional analysts, is
intended to be an illustration of a few basic principles from 'noncommutative functional analysis', more specifically the new field of operator spaces. In our illustration we show how the classical
characterization of (possibly non-surjective) isometries between function algebras generalizes to operator algebras. We give some variants of this characterization, and a new proof which has some
1. Introduction
The field of operator spaces provides a new bridge from the world of Banach spaces and function spaces, to the world of spaces of operators on a Hilbert space. For researchers in the new field, the
philosophical starting point is the combination of the following two obvious facts. Firstly, by the Hahn-Banach theorem any Banach space X is canonically linearly isometric to a closed linear
subspace of C(K), where K is the compact space Ball(X*). Secondly, C(K) is a commutative C*-algebra. Thus one defines a noncommutative Banach space, or operator space, to be a closed linear subspace
X of a possibly noncommutative C* -algebra A. This simplistic idea becomes much more substantive with the addition of some additional metric structure. The point is that if A is any C*-algebra, then
the *-algebra Mn(A) of nx 11, matrices with entries in A has a unique norm 11·lln making it a C*-algebra (this follows from the well known nnicity of C*-norms on a *-algebra). If X c A then Mn(X)
inherits this norm II· lin, and more precisely we think of an operator space as the pair (X, {11·lln}n). We usually insist that maps between operator spaces are completely bounded, where the
adjective 'completely' means that we are applying our maps to matrices too. Thus if T : X -4 Y, then T is completely contractive if Tn is contractive for all n E N, where Tn is the map [Xij] 1--+ [T
(x;j )]. Similarly T is completely isometric if II[T(xij)]1I = I [Xij] I for all n E Nand [Xij] E 1I1n(X). It is an easy exercise (using one of the common expressions for the operator norm of a
matrix in Mn = .Mn(C)) to prove that a linear map T : X -4 Y between 1991 Mat/;tematics Subject Classification. Primary 46L07, 46L05, 47L30; Secondary 46JlO. This research was supported in part by a
grant from the National Science Foundation.. © 85
2003 American Mathematical
DAVID P. BLECHER AND DAMON M. HAY
subspaces of C(K) spaces is completely contractive if and only if it is contractive. Consequently such a T is isometric if and only if it is completely isometric. The identification of the term
'noncommutative Banach space' with 'operator space' may be thought of as a relatively recent entry in the well known 'dictionary' translating terms between the 'commutative' and 'noncommutative'
worlds. We spend a paragraph describing some other entries in this dictionary. Although t.hese items are for the most part well known to the point of being tedious, it will be helpful to collect them
here for the dual purpose of establishing notation, and for ease of reference later in the paper. The most. well known item is of course the fact that the noncommutative version of a C(K) space is a
unital C*-algebra B. The noncommutative version of a unimodular function in C(K) is a unitary u E B (Le. u*u = uu* = 1). The noncommutative version of a function algebra A C C(K) 'containing constant
functions' is a closed subalgebra A of a C* -algebra B, with 18 E A. We call such A a unital operator algebra. For a unital subset S of a C*algebra B, we will take as a simple noncommutative version
of the assertion'S C C(K) separates points of K', the assertion 'the C*-subalgebra of B generated by S (namely, the smallest C*-subalgebra of B containing S) equals B'. The analogue of a closed
subset E of a compact set K is a quotient B / I, where I is a closed twosided ideal in a unital C* -algebra B. More generally, unital *-homomorphisms 7r between unital C* -algebra.'i are the
noncommutative version of continuous functions T between compact spaces. Indeed clearly any such T : Kl -> K2 gives rise to the unital *-homomorphism C(K2 ) -> C(Kd of 'composition with T', and
conversely it is not much harder to see that any unital *-homomorphism C(K2 ) -> C(Kd comes from a continuous T in this way. Moreover such 7r is 1-1 (resp. onto) if and only if the corresponding T is
onto (resp. 1-1). Thus the noncommutative version of a homeomorphism between compact spaces is a (surjective 1-1) *-isomorphism between unital C*-algebras. Coming back to 'noncommutative functional
analysis', it is convenient for some purposes (but admittedly not for others) to view 'complete isometries' as the noncommutative version of isometries. It is very important in what follows that a
1-1 *-homomorphism 7r : A - B between C*-algebras, is by a simple and well known spectral theory argument, automatically an isometry, and consequently (by the same principle applied to 7r n ), a
complete isometry. Similarly, a *-homomorphism 7r : A - B (which is not a priori assumed continuous) is aut.omatically completely contractive, and has a closed range which is a C* -algebra
*-isomorphic to the C* -algebra quotient of A by the obvious two-sided ideal, namely the kernel of the *-homomorphism. The entries we have just described in this 'dictionary' are all easily justified
by well known theorems (for example Gelfand's characterization of commutative C*algebras). That is, if one applies the noncommutative definition in the commutative world, one recovers exactly the
classical object. Similarly one sometimes finds oneself in the very nice 'ideal situation' where one can prove a theorem or establish a theory in the noncommutative world (i.e. about operator spaces
or operator algebras), which when one applies the theorem/theory to objects which are Banach spaces or function algebras, one recovers exactly the classical theorem/theory. An illustration of this
point is the Banach-Stone theorem. The following is a much simpler form of Kadison's characterization of isometries between C*-algebras [17]:
THEOREM 1.1. (Folklore) A surjective linear map T : A --+ B between unital C* -algebras is a complete isometry if and only if T = U7r('), for a unitary u E B and a *-isomorphism 7r : A --+ B. PROOF.
(Sketch.) The easy direction is essentially just the fact mentioned earlier that 1-1 *-homomorphisms are completely isometric. The other direction can be proved by first showing (as with Kadison's
theorem) that T(I) is unitary, so that without loss of generality T(I) = 1. The well known Stinespring theorem has as a simple consequence the Kadison-Schwarz inequality T(a)*T(a) ~ T(a*a). Applying
this to T- 1 too yields T(a)*T(a) = T(a*a), and now the result follows immediately from the 'polarization identity' a*b = ~ E~=o(a + ikb)*(a + ikb). 0 Note that if one takes A = C(Kd and B = C(K2 )
in Theorem 1.1, and consults the 'dictionary' above, then one recovers exactly the classical Banach-Stone theorem. Indeed as we remarked earlier, in this case complete isometries are the same thing
as isometries, unit aries are unimodular functions, and a *-isomorphism is induced by a homeomorphism between the underlying compact spaces. Indeed consider the following generalization of the
Banach-Stone theorem: THEOREM 1.2. [15,22, 1, 20] Let fl be compact and Hausdorff, and A a unital function algebra. A linear contraction T : A --+ C(fl) is an isometry if and only if there exists a
closed subset E of fl, and two continuous functions "f : E --+ '][' and r.p: E --+ 8A, with r.p surjective, such that for all y E E T(f)(y) = "f(y)f(r.p(y))·
Here 8A is the Shilov boundary of A (see Section 2). We have supposed that. T maps into a 'selfadjoint function algebra' C(fl); however since any function algebra is a unital subalgebra of a
'selfadjoint.' one, the theorem also applies to isometries between unital function algebras. If A is a C(K) space too, then 8A = K and then t.he theorem above is called Holsztynski's theorem. We
refer the reader to [16] for a survey of such variants on the classical Banach-Stone theorem. Often the transition from the 'classical' to the 'noncommutative' involves the introduction of much more
algebra. Next we appeal to our dictionary above to give an equivalent restatement of Theorem 1.2 in more algebraic language. THEOREM 1.3. (Restatement of Theorem 1.2) Let A, B be unital function
algebras, with B selfadjoint. A linear contraction T : A --+ B is an isometry if and only if (A) there exists a closed ideal I of B, a unitary u in the quotient C*-algebra B / I, and a unital 1-1
*-homomorphism 7r : A --+ B / I, such that qI (T( a)) = u7r(a) for all a E A. Here qI is the canonical quotient *-homomorphism B
In light of Theorems 1.1 and 1.3 one would imagine that for any complete isometry T : A --+ B between unital operator algebras, the condition (A) above should hold verbatim. This would give a pretty
noncommutative generalization of Theorem 1.3. Indeed if Ran T is also a unital operator algebra, then this is true (see ego B.l in [3]). However, it is quite easily seen that such a result cannot
hold generally. For example, let Mn = Mn(C); for any x E Mn of norm 1, the map A 1---4 Ax is a complete isometry from C into Mn. Now Mn is simple (Le. has no
DAVID P. BLECHER AND DAMON M. HAY
nontrivial two-sided ideals), and so if the result above was valid then it follows immediately that x = u. This is obviously not satisfactory. To resolve the dilemma presented in the last paragraph,
we have offered in [5] several alternatives. For example, one may replace the quotient B / I by a quotient of a certain *-subalgebra of B. The desired relation qJ(T(a)) = U7r(a) then requires u to be
a unitary in a certain C* -triple system (by which we mean a subspace X of a C* -algebra A with X X* X c X). Or, one may replace the quotient B / I by a quotient B / (J + J*), where J is a one-sided
ideal of B. Such a quotient is not an algebra, but is an 'operator system' (such spaces have been important in the deep work of Kirchberg (see [18, 19] and references therein). Alternatively, one may
replace such quotients altogether, with certain subspaces of the second dual B** defined in terms of certain orthogonal projections of 'topological significance' (Le. correspond to characteristic
functions of closed sets in K if B = C(K)) in the second dual B** (which is a von Neumann algebra [25]). The key point of all these arguments, and indeed a key approach to Banach-Stone theorems for
linear maps between function algebras, C* -algebras or operator algebra.-" is the basic theory of C* -triple systems and triple morphisms, and the basic properties of the noncommutative Shilov
boundary or triple envelope of an operator space. These important and beautiful ideas originate in the work of Arveson, Choi and Effros, Hamana, Harris, Kadison, Kirchberg, Paulsen, Ruan, and others.
Indeed our talk at the conference spelled out these ideas and their connection with the Banach-Stone theorem; and the background ideas are developed at length in a book the first author is currently
writing with Christian Le Merdy [7] (although we do not characterize non-surjective complete isometries there). Moreover, a description of our work from this perspective, together with many related
results, may be found in [12]. Thus we will content ourselves here with a survey of some related and interesting topics, and with a new and self-contained proof of some characterizations of complete
isometries between unital operator algebras which do not appear elsewhere. This proof has several advantages, for example the projections arising naturally with this approach seem to be more useful
for some purposes. Also it will allow us to avoid any explicit mention of the theory of triple systems (although this is playing a silent role nonetheless). We also show how such noncommutative
results are generalizations of the older characterizations of into isometries between function algebras or C(K) spaces. We thank A. Matheson for telling us about these results. In the final section
we present some evidence towards the claim that (general) isometries between operator algebras are not the correct noncommutative generalization of isometries between function algebras. For the
reader who wants to learn more operator space theory we have listed some general texts in our bibliography.
2. The noncommutative Shilov boundary At the present time the appropriate 'extreme point' theory is not sufficiently developed to be extensively used in noncommutative functional analysis. Although
several major and beautiful pieces are now in place, this is perhaps one of the most urgent needs in the subject. However there are good substitutes for 'extreme point' arguments. One such is the
noncommutative Shilov boundary of an operator space. Recall that if X is a closed subspace of C(K) containing the identity fUIlction lK on
K and separating points of K, then the classical Shilov boundary may be defined to be the smallest closed subset E of K such that all functions J E X attain their norm, or equivalently such that the
restriction map J f-+ JIE on X is an isometry. This boundary is often defined independently of K, for example if A is a unital function algebra then we may define the Shilov boundary as we just did,
but with K replaced by the maximal ideal space of A. In fact we prefer to think of the classical Shilov boundary of X as a pair (aX, i) consisting of an abstract compact Hausdorff space ax, together
with an isometry j : X -+ C(aX) such that j(IK) = lax and such that j(X) separates points of ax, with the following universal property: For any other pair (0, i) consisting of a compact Hausdorff
space 0 and a complete isometry i : X -+ C(O) which is unital (i.e. i(IK) = IA), and such that i(X) separates points of 0, there exists a (necessarily unique) continuous injection r : ax -+ 0 such
that i(x)(r(w)) = j(x)(w) for all x E X, w E ax. Such a pair (aX, i) is easily seen to be unique up to an appropriate homeomorphism. The fact that such ax exists is the difficult part, and proofs may
be found in books on function algebras (using extreme point arguments). Consulting our 'noncommutative dictionary' in Section 1, and thinking a little about the various correspondences there, it will
be seen that the noncommutative version of this universal property above should read as follows. Or at any rate, the following noncommutative statements, when applied to a unital subspace X c C(K),
will imply the universal property of the classical Shilov boundary discussed above. Firstly, a unital operator space is a pair (X, e) consisting of an operator space X with fixed element e EX, such
that there exists a linear complete isometry Il, from X into a unital C*-algebra C with Il,(e) = Ie. A 'noncommutative Shilov boundary' would correspond to a pair (B,j) consisting of a unital
C*-algebra B and a complete isometry j : X -+ B with j(e) = IB, and whose range generates B as a C* -algebra, with the following universal property: For any other pair (A, i) consisting of a unital
C* -algebra and a complete isometry i : X -+ A which is unital (Le. i(e) = IA), and whose range generates A as a C* -algebra, there exists a (necessarily unique, unital, and surjective)
*-homomorphism IT : A -+ B such that IT 0 i = j. Happily, this turns out to be true. The existence for any unital operator space (X, e) of a pair (B,j) with the universal property above is of course
a theorem, which we call the Arveson-Hamana theorem [2, 13] (see [3] for complete details). As is customary we write C;(X) for B or (B,j), this is the 'C*-envelope of X'. It is essentially unique, by
the universal property. If X = A is a unital operator algebra (see Section 1 for the definition of this), then j above is forced to be a homomorphism (to see this, choose an i which is a
homomorphism, and use the universal property). Thus A may be considered a unital subalgebra of C;(A). If A is already a unital C* -algebra, then of course we can take C; (A) = A. To help the reader
get a little more comfortable with these concepts, we compute the 'noncommutative Shilov boundary' in a few simple examples. Example 1. Let Tn be the upper triangular n x n matrices. This is a unital
subspace of M n , and no proper *-subalgebra of Mn contains Tn. Let (B,j) be the C* -envelope of Tn. By the universal property of the C* -envelope, there is a surjective *-homomorphism IT : Mn -+ B
such that IT(a) = j(a) for a E Tn. The kernel of IT is a two-sided ideal of Mn. However Mn has no nontrivial two-sided ideals. Hence IT is 1-1, and is consequently a *-isomorphism, and we can thus
identify Mn with B. Thus Mn is a C*-envelope of Tn.
DAVID P. BLECHER AND DAMON M. HAY
Example 2. Consider the linear subspace X of M3 with zeroes in the 1-3, 2-3, 2-1, 3-1 and 3-2 entries, and with arbitrary entries elsewhere except for the 3-3 entry, which is the average of the 1-1
and 2-2 entries. It is easy to see that the C* -algebra generated by X inside M3 is M2 EB C. However this is not the C*-envelope. Indeed the 3-3 entry here is redundant, since the norm of x E X is
the norm of the upper left 2 x 2 block of x. The canonical *-homomorphism from M2 EB C onto Nh when restricted to X is a unital complete isometry from X onto T2 (see Example 1). Thus if one takes the
quotient of M2 EB C by the kernel of this homomorphism, namely the ideal 02 EB C, then one obtains M 2 , which by Example 1 is the C* -envelope. Indeed this is typical when calculating the C*
-envelope of a unital subspace X of Mn. The C*-algebra generated by X is a finite dimensional unital C*-algebra. However such a C* -algebras is *-isomorphic to a finite direct sum B of full 'matrix
blocks' M nk • Some of these blocks are redundant. That is, if p is the central projection in B corresponding to the identity matrix of this block, then x t-+ x(IBp) is completely isometric. If one
eliminates such blocks then the remaining direct sum of blocks is the C* -envelope. Example 3. Let B be a unital C*-algebra. Consider the unital subspace S(B) of the C*-algebra M 2 (B) consisting of
for all x,y E B and >.,Ji, complex scalars. We claim that M2(B) is the C*-envelope C of S(B), and we will prove this using a similar idea to Example 1 above. Namely, first note that M 2(B) has no
proper C*-subalgebra containing S(B), Thus by the Arveson-Hamana theorem there exists a *-homomorphism 11' : M 2 (B) ---+ C which possesses a property which we will not repeat, except to say that it
certainly ensures that 11' applied to a matrix with zero entries except for a nonzero entry in the 1-2 position, is nonzero. It suffices as in Example 1 to show that Ker 11' = {a}. Suppose that 11'
(x) = 0 for a 2 x 2 matrix x E M 2(B). Let Eij be the four canonical basis matrices for M 2, thought of as inside M2(B). Then 11'(E1i XEj2 ) = 11'(Eli)11'{X)11'(Ej2) = 0 for i,j = 1,2. Thus by the
fact mentioned above about the 1-2 position, we must have EliXEj2 = O. Thus x = O. In fact a variant of the C*-envelope or 'noncommutative Shilov boundary' can be defined for any operator space X.
This is the triple envelope of Hamana (see [14]). This is explained in much greater detail in [3], together with many applications. For example it is intimately connected to the 'noncommutative
.M-ideals' recently introduced in [4]. This 'noncommutative Shilov boundary' is, as we mentioned in Section 1, a key tool for proving various Banach-Stone type theorems. However in the present
article we shall only need the variant described earlier in this section. 3. Complete isometries between operator algebras We begin this section with a collection of very well known and simple facts
about closed two-sided ideals] in a C* -algebra A, and about the quotient C* -algebra AI]. We have that ]1.1. is a weak* closed two-sided ideal in the von Neumann algebra A**, and there exists a
unique orthogonal projection e in the center of A** with ]1.1. = A**(1 - e). The projection 1 - e is called the support projection for I, and
1 - e may be taken to be the weak* limit in A ** of any contractive approximate identity for I. Thus it follows that A ** / I J..J.. ~ A ** e as C* -algebras. Therefore also A/Ie (A/ 1)** ~ A** /
IJ..J.. ~ A**e
as C*-algebras. Explicitly, the composition of all these identifications is a 1-1 *homomorphism taking an a + I in A/I, to ae = eae in A **. Here' is the canonical embedding A -+ A** (which we will
sometimes suppress mention of). Thus A/I may be regarded as a C*-subalgebra of A**, or of the C*-algebra eA**e. We next illustrate the main idea of our theorem with a simple special case. (The
following appeared as part of Corollary 3.2 in the original version of [5], with the proof left as an exercise). Suppose that T : A -+ B is a complete isometry between unital C*algebras, and suppose
that T is unital too, that is T(I) = 1. Let C be the C*-subalgebra of B generated by T(A). Applying the Arveson-Hamana theorem 1 we obtain a surjective *-homomorphism () : C -+ A such that (}(T(a)) =
a for all a E A. If I is the kernel of the mapping (), then C / I is a unital C* -algebra *-isomorphic to A. Indeed there is the canonical *-isomorphism 'Y : A -+ C / I induced by (), taking a to T
(a) + I. The next point is that C/I may be viewed as we mentioned a few paragraphs back, as a C* -subalgebra of C** , and therefore also of B**. Indeed if e is the central projection in C** mentioned
there, then C / I may be viewed as a C* -subalgebra of eC** e C eB** e C B**. In view of the last fact, the map 'Y induces an 1-1 *-homomorphism 7r : A -+ B** taking an element a E A to the element
of B** which equals (1)
-- -- --
T(a)e = eT(a)
(these are equal because e is central in C**). Conversely, if T : A -+ B is a complete contraction for which there exists a projection e E B** such that eT(a)e is a 1-1 *-homomorphism 7r, then for
all a E A,
~ Ile~ell
117r(a)11 = Iiall
using the fact mentioned earlier that 1-1 *-homomorphisms are necessarily isometric. Thus T is an isometry, and a similar argument shows that it is a complete isometry. Thus we have characterized
unital complete isometries T : A -+ B. If H is a Hilbert space on which we have represented the von Neumann algebra B** as a weak* closed unital *-subalgebra, then B may be viewed also as a unital
C*-subalgebra of B(H), whose weak* closure in B(H) is (the copy of) B**. In this case we shall say that B is represented on H universally. (The explanation for this term is that the well-known
'universal representation' tru of a C*-algebra is 'universal' in our sense, and conversely if 7r is a representation which is 'universal' in our sense then 7r(B)" is isomorphic to 7ru(B)" ~ B**. See
[27] Section 1.) If, further, e E B** is a projection for which (1) holds, then with respect to the splitting H = eH EEl (1- e)H we may write T(a) =
SO '
for all a E A. We will see that this is essentially true even if T(IA) -lIB: 1We remark in passing that one does not need the full strength of the Arveson-Hamana theorem here, one may use the much
simpler [8] Theorem 4.1.
DAVID P. BLECHER AND DAMON M. HAY
THEOREM 3.1. Let T : A --> B be a completely contractive linear map from a unital operator algebra into a unital C* -algebra. Then the following are equivalent:
(i) T is a complete isometry, (ii) There is a partial isometry u E B** with initial projection e E B**, and a (completely isometric) 1-1 *-homomorphism 1r : C;(A) 1r(1) = e, such that for all a E A T
= tL1r(a)
and 1r(a)
eB**e with
= u*T(a).
Moreover e may be taken to be a 'closed projection' (see [25] 3.11, and the discussion towards the end of our proof). (iii) If H is a Hilbert space on which B is represented universally, then there
exist two closed subspaces E, P of the Hilbert space H, a 1-1 *-homomorphism 1r : C;(A) --> B(E) with 1r(1) = IE, and a unitary u : E --> P, such that T(a)IE = u1r(a), and T(a)IE.L C p.l., for all a
E A. Here E.l. for example is the orthocomplement of E in H. (iv) If H is as in (iii), then there exists two closed subspaces E, P of H, a unital 1-1 *-homomorphism 1r : C;(A) --> B(E), a complete
contraction S : C; (A) --> B( E.l. , p.l.), and unitary operators U : E EEl p.l. --> Hand V : H --> E EEl E.l., such that T( ) - U [ 1r(a)
a -
] V
for all a E A.
(v) There is a left ideal J of B, a 1-1 *-homomorphism 1r from C;(A) into
a unital subspace of B I (J + J*) which is a C* -algebra, and a 'partial isometry' u in B I J such that qJ(T(a)) = u1r(a)
for all a E A, where qJ is the canonical quotient map B
Before we prove the theorem, we make several remarks. First, we have taken B to be a C* -algebra; however since any unital operator algebra is a unital subalgebra of a unital C* -algebra this is not
a severe restriction. We also remark that there are several other items that one might add to such a list of equivalent conditions. See [5, 6]. Items (ii)-(iv), and the proof given below of their
equivalence with (i), are new. We acknowledge that we have benefitted from a suggestion that we use the Paulsen system to prove the result. This approach is an obvious one to those working in this
area (Ruan and Hamana used a variant of it in their work in the '80's on complete isometries and triple morphisms [28, 14]). However we had not pushed through this approach in the original version of
[5] because this method does not give several of the results there as immediately. Statement (v) above has been simply copied from [5, 6] without proof or explanation. We have listed it here simply
because Theorem 1.3 may be particularly easily derived from it as the special case when A and B are commutative (see comments below). Note that (iii) above resembles Theorem 1.2 superficially.
PROOF. The fact that the other conditions all imply (i) is easy, following the idea in the paragraph above the theorem, namely by using the fact that a 1-1 *-homomorphism is completely isometric. In
the remainder of the proof we suppose that T is a complete isometry. We view A as a unital subalgebra of C;(A) as outlined in Section 3. We define a subset S(B) of M 2(B) as in Example 3 in Section
2. Similarly define a subset S(T(A)) of S(B) using a similar formula (note that S(T(A)) has 1-2 entries taken from T(A) and 2-1 entries taken from T(A)*). Similarly we define the subset S(A) of the
CO-algebra M2(C;(A)) (Le. S(A) has scalar diagonal entries and off diagonal entries from A and A*). We write 1 EEl 0 for the matrix in S(A) with 1 as the 1-2 entry and zeroes elsewhere. Similarly for
0 EEl 1. We also use these expressions for the analogous matrices in S(B). The map q, : S(A) -+ S(T(A)) c M 2(B) taking
:1] ~
[T~)* ~~~)]
is well known to be a unital complete isometry (this is the well known Paulsen lemma, see the proof of 7.1 in [23]). Let C be the C' -subalgebra of M2 (B) generated by S(T(A)). The CO-envelope of S
(A) is well known to be M2(C;(A)) (see Example 3 in Section 2 where we proved this in the case that A is already a C* -algebra, or for example [3] Proposition 4.3 or [30]). Thus by the Arveson-Hamana
theorem we obtain a surjective *-homomorphism () : C -+ 1V/z(C;(A)) such that () 0 q, is simply the canonical embedding of S(A) into M2(C;(A)). As in the special case considered above the theorem, we
let 10 be the kernel of the mapping (), then C /10 is a unital CO-algebra *-isomorphic to M2(C;(A)). Indeed there is the canonical *-isomorphism "(: M2(C;(A)) -+ Clio induced by (), taking
:1] ~
[T~:)* T~~)]
10 ,
As in the simple case above the theorem, C /10 may be viewed as a C* -subalgebra of PoC"Po, for a central projection Po E C** (namely, the complementary projection to the support projection of 10)'
Now PoC**Po C C** C M2(B)**, and it is well known that M2(B)** ~ M2(B**) as CO-algebras. Thus we may think of C** as a C* -subalgebra of .1112 (B**). Also, C'* contains C as a C* -subalgebra, and the
projections 1 EEl 0 and 0 EEl 1 in C correspond to the matching diagonal projections 1 EEl 0 and 0 EEl 1 in M2(B**). These last projections therefore commute with Po, since Po is central in C**,
which immediately implies that Po is a diagonal sum fEEl e of two orthogonal projections e, f E B**. Thus we may write the C* -algebra POM2(B**)po as the C*-subalgebra [ fB** f eB** f
f B**e ] eB*'e
of M2(B**). We said above that Clio may be regarded as a C*-subalgebra of the subalgebrapoM2(B**)po of M2(B*'). Thus the map "( induces a 1-1 *-homomorphism III : M2(C;(A)) -+ .!I1z(B**). It is easy
to check that 1lI(1 EEl 0) = fEEl 0 and III (0 EEl 1) = 0 EEl e. Since III is a *-homomorphism it follows that III maps each of the four corners of M 2(C;(A)) to the corresponding corner of POM2(B**)
po C M2(B**). We let R: C;(A) -+ fB**e be the restriction of III to the '1-2-corner'. Since III is 1-1, it follows that R is 1-1. If 7r is the restriction of III to the '2-2-corner', then 7r is a
*-homomorphism C;(A) -+ eB**e taking lA to e. Applying the *-homomorphism
DAVID P. BLECHER AND DAMON M. HAY
111 to the identity
[~ ~][~ ~]=[~ ~] we obtain that u = R(I) is a partial isometry, with u*u = 71"(1) = e. Similarly uu* = f. A similar argument shows that R(a) = R(I)7I"(a) for all a E C;(A). Thus u* R(a) = u*u7l"(a) =
7I"(a) for all a E C;(A). Next, we observe that 111 takes the matrix z which is zero except for an a from A in the 1-2-corner, to the matrix w = Pocp(z)Po. Since cp(z) E C** and Po is in the center
of that algebra, we also have w = cp(z)Po = Pocp(z). Also w viewed as a matrix in M2(B**) has zero entries except in the .1-2-corner, which (by the last sentence) equals
---- ----
ff(;;) e = f(;;) e = ff(;;). Also using these facts and a fact from the end of the last paragraph we have --...
----* -
u*T(·) = R(I)*T(·) = (fT(I)e)*T(·) = eT(I) fT(·) = eT(I) T(·)e = u* R(·) = 71". Thus
T(·)e = fT(·) = uu*T(·) = U7l"(')' We have now also established most of (ii). One may deduce (iii) from (ii) by viewing B c B** c B(H), and setting E = eH, and F = (uu*)H. We also need to use facts
from the proof above such as u*u = e. Clearly (iv) follows from (iii). As we said above, we will not prove (v) here. Claim: if e is the projection in (ii) above, then 1 - e is the support projection
for a closed ideal I of a unital *-subalgebra D of B. Equivalently (as stated at the start of this section), there is a (positive increasing) contractive approximate identity (bd for I, with bt ~ 1 -
e in the weak* topology. This claim shows that 1 - e is an 'open projection' in B**, so that e is a closed projection, as will be obvious to operator algebraists from [25] section 3.11 say. For our
other readers we note that for what comes later in our paper, one can replace the assertion about closed projections in the statement of Theorem 3.1 (ii) with the statement in the Claim above. To
prove the Claim, recall from our proof that Po = fEB e = Ie - Pi, where Pi is the support projection for a closed ideal 10 of C. Thus Pi = (1 - f) EB (1 - e). As stated at the start of Section 3, Pi
is the weak* limit in C**, and hence also in M 2(B**), of a contractive approximate identity (et) of 10, By the separate weak* continuity of the product in a von Neumann algebra, it follows that the
net bt = (OEB l)et(OEB 1) has weak* limit (OEB I)Pl (OEB 1) = OEB (1- e). Viewing these as expressions in B, the above says that bt ~ 1 - e weak* in B* *. View (0 EB 1) C (0 EB 1) as a *-subalgebra D
of B, and view (0 EB 1)10(0 EB 1) as a two sided ideal I in D. It is easy to see that (bd is a contractive approximate identity of I. Thus it follows that 1 - e is the support projection of the ideal
I. 0 Some applications of results such as Theorem 3.1 may be found in [6]. Next we discuss briefly the relation between our noncommutative characterization of complete isometries (for example Theorem
3.1 above), and Theorem 1.3. Our point is not to provide another proof for Theorem 1.3 - the best existing proof is certainly short and elegant. Rather we simply wish to show that the noncommutative
result contains 1.3. Indeed Theorem 1.3 quite easily follows from Theorem
3.1 (v). Since however we did not prove Theorem 3.1 (v), we give an alternate proof. COROLLARY 3.2. Let A, B be a unital function algebras, with B selfadjoint. Then condition (ii) in Theorem 3.1
implies condition (A) in Theorem 1.3. PROOF. By hypothesis, T(·)e = U1l"('), and u*u = e = 11"(1) so that u = u1l"(1) = T(I)e. Thus eT(I)*T(·)e = u*U1l"(') = 11"(1)11"(') = 11", so that Ran 11" C eBe
= Be (note B** is commutative in this case). From [25] 3.11.10 for example, the 'closed projection' e in B** corresponds to a closed ideal J in B whose support projection is 1 - e. Alternatively, to
avoid quoting facts from [25], we will also deduce this from the 'Claim' towards the end of the proof of Theorem 3.1. If I is the ideal in that Claim, let J be the closed ideal in B generated by I.
Since J = Bl, the contractive approximate identity of I is a right contractive approximate identity of J. Thus J has support projection 1 - e too, by the first paragraph of Section 3 above. By facts
in the just quoted paragraph, we have a canonical unital 1-1 map '11 : B / J ~ B** taking the equivalence class b + J of b E B to ebe. Indeed in this commutative case we see by inspection that '11 is
a *-homomorphism from the C*algebra B/J onto the C*-subalgebra M = eBe of B**. Define O(a) = '11-1(1I"(a)), this is a 1-1 *-homomorphism A ~ B/J. Since 11"(1) = e, 0 is a unital map too. Since uu* =
u*u = e, u is unitary in M, and so 'Y = '11-1(U) is unitary in B/J. Note also that T(a)e = '11(T(a) + J). Applying '11- 1 to the equation T(·)e = U1l"('), we obtain qJ(T(a)) = 'Y O(a), that is,
condition (A) in Theorem 1.3. D If one attempts to use the ideas above to find a characterization analogous to condition (A) from Theorem 1.3 but in the noncommutative case, it seems to us that one
is inevitably led to a condition such as (v) in Theorem 3.1. We address a paragraph to experts, on generalizations of the proof of Theorem 3.1. Consider a complete isometry between possibly
non-unital C* -algebras. Or much more generally, suppose that T is a complete isometry from an operator space X into a C* -triple system W. One may form the so called 'linking C* -algebra' of W, with
the identities of the 'left and right algebras of W' adjoined. Call this C'(W). As in the proof of Theorem 3.1 we think of S(W) c .c'(W). Similarly, if Z is the 'triple envelope' of X (or if X = Z is
already a C*-algebra or C*-triple system), then we may consider S(X) C S(Z) c .c'(Z). As in the proof of Theorem 3.1 we obtain firstly a unital complete isometry : S(X) ~ S(T(X)) c .c'(Z), and then a
unital 1-1 *-homomorphism 11" : .c'(Z) ~ .c'(W)**. By looking at the 'corners' of 11" we obtain projections e, f in certain second dual von Neumann algebras, so that fT( ·)e is (the restriction to X
of a completely isometric) a 1-1 triple morphism into W**. In fact we have precisely such a result in [5] (see Section 2 there), but the key point is that the new proof gives different projections e,
f, which are more useful for some purposes.
4. Complete isometries versus isometries Finally, as promised we discuss why we believe that in this setting of nonsurjective maps between C* -algebras say, general isometries are not the
'noncommutative analogue' of isometries between function algebras. The point is simply this. In the function algebra case we can say thanks to Holsztynski's theorem that the isometries are
essentially the maps composed of two disjoint pieces Rand S, where R is
DAVID P. BLECHER AND DAMON M. HAY
isometric and 'nice', and S is contractive and irrelevant. However at the present time it looks to us unlikely that there ever will be such a result valid for general nonsurjective isometries between
general C* -algebras. The chief evidence we present for this assertion is the very nice complementary work of Chu and Wong [9] on isometries (as opposed to complete isometries) T : A -+ B between
CO-algebras. They show that for such T there is a largest projection p E B** such that T(·)p is some kind of Jordan triple morphism. This appears to be the correct 'structure theorem', or version of
Kadison's theorem [11], for nonsurjective isometries. However as they show, the 'nice piece' R = T(·)p is very often trivial (Le. zero), and is thus certainly not isometric. Thus this approach is
unlikely to ever yield a characterization of isometries. A good example is A = M 2 , the smallest noncommutative C* -algebra. Simply because A is a Banach space there exists, as in the discussion in
the first paragraph of our paper, a linear isometry of A into a C (K) space. However it is easy to see that there is no nontrivial *-homomorphism or Jordan homomorphism from A into a commutative C*
-algebra. Such an isometry is uninteresting, and this is perhaps because the interesting 'nice part' is zero. Thus we imagine that the 'good noncommutative notions of isometry' are either complete
isometries or the closely related class of maps for which the piece T(·)p from [9] is an isometry. This leads to three questions. Firstly, can one independently characterize the last mentioned class?
Secondly, if T is a complete isometry, then is the projection p in the last paragraph equal (or closely related) to our projection e above? Finally, H. Pfitzner has remarked to us, there is already a
gap between the isometry and the 2-isometry cases (not only isometries and complete isometries). It would be interesting if there were a characterization of 2-isometries.
References [1] J. Araujo and J. J. Font, Linear isometries between subspaces of continuous functions, Trans. Amer. Math. Soc. 349 (1997), 413-428. [2] W. B. Arveson, Subalgebms ofC*-algebms, Acta
Math. 123 (1969), 141-224; II, 128 (1972), 271-308. [3] D. P. Blecher, The Shilov boundary of an opemtor space, and the chamcterization theorems, J. Funct. An. 182 (2001),280-343. [4] D. P. Blecher,
E. G. Effros and V. Zarikian, One-sided M-Ideals and multipliers in opemtor spaces. 1. To appear Pacific J. Math. [5] D. P. Blecher and D. Hay, Complete isometries into C* -algebms, http://
front.math.ucdavis.edu/math.OA/0203182, Preprint (March '02). [6] D. P. Blecher and L. E. Labuschagne, Logmodularity and isometries of opemtor algebms, To appear, Trans. Amer. Math. Soc .. [7] D. P.
Blecher and C. Le Merdy, Opemtor algebms and their modules - an opemtor space approach, To appear, Oxford Univ. Press. [8] M. D. Choi and E.G. Effros, The completely positive lifting problem for
C·-algebms, Ann. Math. 104 (1976), 585-609. [9] C-H. Chu and N-C. Wong, Isometries between C· -algebms, Preprint, to appear Revista Matematica Iberoamericana. [10] J. B. Conway, A Course in Opemtor
Theory, AMS, Providence, 2000. [11] E. G. Effros and Z. J. Ruan, Opemtor Spaces, Oxford University Press, Oxford (2000). [12] R. J. Fleming and J. E. Jamison, Isometries on Banach spaces: function
spaces, Book to appear, CRC press. [13] M. Hamana, Injective envelopes of opemtor systems, Pub!. R.I.M.S. Kyoto Univ. 15 (1979), 773-785.
[14] M. Hamana, Triple envelopes and Silov boundaries of operator spaces, Math. J. Toyama University 22 (1999), 77-93. [15] W. Holsztynski, Continuous mappings induced by isometries of spaces of
continuous functions, Studia Math. 24 (1966), 133-136. [16] K. Jarosz and V. Pathak, Isometries and small bound peturbations of function spaces, In "Function Spaces", Lecture Notes in Pnre and
Applied Math. Vol. 136, Marcel Dekker (1992). [17] R. V. Kadison, Isometries of operator algebras, Ann. of Math. 54 (1951), 325-338. [18] E. Kirchberg, On restricted peturbations in inverse images
and a description of normalizer algebras in C*-algebras, J. Funct. An. 129 (1995), 1-34. [19] E. Kirchberg and S. Wassermann, C*-algebras generated by operator systems, J. Funct. Analysis 155 (1998),
324-351. [20] A. Matheson, Isometries into function algebras, To appear. [21] M. Nagasawa, Isomorphisms between commutative Banach algebras with an application to rings of analytic functions, Kodia
Math. Sem. Rep. 11 (1959), 182-188. [22] W. P. Novinger, Linear isometries of subspaces of continuous functions. Studia Math. 53 (1975), 273-276. [23] V. I. Paulsen, Completely bounded maps and
dilations, Pitman Research Notes in Math., Longman, London, 1986. [24] V. I. Paulsen, Completely bounded maps and operator algebras, To appear Cambridge University Press. [25] G. Pedersen,
C*-algebras and their automorphism groups, Academic Press (1979). [26] G. Pisier, Introduction to operator space theory, To appear Camb. Univ. Press. [27] M. Rieffel, Morita equivalence for
C*-algebras and W*-algebras, J. Pure Appl. Algebra 5 (1974), 51-96. [28] Z. J. Ruan, Subspaces ofC*-algebras, Ph. D. thesis, U.C.L.A., 1987. [29] E. L. Stout, The theory of uniform algebras, Bogden
and Quigley (1971). [30] C. Zhang, Representations of operator spaces, J. Oper. Th. 33 (1995), 327-351. DEPARTMENT OF MATHEMATICS, UNIVERSITY OF HOUSTON, HOUSTON.
E-mail address, David P. Blecher: dblecher~ath. uh. edu E-mail address, Damon Hay: dhayOmath.uh.edu
TX 77204-3008
Contemporary Mathematics Volume 328, 2003
Some Recent Trends and Advances in Certain Lattice Ordered Algebras Karim Boulabiar, Gerard Buskes, and Abdelmajid Triki ABSTRACT. In this paper we give a survey, intended as both a supplement as
well as an update to a survey by Huijsmans [57], with results that have been obtained in the last ten years on Archimedean lattice ordered algebras. Special attention is paid to I-algebras, almost
I-algebras and d-algebras and problems that were posed in the survey by Huijsmans about these special classes of lattice ordered algebras.
l. Introduction 2. Definitions and elementary properties 3. i-algebra multiplications in C (X) 4. Multiplication by an element as an operator 5. Uniform completion and Dedekind completion 6. Powers
in i-algebras 7. Functional Calculus on f-algebras 8. Relationships between i-algebra multiplications 9. Connection between algebra and Riesz homomorphisms 10. Positive derivations 11. Cauchy-Schwarz
inequalities 12. Order biduals 13. Ideal theory 14. Representation of f-algebras 15. Linear biseparating maps on f-algebras References 1991 Mathematics Subject Classification. 06F25, 13J25, 16W80,
46A40, 46840, 46842, 46E25, 47L07, 47847, 47865. Key words and phrases. almost I-algebra, algebra homomorphism, I-algebra, d-algebra, lattice ordered algebra, order ideal, orthomorphism,
representation theory, Riesz homomorphism, ring ideal, space of continuous functions, uniformly complete Riesz space. The second named author gratefully acknowledges support from an Office of Naval
Research Grant with number N00014-01-1-0322. Part of this survey was written while the first named author was visiting the University of Mississippi in the Spring of 2002. © 2003 American
Mathematical Society
1. Introduction
ftehistory of lattice ordered vector spaces (so called Riesz spaces or vector
lattices) goes back to Riesz and the International Congress of Mathematicians in Bologna in 1928. A study of the most important class of lattice ordered algebras (but not their name), J-algebras, was
initiated by Nakano in [75] for a-Dedekind complete ordered vector space in 1951, subsequently in 1953 by Amemiya in [3], and finally with its present definition and name in 1956 by Birkhoff and
Pierce in [22]. A precise date for the very first definition of lattice ordered algebras in general is very hard to pinpoint, but originated at around the same time as the previous three references.
Indeed, in his review [44] of Birkhoff's 1950 address to the International Congress of Mathematicians in Cambridge, Massachusetts, Fl.·ink observed that a general study of lattice ordered rings seems
to be needed to study what are now called averaging or Reynolds operators. A call for lattice ordered rings also had gone out by Birkhoff himself in the form of a listed problem at the end of his
seminal 1942 paper [20] on lattice ordered groups. Thus lattice ordered algebras and J-algebras seem to have multiple origins, including a study of averaging operators, which themselves sprang forth
from problems in fluid mechanics. An appearance at about the same time of J-rings and J-algebras has not resulted in an historical development on complete common ground for these objects. This is not
unlike the development of lattice ordered groups versus the development of Riesz spaces. Where the latter have attracted attention from researchers in analysis, the former have been more widely
investigated by algebraists. A similar divided attention from analysis versus algebra seems to underlie the connected but somewhat separate tracks of lattice ordered rings versus lattice ordered
algebras. Though this separation of tracks is to some extent unavoidable, where each track does have ground that is truly its own, some overlap in results does exist, resulting in difficulties making
accurate literature attributions in a survey like ours. We are grateful to two referees for pointing at some references that were missing in our manuscript, though we take full responsibility for
possible remaining omissions in the reference list. This survey places itself almost completely on the track of lattice ordered algebras and our only apology for not linking algebra facts in a
systematic way to ring results is that all three of us authors were trained as analysts. There is a natural back and forth between the two theories, in one direction by forgetting some of the
structure, and in the other by finding, so to speak adjointly, an enveloping algebra. A nice survey on J-rings was written by Henriksen in 1995 (see [50]), to which we refer the interested reader for
linkage to some of what follows in this survey. Historically, a lot of the credit for a revival of the theory of J-algebras points to the highly motivating Arkansas Lecture Notes by Luxemburg [67]
and the 1982 Ph.D. thesis of de Pagter [76], who systematically explored both the existing literature as well as new directions. Another impetus to research in the area of J-algebras derived from the
desire of Zaanen, who in the late seventies started to develop a program to prove many of the elementary results in the theory of Riesz spaces without using representation theorems for vector
lattices. This desire is directly linked with a preference not to use the Axiom of Choice unnecessarily. The present survey is intended as an update to the one by Huijsmans [57]. We hasten to point
out that we do not intend this survey to replace the one by Huijsmans, but rather that we think of it as augmenting part of it. Since [57] appeared, much progress has been made and several of the
problems explicitly phrased in
[57] have been solved. At the same time, some topics like ideal theory, connections between Riesz homomorphisms and algebra homomorphisms, and representations of f-algebras were absent in [57]. Thus
an update as well as a supplement was needed. However, some sections of [57] receive no attention at all in this survey. We have not included important topics like the role of f-algebras in positive
operator theory (e.g., we do not even include results on the previously mentioned averaging operators) and probability theory. We are rather focused on placing this update, as much as possible, in
the setting of spaces of functions, hoping to interest as large an audience as possible via this approach. Moreover, we feel that the great source of inspiration was and continues to be the beautiful
book by Gillman and Jerison [47], which certainly inspired and continues to inspire a large part of the research in lattice ordered algebras and rings. Last but not least we focus on what could be
called distortions of f-algebras, in particular on Archimedean almost falgebras and d-algebras. Though these distortions have the potential to be seen as aberrations by some, we believe they point
the way to techniques that are needed for the broader theory of lattice ordered algebras, as well as for illumination of various aspects in the theory of f-algebras. Note that the distortions
disappear if the lattice ordered algebra under consideration has a multiplicative identity which is a weak order unit. Indeed, such algebras are automatically f-algebras. It should be mentioned that
classes of lattice ordered algebras other than the ones that appear in this survey have been studied. Notably, the papers [85], [86], [87], and [88] by Steinberg discuss lattice ordered algebras in
which every square is positive, a class of algebras that includes all almost f-algebras. We also pay no attention at all to non-Archimedean lattice ordered algebras. Finally, we point out that
several results in this survey rely heavily on the (relative) uniform topology on Riesz spaces. In particular, since in Archimedean Riesz spaces uniform limits are unique, we shall include the
'Archimedean' property in the definition of uniformly complete Riesz spaces. A complete investigation of that topology can be found in Sections 16 and 63 of [69]. For terminology and concepts not
explained or proved in this survey we refer the reader to the standards books [2], [47], [69], [72], [92] and [93].
2. Definitions and elementary properties A (real) Riesz space A is called a lattice ordered algebra (briefly, an i-algebra) if A also is an algebra and the positive cone A+ = {J E A : f ~ O} is
closed under multiplication, that is, if f,g ~ 0 then fg ~ 0 (equivalently, if Ifgl :::; Ifllgl for all f,g E A).
We make the following blanket assumption: all Riesz spaces under consideration in this paper are assumed to be Archimedean (however, the latter blanket assumption has not stopped us to explicitly add
the word Archimedean to the list of conditions in various results below). After B.irkhoff and Pierce (see [22, p. 55]), we define an i-algebra A to be an f-algebra if for every f, 9 E A, the
condition f /\g = 0 implies Uh) /\g = (hi) /\g = 0 for all h E A+
holds. We call the E-algebra A an almost f-algebra after Birkhoff in [21, Section 6] if f /\ g
= 0 ill A implies
= O.
An E-algebra A for which f /\ 9 = 0 in A and h E A+ imply (.fh) /\ (gh)
(hf) /\ (hg) = 0
is called a d-algebra. The notion of d-algebra goes back to Kudlacek in [65]. Our focus in this survey on E-algebras is almost exclusively on f -algebras, almost falgebras and d-algebras. In this
paragraph, we recall some properties of f-algebras. Using the Axiom of Choice, Birkhoff and Pierce in [22, Theorem 13] proved that any f-algebra is commutative. A constructive proof of this fact, due
to Zaanen, can be found in [61, Theorem 2.1] or [92, Theorem 140.10]. All squares in an f-algebra are positive. Also, Ifgl = Ifllgl for all f,g in an f-algebra A. The multiplication by an element in
the f -algebra A is order continuous, i.e., if inf {fT : T} = 0 in A then inf {g fT : T} = o for all 9 E A +. Phrased more generally, the multiplication 7rf by an element f E A (7rf (g) = f 9 for all
9 E A) is an orthomorphism and all orthomorphisms are order continuous. Recall that an orthomorphism on a Riesz space L is an order bounded linear operator 7r such that 17r (f)I/\ Igl = 0 whenever
Ifl/\ Igl = 0 in L (the reader is referred to [2] or [92] for elementary properties of orthomorphisms). There is another important relationship between orthomorphisms and f-algebras, which we mention
next. Indeed, let Orth (L) be the set of all orthomorphisms on a Riesz space L. Under the operations and the ordering inherited from £b (L), the ordered algebra of all order bounded operators on L,
and under composition as multiplication, Orth (L) is an Archimedean f-algebra with the identity map h on L as unit element. The details of the facts recalled above can all be found in [2], [76] or
[92]. Next we present some properties of almost f-algebras. Almost f-algebras, like f-algebras, are commutative too. The latter fundamental property was first established by Scheffold in [80, Theorem
2.1] for almost f-algebras that are Banach lattices. Using both Scheffold's result and the Axiom of Choice, Basly and Triki were the first to prove commutativity for arbitrary almost f-algebras [10,
Thorme 1.1]. The first proof of the commutativity for almost f-algebras within ZermeloFraenkel set theory was given by Bernau and Huijsmans in their paper [13, Theorem 2.15]. Recently, a shorter
constructive proof was published in [34, Corollary 3] by Buskes and van Rooij. Another property of f-algebras holds for almost f-algebras, namely the positivity of squares. Also, if A is an almost
f-algebra then f2 = Ifl2 for all f E A. However, contrary to the order continuity of the multiplication in falgebra..'l, the multiplication by a fixed element in an almost f-algebra is not always
order continuous as is shown in the following example. EXAMPLE 2.1. Write A = C ([0,1]), the vector space of all real-valued continuous functions on [0, 1]. With respect to the pointwise ordering (i.
e., f :::; 9 in A if an only if f (x) :::; 9 (x) for all x E [0,1]), A is an Archimedean Riesz space. Define a multiplication • in A by
(f. g)(x)
f (1/2) 9 (1/2)
(0 :::; x :::; 1/2) ; (1/2 :::; x :::; 1)
lor all I, 9 E A. Then A is an almost I-algebm with respect to the multiplication •. For every natuml number n :::: 1, define the function In E A by (O :::; x :::; 1/2 - l/n) ; (1/2 - l/n :::; x :::;
1/2) ; (1/2:::; x :::; 1/2 + l/n) ; (1/2+1/n:::;x:::;1).
I f (:1") _ { n/2 - n:1: " ,-n/2 + nx 1
Then sup Un : n = 1,2, ... } exists in A and equals the function e defined bye (x) = 1 lor all x E [0,1]. On the other' hand,
(O :::; x :::; 1/2) ; (1/2:::;x:::;1)
lor all n E {I, 2, ... }, and clearly the set {e. In : n = 1,2, ... } does not have a supremum in A. We conclude that. is not order continuous. For more information about elementary theory of almost
f-algebras, the reader is encouraged to consult [13], [23], [34], and [35]. At this point, we turn our attention to some properties of d-algebras. It follows directly from the definition of
d-algebras that an i-algebra A is a d-algebra if and only if the multiplication map induced by any fixed element in A + is a Riesz (or lattice) homomorphism. It follows that a necessary and
sufficient condition for an i-algebra A to be a d-algebra is that the identity If gl = 1/IIgi holds for all f, 9 in A. Contrary to (almost) I-algebras, d-algebras need not be commutative nor have
positive squares. Next we give an example of a non-commutative d-algebra in which not all squares are positive. EXAMPLE
2.2. Let in this example A be the algebra of real {2 x 2)-matrices of
the form
with the usual addition, scalar multiplication, matrix product and partial or·dering. It is not hard to see that A is an Archimedean d-algebra. But A is not commutative and not all squares in A are
positive. Indeed, if p=
(~ ~)
and q =
(~ ~)
then pq
and qp = O.
Moreover, the square
is not positive. As for almost f-algebras, multiplication by a fixed element in a d-algebra is, in general, not order continuous. Point in case is the almost I-algebra that we considered in Example
2.1, which also is a d-algebra. Our main reference about d-algebras is [13]. In what follows, we look at some of the connections between the three kinds of i-algebras that we consider in this paper.
It is immediate that any I-algebra is both, an almost f-algebra and a d-algebra. Almost f-algebras need not be d-algebras as we see in the next example.
EXAMPLE 2.3. Take A as in Example 2.1, and let () E A be the function () (x) = { 1/2 - x x - 1/2
(0::; x ::; 1/2) ; (1/2::; x ::; 1).
For f, 9 E A, define (f
)( ) _ {
x -
/,1-X f(s)g(s)ds
(0::; x ::; 1/2);
(1/2::; x ::; 1).
() (x) f (x) 9 (x)
Then A is an almost f -algebra under the multiplication -. However, A is not a d-algebra. Indeed, let e, f in A be defined by e(x)=l and
f (x)
-4x + 1 4x-3
(0::; x ::; 1/2) ; (1/2::;x::;1).
If - el (0) = 0 and (If I- e) (0) = If I- Igl fails in A.
Observe that
If - gl =
If(s)1 ds -; O. Thus the property
Since every almost f -algebra is commutative and has positive squares, Example 2.2 shows that d-algebras need not be almost f-algebras. However, if a d-algebra A is commutative or has positive
squares then A is automatically an almost f-algebra [22, p. 60]. Summarizing part of the relations, we have the following diagram f-algebra::::} commutative d-algebra ::::} almost f-algebra. For more
detail, see [22], [13] and [57]. The next lines deal with nilpotent element in i-algebras. The set of all nilpotent elements in the i-algebra A is denoted by N (A). In other words,
N (A)
= {f E A : r = 0 for some n =
1, 2, ... } .
Given a natural number p, we define Np (A)
= {f
E A : fP
= O} .
The i-algebra A is said to be semiprime if 0 is the only nilpotent element in A, that is, if N (A) = {O} . If A is an f-algebra then the following equalities hold N (A)
= N2 (A) = {f E A
: fg
= 0 for all 9 E A}.
(see [76, Proposition 10.2] or [92, Theorem 142.5]). If A is an almost f-algebra then N (A)
= N3 (A) = {J E A
: fg2
= 0 for all 9 E A}
= {f E A : fgh = 0 for all g, hE A}
(see [51, Theorem 3.11]) and, as for f-algebras, N2 (A)
= {f
E A : fg
= 0 for
all 9 E A}
(see [23, Lemma 5.3]). If A is a d-algebra then N (A)
= N3 (A) = {f E A
: gfh
= 0 for all 9
• A}
(see [13, Theorem 5.5] or [29, Theorem 5]). However, and co we documented for f-algebras and almost f-algebras, in d-algebl
ry to what he equality
N2 (A) = {J E A: fg = 0 for all 9 E A} does not necessarily hold as is illustrated by the following example.
EXAMPLE 2.4. Take A as in Example 2.1 and define the multiplication. in A by (f. g)(x) = f (0) 9 (1) for all f, 9 E A. Let f E A be the function defined by f (x) = 1 - x for all x E [0,1]. Clearly,
f. f = 0 but f. e i= 0 where e E A is defined bye (x) = 1 for all x E [0,1].
Finally, note that any f-algebra with multiplicative identity is semiprime and any semiprime almost f-algebra or semiprime d-algebra is automatically an falgebra (see Section 1 in [13]). As a final
comment we remark that an l-algebra which has positive squares and has a multiplicative identity need not be an falgebra.
3. i-algebra multiplications in C (X) Let C (X) be the set of all real-valued continuous functions on a compact Hausdorff' topological space X. Under pointwise addition and scalar multiplication, C
(X) is a real vector space. Moreover, C (X) is an Archimedean Riesz space with respect to the pointwise ordering (i.e., f ~ 9 in C (X) if and only if f (x) ~ 9 (x) for all x E X). By defining the
multiplication in C (X) pointwise as well (Le., (fg) (x) = f(x)g(x) for all f,g E C(X) and all x E X), the space C(X) is easily seen to have the structure of an f-algebra with e as unit element,
where e (x) = 1 for all x E X. Now consider another associative multiplication. in C (X). The main topic of this section is to produce necessary and sufficient conditions for C (X) to be an f-algebra
(respectively, an almost f-algebra, a d-algebra) with respect to this new multiplication •. The first theorem in this direction goes back to Conrad (see [38, Theorem 2.2]), who obtained the
following. THEOREM 3.1. Let. be an associative multiplication in C (X). Then C (X) is an f -algebra with respect to • if and only if there exists a positive function w E C (X) such that (f.g)(x) =w
(x)f(x)g(x) for all f,g E C (X) and all x EX.
In fact, Conrad established the theorem above for any Archimedean f-ring with unit element. The representation formula given in Theorem 3.1 above was obtained in an alternative way by Scheff'old in
[80, Korollar 1.4]. While Conrad's proof is purely algebraic and order theoretic, the proof presented by Scheff'old relies on analytic tools like the Riesz representation theorem. With the same
analytic tools Scheff'old also obtained the following representation theorem for almost f-algebra multiplications in C (X) (see [80, Theorem 1.2]). THEOREM 3.2. Let. be an associative multiplication
in C (X). Then C (X) is an almost f -algebra with respect to • if and only if there exists a family (/Jx : x E X) of positive measures such that (f. g)(x) for all f, 9 E C (X) and all x E
f(s) 9 (s) d/Jx (s)
Since every commutative d-algebra is an almost I-algebra, the previous theorem remains valid for commutative d-algebra multiplications in C (X) as well. Recently in [25, Corollary 3.2], Boulabiar
proved the following representation formula for any (not necessarily commutative) d-algebra multiplication in C (X). THEOREM 3.3. Let. be an associative multiplication in C (X). Then C (X) is a
d-algebra with respect to • if and only if there exist (i) a positive function wE C (X), and (ii) functions h,k: X -+ X (continuous on coz(w) = {x EX: w(x)"! O}) such that (J. g) (x) = w (x) I (h
(x)) 9 (k (x)) for all I,g E C(X) and all x E X.
Notice that if C (X) is a d-algebra with respect to the multiplication. then • is commutative if and only if the functions hand k coincide on coz (w) (where 11" k and ware as in Theorem 3.3). The
latter observation yields. in addition to the formula cited in Theorem 3.2 above, another representation for commutative d-algebra multiplications on C (X) . More abstract versions of the results
above will be given in Section 8 below. 4. Multiplication by an element as an operator Let A be an (-algebra and recall that C b (A) denotes the ordered algebra of all order bounded operators on A.
For every f E A, we define the map 7rf on A by 7rf (g) = f 9 for all 9 E A. Clearly, 7rf is an order bounded operator on A for all lEA. The map p : A -+ Cb (A) defined by p (J) = 7rf for all I E A is
obviously an algebra homomorphism, that is, p (J g) = p (J) p (g) for all I, 9 E A. Hence the range p (A) of p is a subalgebra of Cb (A). In this section, we will see that if A is an almost f-algebra
then p (A) can canonically be equipped with an ordering, under which p (A) is an Archimedean I-algebra. A corresponding result will also be given for commutative d-algebras and f-algebras. Let A be
an almost I-algebra. Since
N2 (A) = {f E A : Ig = 0 for all 9 E A}, = 7r9 if and only if f by putting
9 E N 2 (A). This allows us to define an ordering on p (A)
(0) The ordering defined by (0) coincides with the ordering inherited from Cb (A), namely, 7rf is positive with respect to (0) if and only if 7rf is a positive operator on A. Under the usual addition
and composition of operators, and with the ordering defined by (0), p (A) is an Archimedean ordered subalgebra of Cb (A). In fact, we have the following theorem (see Theorem 4.2 and Theorem 4.4 in
[23]). THEOREM 4.1. Let A be an Archimedean almost I -algebra. Then p (A) is an Archimedean I -algebra with respect to the addition and composition of operators, and the ordering inherited from Cb
(A). The lattice operations in p (A) are given by 7r f
V 7r9
= 7rfV 9'
7r f 1\ 7r9
= 7rf /1.9
for all f, 9 EA.
In particular, (7rf)+
= 7rf+,
= 7rf-'
= 7rlfl
for all lEA.
In other words, p defines a surjective Riesz homomorphism from A onto p (A).
Theorem 4.1 of course holds in commutative d-algebras. Moreover, let A be a commutative d-algebra. For f E A and 9 E A + the equalities 17I'fl (g)
= 71'tft (g) = Iflg = sup{lfllhl : Ihl
::; g}
= sup {Ifhl : Ihl ::;g} = sup {17I'f (h)1 : Ihl ::;g} imply that 71'f has an absolute value in Lb (A), which coincides with its absolute value in p (A). We collect the latter observations for
commutative d-algebras. THEOREM 4.2. Let A be an Archimedean commutative d-algebra. Then p (A) is an Ar'chimedean f -algebra when equipped with the addition and composition of operators, and the
ordering inherited from Lb (A). Moreover, the absolute value 71'tft of 71'f in p (A) coincides with the absolute value of 71' f in Lb (A) for all f E A. that is, 17I'fl (g) = 71'tft (g) = sup {17I'f
(h)1 : Ihl ::; g} for all 9 E A+.
We obtain the f-algebra case as a corollary. COROLLARY 4.3. Let A be an Archimedean f-algebra. Then p (A) is an fsubalgebra of the Archimedean f-algebra Orth (A) of all orthomoTphisms of A.
The fact that the range of p in Corollary 4.3 is an f -algebra was first proved in [22, Corollary 3, p. 57] by Birkhoff and Pierce, while the fact that Orth (A) itself is an f-algebra has been proved
in [18] by Bigard and Keimel and in [39] by Conrad and Diem. This topic was also discussed in great detail by de Pagter in his thesis [76, Proposition 12.1]. Note that if A is an f-algebra then A is
semiprime if and only if p is one-to-one as a map from A into Orth (A). In this case, A and p (A) are isomorphic a.'! f-algebras. Also, if A is an f-algebra then A has a multiplicative identity if
and only if the map p is one-to-one and onto as a map A ---> Orth (A), and consequently A and Orth (A) are isomorphic as f-algebra.'!.
5. Uniform completion and Dedekind completion Let A be an Archimedean i-algebra. The closure A1'U of A in its Dedekind completion A8 with respect to the uniform topology is a uniformly complete Riesz
space. Using Quinn's Definition 2.12 in [79], A1'U is the uniform completion of A. The following theorem was obtained by n'iki in [91]. THEOREM 5.1. Let A be an Archimedean f-algebra (respectively,
almost falgebra, d-algebra, f -algebra). Then the multiplication in A extends uniquely to a multiplication in A1'U such that A1'U is a uniformly complete i-algebra (respectively, almost f -algebra,
d-algebra, f -algebra) with respect to this extended multiplication. Moreover, if A is semiprime (respectively, has a unit element e) then ATU is semiprime (respectively, has e as unit element).
We now turn our attention to the Dedekind completion of the i-algebra A. Johnson in his paper [64] proved that if A is an f-algebra (or even an Archimedean f-ring), then the multiplication in A
extends uniquely to an f-algebra multiplication in A8. The uniqueness of such an extended multiplication in A8 of course arises from the order continuity of the multiplication in the f-algebra A.
Alternative proofs of this extension can be found in [76, pp. 66-67] and [59, p. 166].
THEOREM 5.2. Let A be an Archimedean f -algebra. Then the multiplication in A extends uniquely to a multiplication in A" such that Ad 'is a Dedekind complete falgebra with respect to this extended
multiplication. FUrthermore, if A is semiprime (respectively, A has a unit element e) then A" is semiprime (respectively. has e as unU element).
The corresponding results for (I-algebras in general, or almost f-algebras and d-algebras in particular, is much harder because of the absence of order continuity of the multiplication. Nonetheless,
extensions of the multiplication to the Dedekind completion often exist, though such extensions are no longer necessarily unique. For almost f-algebras Buskes and van Rooij proved the following (see
[35, Theorem 10]). THEOREM 5.3. Let A be an Archimedean almost f-algebra. Then the multiplication in A extends to a multiplication 'in A" such that A" is a Dedekind complete almost f -algebra with
respect to that extended m'ultiplication.
Using the previous result as a starting point, Boulabiar and Chil in [28, Corollary 3] proved that from amongst the extensions provided, A" can be equipped with a commutative d-algebra multiplication
whenever A is a commutative d-algebra. Then in [37, Theorem 7], Chil wa.c; able to drop the commutativity condition and prove the following theorem. THEOREM 5.4. Let A be an Archimedean d-algebra.
Then the multiplication in A extends to a multiplication in Ad such that A" is a Dedekind complete d-algebra with respect to that extended multiplication.
In summary, all but one of the problems concerning Dedekind completions that Huijsmans raised in his survey paper [57] have now been solved. The remaining problem, though admittedly outside the scope
of this survey, is the following. PROBLEM 5.5. Let A be an Ar'chimedean (I-algebra. Does the multiplication in A extend to a multiplication in A" so that A" is a Dedekind complete (I-algebra?
6. Powers in i-algebras Let A be a uniformly complete i-algebra and let P E lR+ [Xl, ... , Xn] be a homogeneous polynomial of degree a non zero natural number p. In their paper [16]' Beukers and
Huijsmans considered the following problem: does there exist in A a 'p-th root' of P(iI, ... ,fn) for iI, ... ,fn in A+? They gave an affirmative answer in the case where A is a semiprime f-algebra.
More precisely, they prowd the following theorem (see [16, Theorem 5]). THEOREM
6.1. Let A be a uniformly complete semiprime f-algebra and let
P E lR+ [Xl, ... , Xn] be a homogeneo1Ls polynomial of degree a non zero natural numbe1'p. Thenfo1' every fl, ... ,fn E A+ there exists a 'unique f E A+ such that fP = P (iI, ... , fn). As a
consequence, one has the following corollary (see Corollary 6 in [16]). COROLLARY 6.2. Let A be a uniformly complete f -algebra with unit element and p E {I, 2, ... }. Then for each f E A +, there
exists a unique 9 E A + such that
= f.
Note that the previous result was first proved for p = 2 in [17, Theorem 4.2 and Cororllary 4.3] by Beukers, Huijsmans and de Pagter. Also, it should be noted that Theorem 6.1 above is proved
alternatively by Buskes, de Pagter, and van Rooij in [31, Corollary 4.11], a paper that deals with a more general functional calculus on Riesz spaces and f-algebrar; to which we will return in the
next section. The problem corresponding to Theorem 6.1 for almost f-algebras was considered by Boulabiar and follows next (see Theorem 3 in [24]). jR+
THEOREM 6.3. Let A be a uniformly complete almost f -algebra and let P E [Xl, ... , Xn] be a homogeneous polynomial of degree a natural number p. Then
for every II, ... , fn E A+ there exists a (not necessarily unique) f E A+ such that fP = P (II, ... , fn)·
Observe that, where roots are unique in semiprime f-algebras, this is no longer always the case for almost f-algebras. We illustrate this with an example. EXAMPLE 6.4. Let A = C ([-1,1]) be the
uniformly complete Riesz space of all real-valued continuous functions on [-1, 1] and define w E A by
W(x)={ O-x
(-1::;x::;0); (0::;x::;1).
For every f, g E A, we put
(f. g) (x) =
(-1::; x::; 0);
(0::; x::; 1).
Then A is an almost f -algebra with respect to the multiplication.. h, g, Ct, {3 E A defined by g(x) =
h (x)
= exp (x),
and Ct
= Jx 2 + exp (2x)
, and {3 (x)
for all x E [-1,1]' where X[O,I] (x) Then Ct • Ct
= X.X[O,I] (x) + Jx 2 + exp (2x)
if x E [0,1] and X[O,I] (x)
if x E [-1,0).
= {3 • {3 = 9 • g + h • h.
At this point, we define for each non zero natural number p, Ap =
{II ... fp : II, ... , fp
E A}.
In what follows, we will investigate the order structure as well as the algebra structure of Ap (since Al = A, we suppose that p ~ 2). The sets A2 and A3 were first considered in [35] by Buskes and
van Rooij and then in great detail by Boulabiar in [24] from which we summarize the results in the following theorem (see Theorems 4, 5, and 6 in [24]). THEOREM 6.5. Let A be a uniformly complete
almost f -algebra and let p ~ 3 be a natural number. Then Ap is a uniformly complete semiprime f -algebra under the ordering and multiplication inherited from A. The positive cone At of Ap is defined
The lattice operations I\p and V p in Ap are given by
fP I\p gP = (f 1\ g)P and the absolute value
1.l p
Vp gP
= (f V g)P
for all 0 ~ f,g E A,
in Ap is defined by If Pip
Ifl P
for all f E A.
Contrary to Ap (p 2: 3), A2 need not be a Riesz space under the ordering inherited from A as is proved by the next example. EXAMPLE 6.6. Consider A = C ([0,1]) with the pO'intwise addition, scalar
rrmltiplication and partial ordering. For f, 9 E A, define
(f _ g)(x) = {
(0 x - 1/2
x ~ 1/2);
f(s)g(s)ds (1/2 < x ~ 1). o Then A is a uniformly complete almost f -algebra under the multiplication - and h is an element of A2 if and only if h(x) = 0 for all x E [0,1/2] and the restriction of h
to [1/2,1] belong to C 1 ([1/2,1]). Hence A2 is not a Riesz space under the order inherited from A.
The following example proves that though Ap (p 2: 3) is a Riesz space, in general it is not a Riesz subspace of A. EXAMPLE 6.7. Take A = C ([-1, 1]) with the pointwise addition, scalar multiplication
and ordering, and define w EA· by
w (x)
{ -x 0'
(0 ~ x ~ 1) .
For f, 9 E A, define
(f - g) (x) =
Clearly, A is a uniformly complete almost f -algebra under the multiplication _. Define 0 E A by o (x)
= 2x + 1
for all x E [-1, 1].
It follows that
10 - 0 - 01 (1) =
=110 - 0 - ob (1) = (101-101-101) (1) =
If, however, A is a commutative d-algebra then some of the unpleasantness of the preceding example disappears. COROLLARY 6.8. Let A be a uniformly complete commutative d-algebra and p 2: 2 be a
natural number. Then Ap is a uniformly complete f -subalgebra of A. If in addition p 2: 3 then Ap is semiprime.
In spite of the improvement in the conclusion of Corollary 6.8 over the conclusion for the more general situation of almost f-algebras, A2 still need not be semiprime. This is illustrated in the next
EXAMPLE 6.9. Let A be the coordinatewise ordered vector space R3 with the multiplication defined by:
n; R}
Then A is a uniformly complete commutative d-algebm and
X,Y E
Dbuiou,ly, A, is an f-""balg,bm of A. FUrth,""ore, (
~ ) ' ~ 0 and A, is not
semiprime. For f -algebras we have the following corollary. COROLLARY 6.10. Let A be a uniformly complete f -algebm and let p 2: 2 be a natuml number. Then Ap is a uniformly complete semiprime f
-subalgebm of A.
There is a universal way in which A 2 , or more generally Ap for any p 2: 2 can be described. We provide the details of that description for A2 next (see [36]). Let E and F be Riesz spaces. A
bilinear map q> : E x E ~ F is called orthosymmetric if whenever f I\g = 0 for f,g E E we have q>(f,g) = 0 (the notion of orthosymmetric bilinear map was introduced by Buskes and van Rooij in [34]).
The bilinear map q> is a Riesz bimorphism if it is a Riesz homomorphism in each variable separately (more about Riesz bimorphisms can be found in [29]). Let E be a Riesz space. The pair (E8, 8) is
called a square of E, if E8 is a Riesz space and if (1) 8: E x E ~ E8 is an orthosymmetric Riesz bimorphism, and (2) for every Riesz space F, whenever q> : E x E ~ F is an orthosymmetric Riesz
bimorphism there exists a unique Riesz homomorphism q>8 : E8 ~ F such that q>8 08 = q>. The existence and uniqueness of squares for any Riesz space follows easily from the Riesz space tensor product
as constructed by Fremlin in [43]. To understand the structure of the square of a Riesz space is best not done via this tensor product. The set A2 described above is often more helpful. The
connection between semi prime f-algebras and squares of uniformly complete Riesz spaces is described in the next theorem. After reading that theorem the reader might feel like moving the lower index
2 in A2 to an upper index. THEOREM 6.11. Let E be a uniformly complete Riesz subspace of an Archimedean semiprime f -algebm G whose multiplication is indicated by a period e. Put E2 := {x e y : x,y E
E} as before. Then E2 is a Riesz subspace of G and (E2,e) is a square of E.
7. Functional Calculus on f-algebras The theorem that we presented in Section 6 on the existence of p-th roots of homogeneous polynomials in f-algebras is a very special case of a rich functional
calculus on uniformly complete f-algebras. The idea behind functional calculus
for Riesz spaces in general is straightforward. For elementary functions on JRN one ought to be able to simply substitute elements of the Riesz space into these functions and get elements of the
Riesz space as output. The idea of how to execute this substitution of elements in sufficiently simple functions essentially goes back to Yudin and in the form that we represent it to Lozanovsky
[70]. The technical problem surmounts to what the class of sufficiently simple functions really looks like. Let n EN. We denote by 1i(JRN) the Riesz space of all continuous functions r.p : JRN ---+
JR for which
r.p(tx) = tr.p(x) for all x E JRN and all t ~ O. Let E be a Riesz space, r.p E 1i(JRN) and
iI, ... , fn
E E. We say that
r.p(iI, ... '/n) exists in E if there is an element 9 of E such that
w(g) = r.p(w(iI), ... ,w(fn)) for every real-valued Riesz homomorphism w on the Riesz subspace of E generated by iI, ... , f n, g. For any given E, r.p and iI, ... ,/n there exists at most one 9 with
this property. This 9 is also indicated by
r.p(iI, ... , fn). In this situation we have the following theorem (see Lozanovsky [70]). THEOREM 7.1. Let E be a uniformly complete Riesz space and Then r.p(iI, ... , fn) exists fOT every r.p E 1i
(JRN). The map
iI, ... , fn
r.p(iI, ... , fn) (r.p E 1i(JRN)) is a Riesz homomorphism from 1i(JRN) into E. r.p
Remark. In a way, r.p(iI, ... , fn) is independent of E. Indeed, if D is any Riesz subspace of E that is uniformly complete and contains iI, ... , fn then r.p(iI, ... , fn) relative to D means the
same as r.p(iI, ... , fn) relative to E. In particular, every Riesz subspace of E that is uniformly complete and contains iI, ... , fn must also contain r.p(fl, ... , fn). By A(JR N) we denote the
set of all continuous functions r.p : JRN ---+ JR that are of polynomial growth and for which limt!o rlr.p(tx) exists uniformly on bounded subsets of JRN (the latter condition is equivalent to the
existence of a 't/J E 1i(JRN) such that r.p(x) = 't/J(x) + 0(11 x II) (x ---+ 0)). Observe that A(JRN) is an f-algebra. Let E be a semiprime f-algebra, r.p E A(JRN) and iI, ... , fn E E. We say that
r.p(iI, ... ,fn)exists in E if there is agE E with
w(g) = r.p(w(iI), ... ,w(fn)) for every real-valued multiplicative Riesz homomorphism w defined on the f-subalgebra of E generated by iI, ... , fn,g. There exists only one such g, which is then
called r.p(iI, ... , fn). This definition is in accordance with the one we gave for 1i(JRN) if r.p E 1i(JRN). For 1i(JRN) we have the following theorem (see [31, Theorem 4.10]).
ft, ... , fn
7.2. Let E be a uniformly complete semiprime f-algebra and let
E E. Then cp(ft, ... , fn) exists for every cp E A(~JII). The map
cp-+cp(ft, ... ,fn) (cpEA(~JII)) is a multiplicative Riesz homomorphism from A(~JII) into E.
8. Relationships between i-algebra multiplications
Let A be an i-algebra with multiplication denoted by juxtaposition, and assume that A is equipped with another associative multiplication e. In the first theorem of this section, we present a
relationship between the two multiplications in A, under the conditions that A is a unital f-algebra with respect to the initial multiplication and an (almost) f-algebra with respect to the other
multiplication e. For proofs, see [38, Theorem 2.2] and [23, Theorem 5.2]. THEOREM 8.1. Let A be an Archimedean f -algebra with identity element e and assume that A is furnished with another
associative multiplication e. Then (i) A is an f-algebra under e if and only if
for allf,g E A, and
(ii) A is an almost f-algebra under e if and only 'if f e9
= e e (I g)
for all
J, 9 E A.
The corresponding problem in the case where A is d-algebra with respect to e is rather more difficult. Indeed, since then e need not be commutative, one cannot write the product f e 9 as a function
of the product fg (I, 9 E A). However, there exists another way (involving f, 9 and the initial multiplication in A) to express the product f e g. This is the subject of the next result. First recall
that the maximal ring of quotients Q (A) of the Archimedean f -algebra A with unit element e is again an Archimedean J-algebra with the same e as multiplicative identity. Moreover, A is an
f-subalgebra of Q (A), a fact proved by Anderson in [4] (see also the recent paper [71, Cororllary 2.7.1] by Martinez). For the definition of the maximal ring of quotients of a ring, the reader can
consult e.g. [66]. The proof of the following theorem can be found in [25, Theorem 4.3]. THEOREM 8.2. Let A be an Archimedean f-algebra with 'identity element e and let e be another associat'ive
multiplication in A. Then A is a d-algebra with respect to e if and only if there exist two algebra and Riesz homomorphisms cp and 'l/J from A into its maximal ring of quotients Q (A) such that
f e 9 = (e e e) cp (I) 'l/J (g)
for all f, 9 E A.
As mentioned at the end of Section 3, the two preceding theorems are abstract versions of the corresponding results, given in that section, for the C (X)-case. In the second part of this section, we
are interested in A being a commutative dalgebra with respect to the initial multiplication rather than an f-algebra with unit element. However, we will impose the additional assumption that A is
uniformly complete. Uniform completeness is not needed for all our results but we will use the set A2 and remind the reader of the special nature of that set under the extra condition of uniform
completeness (see Corollary 6.8). If there exists a positive operator T OIl A2 such that
f e9 =
T (lg)
for all
f, 9 E
then e is an almost f-algebra multiplication and N2 (A) c N; (A), where
N; (A)
= {f
E A :
f e f = O} .
In what follows, we show in detail what happens if we assume that A with the initial multiplication is a (uniformly complete) commutative d-algebra, and the inclusion N2 (A) c N; (A) holds. Under
those circumstances, we then relate a necessary and sufficient condition for the new multiplication to be an almost f-algebra, a d-algebra or an f-algebra to the existence of some posit.ive operator
T satisfying the relation (T). The details follow in the next theorem, the proof of which can be found in [23, Theorems 5.4 and 5.5]. THEOREM 8.3. Let A be a uniformly complete commutative d-algebra
and assume that A is an I!-algebra with respect to another associative multiplication e such that p = 0 implies f e f = O. Then the following statements hold.
(i) A is an almost f -algebra under
e if and only if ther-e exists a positive operator T from A2 into A such that
(ii) A is a commutative d-algebra under
e if and only if there exists a Riesz homomorphism T from A2 into A such that
f e 9 = T (fg)
for all f, 9 E A.
(iii) A is an f-algebra under e if and only if there exists an operator T from A2 into A such that To 7rf E Ort.h (A) for all f E A+, where 7rf (g) = fg for all 9 E A, and f e9
(f g)
for all f, 9 E A.
We remark that A2 in the previous theorem is a Riesz space (see Section 6). Next, we produce an example which shows that in Theorem 8.3 above, the hypothesis 'A is a commutative d-algebra' cannot be
replaced by 'A is an almost f-algebra'. EXAMPLE 8.4. Take A = C ([-1,1]) with the usual operations and order and define a, (3 E A by
-(4x+l) a(x)= { 0 4x -1
(-I~x~-1/4); ~ x ~ 1/4); (1/4 ~ x ~ 1)
and (3 (x) = {
~4x + 1
(-1 ~x~ 1/4); (1/4 ~ x ~ 1).
For f, 9 E A, define a(x)f(x)g(x) (f x g)(x) =
(-1~x~1/4); ~ 3/4);
(1/4 ~ x °J(X-3/4)
and (feg) (x) =f3(x)f(x)g(x)
for all x E [-1,1]. Then A is an almost f-algebra (respectively, an f-algebra) with respect to the multiplication x (respectively, .). It follows that N; (A)
= N 2x
= {f
E A : f (x)
= 0 for
all x E [-1, I/4]}.
Consider <.p, w E A, defined by
"'(X)~{ and oJ (x)
-3x-I x -3x+I
(-1 ::=;x::=; -1/4); (-I/4::=; x::=; 1/4); (i/4::=;x::=;I)
0 4x+ 1 { -4x+I 0
(-I::=;x::=;-I/4); (-I/4::=; x::=; 0); (0::=;x::=;I/4); (i/4::=;x::=;I).
Then and 'Pxw=O 'P. W =f:. O. Therefore, there is no operator T satisfying the condition f • g = T (f x g)
for all f, 9 E A.
We notice that the kind of relationships between two f-algebra multiplications via an operator T as above were first studied in [35] by Buskes and van Rooij. In particular part (i) of the theorem
above has its origins in [35, Theorem 1], where a general representation theorem for almost f-algebras (without any extra conditions) was proved by first dividing out the radical and then showing the
existence of an operator T a.'l above on the square (see the end of Section 6 above) of the resulting f-algebra, thus giving quantified credence to the almost part in the name almost f-algebras. They
also used this representation to discover that the Dedekind completion of an almost f-algebra is an almost f-algebra and to prove (CSP) for almost f-algebras. Finally, we should point out that the
Buskes-van Rooij representation theorem for almost f-algebras itself is, in a way, an abstract reformulation of Theorem 3.2 by Scheffold that we discussed earlier. We end this section with the
following problem, left open in the above discussion. PROBLEM 8.5. Let A be a uniformly complete commutative d-algebra and assume that A is a non comm'utative d-algebra with respect to another
multiplication •. Does there exists a relationship between the two multiplications in A?
9. Connection between algebra and Riesz homomorphisms Since in any f-algebra there is an order structure as well as an algebra structure, it is natural to compare the operators that preserve the
lattice operations, namely the Riesz homomorphisms, with those that preserve the algebra structure, the algebra homomorphisms. Such a comparison was initiated by Ellis [41], who considered the
problem for operators between spaces of continuous functions on compact Hausdorff spaces, and his central result is then one of equivalence: a Markov operator (Le., a positive operator preserving the
identity) between such spaces is a Riesz homomorphism if and only if it is an algebra homomorphism. Some years. after Ellis's paper was published, Hager and Robertson presented in [48] a more
abstract version of Ellis's theorem. More precisely, Hager and Robertson proved that any
Riesz homomorphism between two Archimedean f-algebras with unit elements that preserves the identity is an algebra homomorphism. Later, van Putten established the converse of the result by Hager and
Robertson in his thesis [78]. The aforementioned results were generalized by Huijsmans and de Pagter in [59, Theorem 5.4] as follows. THEOREM 9.1. Let A be an Archimedean f-algebra with unit element
e, B be an Archimedean semiprime f-algebra. and T : A ---> B be a positive operator. Then the following are equivalent (i) T is an algebra homomorphism. (ii) T is Riesz homomorphism with (Te)2 = Te.
Notice that the previous theorem generalizes all of the facts cited above. In the same paper [59], Huijsmans and de Pagter proved that., if A is an Archimedean falgebra with unit element and B is an
Archimedean semiprime f-algebra then any order bounded algebra homomorphism from A into B is automatically a Riesz homomorphism [59, Theorem 5.3]. Very recently in [91, Theorem 4.3], Triki obtained
this result in the more general setting of almost f-algebras. THEOREM 9.2. Let A be an Archimedean almost f-algebra and let B be an Archimedean semiprime f -algebra. Then any order bounded algebra
homomorphism from A into B is a Riesz homomorphism.
Since any conmmtative d-algebra is automatically an almost f-algebra, Theorem 9.2 holds if one replaces 'A is an Archimedean almost f-algebra' by 'A is an Archimedean commutative d-algebra'. However,
in the recent. work [89], Toumi proved that we have the same conclusion even if the d-algebra under consideration is not commutative. THEOREM 9.3. Let A be an arbitrary Archimedean d-algebra and let
B be an Archimedean semiprime f-algebra. Then any order bounded algebra homomorphism from A into B is a Riesz homomorphism.
Even for f-algebras, the condition of order boundedness in Theorems 9.2 and 9.3 cannot be dropped as is shown in the following example. EXAMPLE 9.4. Consider the set A of all real sequences u = {un}
n>l for which there exists a polynomial Pu E IR [X] and a natural number N such that Un = Pu (n) for all n 2: N. Under the usual operations and partial ordering, A is an Archimedean f-algebra (and
thus an almost f-algebra and a d-algebra) and A is not relatively uniformly complete. Define the algebra homomorphism T : A ---> IR by T(u) = Pu (-1) (u E A). For every A E [l,+po[, we define u), =
{U),.n}n~l E A by
u)',n =
{ 0An
(n ~ A);
(n> A) .
lfu = {n 2 } n>l then 0 ~ u), ~ u for all A E [1, +00[. Observe now that T (1L),) Therefore T 1:s not order bounded and hence not a Riesz homomorphism.
= -A.
In [59, Theorem 5.1], Huijsmans and de Pagter further illustrated the close ties between Riesz homomorphisms and algebra homomorphisms on f-algebras as follows. If A and Bare semiprime f-algebras and
A is uniformly complete, then any algebra homomorphism from A into B is a Riesz homomorphism. Their theorem was generalized by Boulabiar in [27, Theorem 3] to the setting of almost f-algebras.
THEOREM 9.5. Let A be a uniformly complete almost f -algebra and let B be an Archimedean semiprime f-algebra. Then any algebra homomorphism from A into B is a Riesz homomorphism.
Trivially, one can replace in Theorem 9.5 above 'almost f-algebra' by'commutative d-algebra'. The case of d-algebras that are not necessarily commutative was considered by Toumi in [89, Theorem 3].
THEOREM 9.6. Let A be an arbitrary uniformly complete d-algebra and let B be an Archimedean semiprime f-algebra. Then any algebra homomorphism from A into B is a Riesz homomorphism.
The condition that A is uniformly complete is not redundant in Theorems 9.5 and 9.6, even if A is an f-algebra. Indeed, A in Example 9.4 above is not uniformly complete. The following theorem by
Triki in [91, Theorem 4.4] is an f-algebra version of the well known Nagasawa's theorem (see [74, Theorem 1]). First, recall that an operator T between two unital f-algebras A and B is said to be
contractive if IT (J)I ~ eB in B whenever If I ~ eA in A, where eA and eB are the unit elements of A and B, respectively. If moreover T is bijective and the inverse T- 1 of T is also contractive then
we say that T is bicontractive. THEOREM 9.7. Let A and B be Archimedean f-algebras with identity elements eA and eB, respectively. For an order bounded bijection T : A ----t B such that T (eA) = eB,
the following are equivalent. (i) T is an algebra homomorphism. (ii) T is a Riesz homomorphism. (iii) T is bicontractive.
The last result of this section again deals with operators between two unital falgebras that preserve identities, so called Markov operators. Denote by M (A, B) the set of all Markov operators from
an f-algebra A with unit element eA into an f-algebra B with unit element eB, that is, M (A,B)
= {T:
B: T linear positive with T(eA)
= eB}.
Obviously, M (A, B) is a convex set. In [77, Theorem 2.1], Phelps proved that a Markov operator T from C (X) into C (Y) (where X and Yare compact Hausdorff topological spaces) is an algebra
homomorphism if and only if T is an extremal point in M (C (X) ,C (Y)). This connection to extreme points predates the paper by Ellis and we refer the reader to Arens and Kelley [8] and A. and C.
Ionescu Thlcea [62]. Combining the aforementioned Phelps's theorem and Ellis's result cited above we get that for T in M (C (X), C (Y)) the following are equivalent. (i) T is a Riesz homomorphism.
(ii) T is an algebra homomorphism. (iii) T is an extremal point in M (C (X) , C (Y)). As a generalization, van Putten in his thesis obtained the following result [78, Theorem 18.8]. THEORSM 9.8. Let
A and B be Archimedean f-algebras with unit elements, and let T E M (A, B). Then the following are equivalent. (i) T is a Riesz homomorphism.
(ii) T is an algebra homomorphism, (iii) T is an extremal point in M (A, B). An elementary proof of van Putten's theorem above, due to Huijsmans and de Pagter, can be found in [59, Theorem 5.7].
10. Positive derivations We recall that an operator D on a commutative algebra A is said to be a derivation if for all f,g EA. D (1g) = f D (g) + gD(1) In this section, we investigate positive
derivations on f-algebras as well as on almost f-algebras. Positive derivations on f-algebras were first considered in great detail by Colville, Davis and Keimel in [40]. Their main result provides
the following necessary and sufficient condition for a positive operator on an f-algebra to be a derivation (see Theorem 5 in [40]), where we recall that if p is a non zero natural number, and A is
an algebra then Ap = {11 ... fp : 11, ... , fp E A}. THEOREM 10.1. Let A be an f-algebra and D be a positive operator on A. Then D is a derivation if and only if
D (1) = 0 for all f E A2
D (1)2 = 0 for all f E A.
Positive derivations on f-rings were considered by Henriksen and Smith in [54]. It straightforwardly follows from Theorem 10.1 above that, if A is in addition semiprime, then there exist no non
trivial positive derivations on A. We turn our attention now to positive derivations on almost f-algebras. The result corresponding to Theorem 10.1 for almost f-algebras is the following (for a
proof, see [26, Theorem 3]). THEOREM
10.2. Let A be an almost f-algebra and D be a positive derivation
on A. Then D (1)
= 0 for
all f E A3
D (1)3
= 0 for
all f E A.
Contrary to f-algebras, Theorem 10.2 does not produce a characterization of positive derivations on almost f-algebras. Also, the third power in Theorem 10.2 is the best possible. The next example
illustrates these facts. EXAMPLE 10.3. Consider A the Cartesian product IR x IR with coordinatewise addition, scalar multiplication and ordering. Define the multiplication. on A by
for all 0,/3,0',/3' E R
(0,/1). (0',/3') = (0,00')
Then A is an Archimedean almost f -algebra with respect to.. Observe that A3 fA satisfies
{o}. Hence the identity map fA
(1) = 0 for all f E A3
(1) 3 = 0 for all f E A,
but fA is not a derivation on A. On the other hand, let D be the positive operator defined on A by for all a, /3 E R D(o,/3) = (0,2/3) Clearly, D is a derivation on A. However, D ((1, 0) • (1,0))
= (0,2)
D (1, 0) • D (1, 0) = (0,1) =I- (0,0) . Theorem 10.2 above of course holds for commutative d-algebras as well. Moreover, for commutative d-algebras the third power is the best possible too. Indeed,
the almost f-algebra considered in Example 10.3 is in fact a commutative d-algebra.
11. Cauchy-Schwarz inequalities We say that an i-algebra A possesses the Cauchy-Schwarz property (abbreviated as (CSP)), if for every vector space V and every bilinear map 'l/J : V x V -> A such that
(i) 'l/J is symmetric, and (ii) 'l/J (I, f) E A+ for all f E V, we have (CSI)
'l/J(I,g)2 S'l/J(I,f)'l/J(g,g)
for all f,g E V
In [61, Corollary 3.5], Huijsmans and de Pagter proved that any Archimedean semiprime f-algebra has (CSP). Later, Bernau and Huijsmans in [15, Theorem 2.6] generalized (CSP) to arbitrary Archimedean
f-algebras. THEOREM 11.1. Let A be an Archimedean f-algebra, V a vector space and 'l/J : V x V -> A a bilinear map such that (i) 'l/J is symmetric, and (ii) 'l/J (I, f) E A+ for all f E V . Then
'l/J (I, g)2 S 'l/J (I, f) 'l/J(g, g)
for all f,g E V.
Some years after Theorem 11.1 was published, Buskes and van Rooij established the corresponding inequality for Archimedean almost f-algebras and therefore for Archimedean commutative d-algebras (see
[34, Corollary 4]). THEOREM 11.2. Let A be an Archimedean almost f-algebra, V a vector space and'l/J : V x V -> A a bilinear map such that (i) 'l/J is symmetric, and (ii) 't/J (I, f) E A + fOT all f
E V . Then
'1/) (I, g)2 S .t/J (I, f) ,¢'(g, g)
for all f, g E V.
An alternative proof of the previous inequality was given by Boulabiar in [23, Theorem 3.9]. Not every d-algebra A has (CSP). An example illustrating this situation is the following. EXAMPLE 11.3.
Let A be the set of all real-valued functions defined on [0,1] equipped with the usual operations and order. Consider the multiplication. defined in A by (I )() - { f (0) g (1) (0 S x S 1/2) ; .g x f
(l)g(l) (1/2SxS1)
for all f, g E A. Then A is an Archimedean non-commutative d-algebra with r-espect to the multiplication •. At this point, let V be the f-algebra C ([0,1]) of all real-valued continuous functions on
[0, 1] , provided with the pointwise addition, multiplication, scalar multiplication and partial ordering. We define a positive operator T from V into A by T (I) (x) = f (1 - x) for all f E V and x E
[0,1]. Finally, let f E V such that f (0) = 2, f (1) = 1 and e E B such that e (x) = 1 for all x E [0,1]. Then (T (Ie) • T (Ie)) (0)
= (T (I) • T
(I)) (0)
(0) f (1)
and Hence the inequality
T (fe). T (fe) :s T (J2) • T (e 2) does not hold in A. By defining '!/J(f,g) = T(fg) (f,g E V), we find a bilinear map V x V --t A for which 1jJ (f, e)2 :s 1jJ (f, f) 1jJ(e, e) does not hold. Though
(CSP) does not hold for all non commutative d-algebras, Boulabiar and Toumi proved, in essence, the following variant of (CSP) in such algebras (see [29, Theorem 6]). THEOREM 11.4. Let V be a vector
space and let A be an Archimedean d-algebra. Consider a bilinear map '!/J : V x V --t A such that (i) 1jJ is symmetric, and (ii) 1jJ (f, f) E A+ for all f E V. Then
and hence 11jJ(f,g)21:s (1jJ(f,f)'!/J(g,g)) V (l/J(g,g)'!/J(f,f)) for all f,g E V. 12. Order biduals We refer the reader to [2] for terminology and notations not explained below. For an Archimedean
C-algebra A, the order dual is denoted by A~ and the order bidual is denoted by A~~. Recall here that the order dual A~ of A is the Dedekind complete Riesz space of all real-valued order bounded
functionals on A. A multiplication can be introduced in A~~ in three steps as follows: for all u, v E A, f E A~ and cp, '!/J E A~~, we define f.u E A~, '!/J.f E A~ and cp.1jJ E A~~ by the following
(f:u) (v) = f (uv)
('!/J.f) (u) = '!/J (f.u)
(cp.'!/J) (f) = cp ('!/J.f)
The multiplication defined by the equation (3) is called the Arens multiplication in A~~ (see [6] and [7]). The order continuous order bidual (A~);: of A is the projection band of all order
continuous elements in A~~. Moreover, (A~);: is closed under the Arens multiplication, that is, cp ..l/J E (A~);: whenever cp, '!/J E (A~);: . The following theorem was established by Huijsmans and
de Pagter (see [60, Theorem 4.1]). 12.1. Let A be an Archimedean C-algebra. Then A~~ (and hence is a Dedekind complete C-algebra with respect to the Arens multiplication.
If the f-algebra A has, in addition, a unit element e then E (A~);: (defined by e(f) = f (e) for all f E A~) is the unit element of A~~. Generally, A~~ need not be commutative even if A is
commutative. An example was provided by Arens in [7]. However, commutativity does carryover from A to (A~);:, which was first established for normed C-algebras by Scheffold in his paper [82] and
generalized by Grobler in [46, Theorem 4] to the more general setting of Archimedean C-algebras.
THEOREM 12.2. Let A be an Archimedean i-algebra. Then the order continuous (A~);: , provided with the Arens multiplication, is commutative.
orde1' bidual
The first results concerning the order bidual of f-algebras are due to Huijsmans and de Pagter (see [60]). They proved, among other results, that if A is an f-algebra with unit element such that A~
separates the points of A (i.e., if u E A and f (u) = 0 for all f E A~ then u = 0) then A~~ = (A~);:. They deduced that A~~ then again is an f-algebra (of course, with unit element) with respect to
the Arens multiplication (note that if A~ separates the points of A then A is Archimedean). Later Huijsmans was able to do away with the condition that A have a unit element, and he showed that the
order bidual of an f-algebra with separating order dual is also an f-algebra with respect to the Arens multiplication (see [56, Theorem 2.8]). Then, without assuming any additional 'separation'
condition, Bernau and Huijsmans proved in [15] that the order bidual of an Archimedean falgebra, equipped with the Arens multiplication, is again an f-algebra (see Theorem 3.2 and Theorem 3.5 in
[15]). THEOREM 12.3. Let A be an Archimedean f-algebra. The order bidual A~~ of A, equipped with the Arens multiplication is an f-algebra and hence 'is commutative.
In the paper [14], Bernau and Huijsmans also dealt with the case of almost f-algebras. THEOREM 12.4. Let A be an Archimedean almost f -algebra. Furnished with the Arens multiplication, the order
bidual A~~ of A is an almost f-algebra and hence is commutative.
Combining Theorem 12.4 and the fact that commutative d-algebras are almost f-algebras, Bernau and Huijsmans deduced the following result (see [14, Theorem 4.2]). COROLLARY
order bidual plication.
12.5. Let A be an Archimedean commutative d-algebra. then the of A is a commutative d-algebra with respect to the Arens multi-
It is interesting to observe that the square of the singular part of the bidual (the orthogonal complement of the order continuous part) vanishes in the case of an almost f-algebra. Contrary to the
case of almost f-algebras, the corresponding problem for non commutative d-algebras remains open and only a partial result has been obtained, again by Bernau and Huijsmans in [14, Theorem 4.1]. Their
result is the following. PROPOSITION 12.6. Let A be an Archimedean d-algebra. Then the order continuous order bidual (A~);: is a d-algebra with respect to the Arens multiplication.
As just observed, the question whether the full order bidual of a non commutative d-algebra is a d-algebra with respect to the Arens multiplication is still open. PROBLEM 12.7. Let A be an
Archimedean d-algebra. Is the order bidual of A a d-algebra with respect to the Arens multiplication?
At the end of this section, we notice for the sake of completeness that all of the results above were obtained independently by Scheffold under the additional . assumption of A being a Banach lattice
(including the result on d-algebras) (see [81] and [83]).
13. Ideal theory In this section, we follow [69J in our terminology and notations. Recall that an order ideal in a Riesz space L is a vector subspace I of L with the extra condition III ::; Igl in
Land gEl imply I E I. We call a ring ideal I in an i-algebra A an i-ideal if I is simultaneously an order ideal in A. Ideal theory in lattice ordered algebras comprises an investigation of ring
ideals, order ideals and the connection between them. In the first part of this section, we focus on the following problem: (P 1) Under what conditions is any order ideal in an I -algebra a ring
ideal? In the second part we consider the converse problem: (P2) Under what conditions is any ring ideal in an I -algebra an order -ideal? The first problem was initiated by Henriksen in [49J, who
studied the ideal theory in I-rings by means of representations. In their work [58J on ideal theory in I-algebras, Huijsmans and de Pagter proved the next result by a representation-free approach
(see [58, Proposition 3.1]). PROPOSITION 13.1. Let A be an Archimedean I-algebra. Then any unilormly closed order ideal is a ring ideal.
The problem (PI) for non uniformly closed order ideals was considered cxtensively by Basly and Triki in [l1J and [12J. In what follows, we present the major results they obtained in that direction.
First, we need to recall some prerequisites. A ring ideal I in a commutative algebra A is said to be modular whenever there exists h E A such that I - I h E I for all I E A. In particular, any ring
ideal in A is modular if A is in addition unital. From now on A is an I-algebra. We know that the map p: A ~ Orth(A)
(where 7rf (g) = I g for all I, g E A) is an algebra and Riesz homomorphism (see Section 4 above). An element I in A is said to be bounded if p (a) = 7ry. E Z (A), where Z(A) = {7r E Orth(A): 17r1::;
>.IA for some real number A} is the centre of A (Le., the principal order ideal generated by IA in Orth (A)). We denote, after Triki (see [90]), the set of all bounded elements in A by Ab. Note that
rather than A b , Henrikscn and Johnson use A* in [53J after a similar usage in Gillman and Jerison's book. Clearly, Ab is an I-subalgehra of A. The I-algebra A is said to be bounded if A = Ab, that
is, if p (A) is a subset of Z (A). In particular, a unital I-algebra A is bounded if and only if its multiplicative identity is also a strong order unit, and in this situation A and Z (A) are
isomorphic as f-algebras. Here we recall that the identity element in an Archimedean unital f-algebra is a weak order unit (see [22, p. 60]). The equivalence of (i) and (ii) in the next theorem is an
immediate consequence of the definitions of an order ideal and a bounded Archimedean I-algebra, while for the proof of the rest of tllis theorem, we refer to Theorem 3 in [l1J and Theorem 4 in [12J.
THEOREM 13.2. Let A be an Archimedean I-algebra. Then the following are equivalent. (i) Every order ideal is a ring ideal. (ii) A is bounded.
If A is in addition uniforrn,ly complete then (i) above is equivalent to each of the following.
(iii) Every maximal modular ring ideal in A is uniformly closed. (iv) Every maximal modular ring ideal is the kernel of a Hiesz and algebra homomorphism from A onto JR. (v) Every maximal modular ring
ideal is a maximal or'der ideal. We turn our attention to semiprime f-algebras. Let L be a Riesz space. A norm on a L is called a Riesz norm if IIfil ::; IIgil whenever If I ::; Igl in L. If a Riesz
norm on L exists then L is said to be a normed Riesz space. If the normed Riesz space L is a Banach space as well, we say that L is a Banach lattice. We call the Riesz norm 11.11 on L an AI-norm if
IIf V gil = max {IIfil , IIgll} for all f,g E L. An Ai-space is an At-normed Banach lattice. We can now state the following theorem (see [11, Theorem 5] and [12, Corollary 5]). 11.11
THEOREM 13.3. Let A be an Archimedean semiprime f -algebra. The following are equivalent.
(i) (ii) (iii) (iv)
Every order ideal in A is a ring ideal in A. A is a isomorphic as an f -algebra to a subalgebra of Z (A). There exists an M -norm in A. There exists a Riesz norm in A.
If, in addition, A is uniformly complete then each of (i), (ii), (iii) and (iv) above is equivalent to
(v) Every maximal modular ring ideal in A is uniformly closed. We note that Problem (PI) was extensively studied by Henriksen, Larson and Smith [52] in the context of f-rings. We move on to discuss
Problem (P2). To this end, we need the notion of a normal Riesz space. The Riesz space L is said to be normal if L = {f+} d+{f-} d for allf E L, where {f+}d = {g E L: Igl.l\f+ = O} and {f_}d = {g E
L: Igl.l\f- = O}. For a completely regular topological space X, the Riesz space C (X) is normal if and only if the sets P(f) = {x EX: f(x) > O} and N(f) = {x EX: f(x) < O} are completely separated
for every fEe (X). For spaces of the type C (X), Problem (P2) has the following solution (see Theorem 14.24 in [47]). THEOREM 13.4. Let X be a completely regular (Hausdorff) topological space. Then
the following are equivalent. (i) Every ring ideal in C (X) is an order ideal. (ii) Every finitely generated ring ideal in C (X) is a principal ring ideal (i.e., X is an F -space). (iii) C (X) is
normal. Huijsmans and de Pagter considered (P2) in f-algebras. They only considered the unital case and their central result is the following generalization of Theorem 13.4 above (see [58, Theorem
6.6] for the proof). THEOREM 13.5. Consider the following conditions for an Archimedean f -algebra A with unit element.
(i) Every ring ideal is an order ideal. (ii) Every finitely generated ring ideal is a principal ring ideal.
(iii) A is normal. (ii) (iii). If, in addition A is uniformly complete then (i) {:} (ii) {:} Then (i) (iii) .
Next we focus on the non unital case, which was considered by Triki in [90]. First we define the notion of a stable f-algebra. The f-algebra A is said to be stable if 1f (f) E (f) (where (f) is the
principal ring ideal generated by f in A) for all 1f E Z (A). It is clear that A is stable if and only if 1f (1) c I for all ring ideals I in A and all 1f E Z (A). We now are in a position to present
the result corresponding to Theorem 13.5 for semiprime f-algebras (see [90, Theorem 5.4]). THEOREM 13.6. Consider the following conditions for an Archimedean scmiprime f -algebra A. (i) Every ring
ideal is an order ideal. (ii) Every finitely generated ring ideal is a principal ring ideal. (iii) A is stable and normal, and {f+} d or {f-} d is a modular ring ideal for all f E A. Then (i) (ii)
(iii). Furthermore, if A is in addition uniformly complete then (i) ¢:> (ii) {:} (iii).
We proceed to the non semiprime case (see Theorem 5.5 in [90]). THEOREM 13.7. Let A be a non semiprime Archimedean f-algebra. Then the following are equivalent. (i) Every ring ideal is an orner
zdeal. (ii) There exist a semiprime f-algebra B such that (a) every 7ing ideal in B is an order ideal, and (b) {b} d is a modular ring ideal in B for every b E H, so that A is isomorphic to the f
-algebra B x JR endowed with the multiplication defined by (f, 0:) (g, 13) = (fg,O) for all i, g E B; 0:, 13 E JR.
Finally, note that the only i-algebras that we have considered in this section are i-algebras. Indeed, a Problem (PI) for more general i-algebras is futile since an i-algebra in which every order
ideal is a ring ideal automatically is an f-algebra (see Page 144 in the classical book [42] or Proposition 1 in [12]). The situation for Problem (P2) is less clear. Indeed, in matrix algebras and
algebras of formal power series in one variable, when ordered coordinatewise, every ring ideal is an order ideal. Thus we phrase Problem (P2) for lattice ordered algebras in general. PROBLEM
13.8. Study Problem
for Archimedean i-algebras other than
i-algebras. 14. Representation of I-algebras
Let A be an f-algebra and recall that Ab denotes the i-subalgebra of all bounded elements in A (see Section 13 above). Several properties are satisfied by A if and only if they are satisfied by Ab
and vice versa (see Sections 3,4 and 5 in [90]). Thus we can study some aspects of A via an investigation of Ab. For instance, if we assume that every ring ideal in A is an order ideal then A b has
the same property and the converse holds if A in addition is uniformly complete (see [84] by Steinberg). In particular, if every ring ideal in A is an order ideal then order
ideals and ring ideals coincide in Ab (see Theorem 13.2). It turns out that under the latter hypothesis, A b has a nice representation as a space of functions. This kind of a representation then is
precisely the topic of this section. First assume that A has an identity element (Le., A is a
k (D) = n {M : M E D} (where it is understood that k (0)
= A). The
hull h (I) of an (-ideal I in A is
h(I) = {M E M (A): I
c M}. = h (k (D)).
The subset D of M (A) is said to be closed if D One thus defines the hull-kernel (or Stone) topology on M (A). It turns out that M (A) with respect to this hull-kernel topology is a compact Hausdorff
space (see [53, Theorem 2.3]). Suppose at this point that A is in addition uniformly complete (instead of 'uniformly complete', Henriksen and Johnson in [53] use 'uniformly closed') and denote the
identity element of A bye. Since A is unital, Ab is precisely the principal order ideal generated bye. Also recall that A and Orth (A) are isomorphic as f-algebra under the given condition. As usual,
the f-algebra of all real-valued continuous functions on the compact Hausdorff space M (A) is denoted by C (M (A)). Henriksen and Johnson in [53, 3.2, p. 84] proved, using the Stone-Weierstrass
theorem, the following variant of Stone's representation theorem. THEOREM 14.1. Let A be a uniformly complete f-algebra with identity element e. Then Ab and C (M (A)) are isomorphic as f-algebras. In
particular, if e is a strong order unit in A then A and C (M (A)) are isomorphic as f-algebras.
The latter result can of course also be obtained via Kakutani's representation theorem. Indeed, being the principal order ideal in A generated bye, Ab is a uniformly complete Riesz space with e as a
strong order unit. It follows that Ab is an M-space with respect to the M-norm 11.ll e defined by IIflle = inf {A > 0 : If I :'S Ae}
for all
E A
(see [72, Proposition 1.2.13]). Kakutani's representation theorem (see, for instance, [72, Theorem 2.1.3]) guarantees the existence of a compact Hausdorff space 0 so that Ab and C (0) are isomorphic
as f-algebras and isometric as M-spaces. From an investigation of the cited proof of Kakutani's theorem, we see that 0 is the set of all algebra homomorphisms from Ab onto JR, or, equivalently, the
set of all real valued Riesz homomorphisms that send e to 1. With respect to the weak topology a ((Abf ,Ab), where (Ab)'" is the order dual of Ab, the set 0 is a a ((Ab)'" ,Ab)-compact Hausdorff
space. In view of Theorem 13.2, we observe that o is homeomorphic to M (Ab) and then to M (A) since M (Ab) and M (A) are also homeomorphic (see [53, Corollary 2.8]). We thus recover Theorem 13.1
above. The reader should also compare the above with Gelfand theory for Banach algebras
[94]. We proceed to representation theorems for non unital f-algebras. For the terminology and notations concerning the topological spaces under consideration, we refer the reader to the book [47] by
Gillman and Jerison. The following two theorems deal with semiprime f-algebras (see Theorems 7.2 and 7.9 in [90]).
THEOREM 14,2, Let A be a uniformly complete semiprime f-algebra with a weak order unit, Then the following are equivalent. (i) Every ring ideal in A is an order ideal in A. (ii) There exists an
almost compact F -space 0 such that A b and Co (0) are isomorphic as f -algebras.
For almost compact. F-spaces, we refer to [47, 6J]. THEOREM 14.3. Let A be a semiprime Dedekind a-complete f -algebra. Then the following are equivalent. (i) Every ring ideal in A is an order ideal
in A. (ii) There exist a basically disconnected locally compact Hausdorff space 0 1 and a basically disconnected almost compact Hausdorff space O2 so that Ab is isomorphic as an f-algebra to one of
the algebras
CK (Ot) ,Co (0 2 ) or CK (Od EB Co (0 2 )
Our next representation theorem represents non semiprime f-algebras (see [90, Theorem 7.10]). THEOREM 14.4. Let A be a uniformly complete non semiprime f-algebra A. Then the following are equivalent.
(i) Every ring ideal in A is an order ideal in A. (ii) There exists a locally compact F'-space 0 such that Ab is 'isomorphic as an f -algebra to the Cartesian product Co (0) x JR, where the
multiplication in the f-algebra Co (0) x JR is defined by
(1,.x) (g, J.t)
= (1g,O)
for all f, g E Co (0) and.x, J.t E JR.
Note that the topological spaces 0 in Theorem 14.2 and in Theorem 14.4 are constructed in the same way as follows: for the uniformly complete f-algebra A (semiprime or not), the centre Z (A) is a
uniformly complete f-algebra and its multiplicative identity IA is a strong order unit. According to Theorem 14.1, Z (A) is isomorphic as an f-algebra to the f-algebra C (M (Z (A))) of all
real-valued continuous functions on the compact Hausdorff topological space M (Z (A)) (the set of all maximal f-ideals in Z (A)). It turns out that 0=
u {coz (1) : f
E C (M (Z (A)))).
In a similar way, the topological spaces 0 1 and O2 in Theorem 14.3 are constructed from a certain uniformly complete f-subalgebra of A (for more information, we refer to [90]). We end this section
with two comments. Observe that representation theorems listed in this section apply to Ab and not to the whole f-algebra A. What one can say about A itself is the following. If A is unital then A
can be embedded as an f-algebra into an algebra of extended functions on M (A) [53, Theorem 2.3], and if A is semiprime then A can be considered as an f-subalgebra of M (Orth (A)) (recall that if A
is semiprime then A can be considered as an f-subalgebra of Orth (A)). In addition, the representation theorems that we presented in this section are not necessarily excessively restrictive due to
the previously mentioned fact about transference of various properties of Ab to A. Our final comment about representing lattice ordered algebras deals with a matter of set theory. Zaanen started an
ambitious program at around 1980, intending to prove all available material in vector lattices in as far as possible in an
elementary way, i.e., without using representation theorems. A commonly quoted reason to adhere to Zaanen's program is the need to not use the Axiom of Choice unnecessarily. It is therefore
interesting to know that one can avoid the Axiom of Choice and still use representation theorems, as long as the constructs that one has in mind depend on say countably many elements of a Riesz
space. For f-algebras one of the most useful theorems in that direction is the following (combine Theorem 2.2 in [33] and Corollary 2.7 in [36]). THEOREM 14.5. Let A be a semiprime f-algebra. If D is
a countable subset of A then the f -subalgebra generated by D in A can be r-epresented within ZermaloF'raenkel set theory as an f -subalgebra of the space of continuous functions on a metric space.
It should be noted at that the connection between the Axiom of Choice and representations of f-rings in terms of sub direct products of totally ordered algebras was discussed in the works [42] by
Feldman and Henriksen, [67] by Luxemburg, and [9] by Banaschewski. Contrary to the possibility of locally representing countably many elements of any semiprime f-algebra as continuous functions
without any choice, the global representation of f-algebras as sub direct products of totally ordered algebras can not be obtained without appealing to some transcendent tool from set theory. The
latter kind of representation theorem is important for more than historical reasons. First of all, many researchers define f-algebras as subdirect products of totally ordered algebras. Secondly,
Birkhoff and Pierce in their seminal paper [22] observed that the Axiom of Choice seems to be involved if one wishes to obtain (with the definition for f-algebras as in this survey) a representation
theorem for f-algebras as a sub direct product of totally ordered algebras, which for convenience we will now name the Birkhoff-Pierce Representation Theorem. The three papers [42], [67], and [9]
independently prove that the Boolean Prime Ideal Theorem is both sufficient as well as needed for the Birkhoff-Pierce Representation Theorem. Thus Stone's Representation Theorem for Boolean algebras
is constructively equivalent to the Birkhoff-Pierce Representation Theorem for f-algebras. In turn, each of the latter representation theorems is constructively equivalent to the Kakutani
Representation Theorem for vector lattices with a strong order unit. In about that same direction, we observe that it is still unknown whether the Boolean Prime Ideal Theorem suffices for that other
main representation tool for vector lattices as vector sublattices of extended real valued continuous functions, the so-called Maeda-Ogasawara Representation Theorem (see [32]).
15. Linear biseparating maps on I-algebras A linear map T between two algebras A and B is said to be separating if T (I) T (g) = 0 in B whenever f 9 = 0 in A. If in addition T is bijective and its
inverse T- 1 is separating as well then T is said to be biseparating. Clearly, if T is one-to-one and onto then T is biseparating if and only if
0 in A <=> T (I) T (g) = 0 in B.
If A and B are assumed to be f-algebras with unit elements, then T is separating if and only if Tis disjointness preserving, that is, If I II Igl = 0 in A implies IT (1)1 II IT (g) I = 0 in B. This
follows directly from the equivalence fg = 0 <=> If I II Igl = 0,
which holds in any semiprime I-algebra. For the same reason, the linear map T between two I -algebras with multiplicative identity is biseparating if and only if T is ad-isomorphism (Le., T is
bijective and both T and T- 1 are disjointness preserving) The reader is encouraged to consult the beautiful memoir [1 J by Abramovich and Kitover for the theory of disjointness preserving linear
maps on Riesz spaces. The study of when linear biseparating maps on algebras of real or complex valued continuous functions are weighted isomorphisms started in 1990 with the paper [63J by Jarosz and
culminated in the work [5J by Araujo, Beckenstein and Narici with the following result. Let C (X) and C (Y) be the algebras of real or complex valued continuous functions on completely regular
topological spaces X and Y, respectively. If T is a linear biseparating map then there exist a nonvanishing wEe (Y) and an homeomorphism h from the realcompactification vX of X onto v Y, such that T
(f) (y) = w (y) I (h (y))
for all lEe (X) and y E Y.
Henriksen and Smith in [55J explored the aforementioned result by Araujo, Beckenstein and Narici in the more general setting of unital I-algebras. They proved that every positive linear biseparating
map T between two unital I-algebras A and B closed under inversion is a weighted isomorphism, that is, there exist an invertible wEB and a Riesz isomorphism S : A ---> B which is simultaneously an
algebra isomorphism, such that
T (f) = wS (f)
for all lEA.
Very recently in [30J, Boulabiar, Buskes, and Henriksen extended the latter result to all order bounded linear biseparating maps on arbitrary (not necessarily closed under inversion) unital
I-algebras over the reals as well as over the complex numbers (for the theory of complex I-algebras, we refer to [17]). The theorem they obtained is the following. THEOREM 15.1. Let A and B be (real
or complex) I-algebras with unit elements, and let T : A ---> B be an order bounded linear biseparating map. Then T is a weighted isomorph'ism.
In the previously mentioned memoir by Abramovich and Kitover, we find the following theorem. A d-isomorphism between two uniformly complete Riesz spaces A and Jyf is automatically order bounded as
soon as every universally a-complete projection band in A is essentially one-dimensional (see [1, Corollary 15.3]). This theorem is used in [30], under the same condit.ions on A, to show that every
linear biseparating map between two uniformly complete I-algebras A and B is a weighted isomorphism. We point out that a complex I-algebra is by definition uniformly complete. Thus the phrase
'uniformly complete unital I-algebra' is understood to mean either a uniformly complete unital I -algebra over the reals, or simply a unital I-algebra over the complex numbers. For t.he proof of the
next theorem, see Proposition 5.1 and Theorem 5.2 in [30J. THEOREM 15.2. Let A and B be unilormly complete unital I -algebras and assume that every universally a-complete projection band in A is
essentially onedimensional. Then every linear biseparating map from A onto B is order bounded and then a weighted 'isomorphism.
To obtain the result by Araujo, Beckenstein and Narici cited above as a consequence of the preceding theorem, Boulabiar, Buskes and Henriksen proved t.hat
every algebra of all scalar-valued continuous functions on a completely regular topological space has the property that every universally a-complete projection band is essentially one-dimensional
(see [30, Theorem 5.5]). THEOREM 15.3. Let X be a completely regular topological space X. Then every universally a-complete projection band in the Riesz space C (X) is essentially onedimensional.
Combining Theorems 15.2 and 15.3, we arrive at the next result. COROLLARY 15.4. Let X and Y be completely regular topological spaces. Then every biseparating linear map T : C (X) -> C (Y) is a
weighted isomorphism. In particular, C (X) and C (Y) are isomorphic as f -algebras if and only if there exists a linear biseparating map from C (X) onto C (Y).
It is well known that if X and Yare completely regular topological spaces and S is an isomorphism from C (X) into C (Y) then there exists an homeomorphism h from vY into vX such that S (f) = f 0 h,
where vX and vY denote the realcompactifications of X and Y, respectively (see Section 10 in [47]). The latter fact, together with Corollary 15.4 above, directly leads to the next corollary, which
was proved earlier in an alternative way by Araujo, Beckenstein and Narici in [5, Proposition 3]. COROLLARY 15.5. Let X and Y be completely regular topological. Then for every l'inear biseparating
map T : C (X) -> C (Y) there exist a non-vanishing function wE C (Y) and an homeomorphism h : vY -> vX such that
T (f) (y) = w (y)
(h (y))
for all f E C(X) and y E Y.
It is shown in [47, Theorem 8.3] that two realcompact X and Yare homeomorphic if and only if C (X) and C (Y) are isomorphic as f-algebras. Another classical result of rings of continuous functions
theory is that if X is a completely regular topological space then C (X) and C (vX) are isomorphic as f-algebras (see Remark 8 (a) in [47]). It follows immediately that if X and Y are two completely
regular topological spaces, and C (X) and C (Y) are isomorphic as f-algebras, then vX and v Yare homeomorphic. The latter implies that, without further assumptions, the conclusion that vX is
homeomorphic to vY in Corollary 15.5 is best possible. Under additional assumptions, however, X and Y may themselves be homeomorphic as is shown in the next Corollary. COROLLARY 15.6. Let X and Y be
completely regular topological spaces and assume either (i) or (ii) below. (i) X and Yare realcompact. (ii) The points of X, as well as those ofY, are Go-points. If there exists a linear biseparating
map from C (X) onto C (Y) then X and Yare homeomorphic.
Under the condition (i), the result in Corollary 15.6 above follows straightforwardly from Corollary 15.5 while under the condition (ii), it stems directly from [73] by Misra. Finally, it should be
noted that the result of Theorem 15.2 above is false if the universally a-complete projection bands in A fail to be essentially one-dimensional. Indeed, Example 7.13 in [1] produces a linear
biseparating map T on the Dedekind
complete (and then uniformly complete) unital i-algebra Lo ([0, 1]) of all equivalence classes of measurable functions on [0,1)' which is not order bounded and thus cannot be a weighted isomorphism.
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BP 51, 2070-LA MARSA, TUNISIA
E-mail address:karim.boulabiar«lipest.rnu.tn DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MISSISSIPPI, UNIVERSITY, MS-38677, USA
E-mail address: mmbuskes«lsunset. olemiss. edu DEPARTEMENT DE MATHEMATIQUES, FACULTE DES SCIENCES DE TUNIS, UNIVERSITE TUNIS EL MANAR, 1060-TUNIS, TUNISIA
E-mail address:abdelmajid.triki«lfst.rnu.tn
Contemporary Mathematics Volume 328, 2003
An extension of a theorem of Wermer, Bernard, Sidney and Hatori to algebras of functions on locally compact spaces Eggert Briem
ABSTRACT. It follows from a theorem of J. Wermer that if A is a uniformly closed algebra of continuous complex-valued functions vanishing at infinity on a locally compact Hausdorff space X, the
property that b2 E Re A for all bE Re A implies A = Co(X). In other words, if the function h(t) = t 2 operates by composition on ReA then A = Co(X). This result was generalized by O. Hatori. He
proved that in place of the function h(t) = t 2 one can put any real-valued function defined in a neighbourhood of 0, thus extending a similar result for the compact case. Here a simple alternative
proof is given for the locally compact case.
1. Introduction
Let X be a compact Hausdorff space and B a uniformly closed subspace of CIR(X), the space of all continuous real-valued functions on X, which separates the points of X and contains the constant
functions. A version of the Stone-Weierstrass theorem says that if b2 E B for all b E B then B = CIR(X). Clearly this result does not hold if, instead of assuming that B is uniformly closed, one
assumes that B is a Banach space in some norm which dominates the sup-norm, as the example of any non-trivial real Banach function algebra shows. However, if B is the real part of a uniform algebra,
a theorem of J. Wermer says that if b2 E B for all b E B then
B = CIR(X). Let us say that a real-valued function h, defined on an interval I of the real line, E B whenever b E Band b maps X into I. Thus b2 E B for all bE B means that h(t) = t 2 operates on B.
The Stone-Weierstrass theorem W8.', generalized by K. de Leeuw and Y. Katznelson (see [4], theorem 4.21), they showed that h(t) = t 2 can be replaced by any continuous non-affine function (Le. a
function not ofthe type h(t) = o:t+!3) defined on an interval, and the theorem of J. Wermer was similarily generalized by A. Bernard, S. Sidney and O. Hatori [1], [5] and [8]. Here one can even do
without the continuity assumption, a function operating on the real part of a uniform algebra of infinite dimension is automatically continuous.
operates on B if hob
1991 Mathematics Subject Classification. Primary 46JlOj Secondary 46E15. Key words and phmses. uniform algebra, locally compact space, functional calculus. © 2003 American Mathematical Society
In the case where X is locally compact, the functional calculus for a uniformly closed subspace of Co(X, JR), the space of all continuous real-valued functions vanishing at infinity on X, may be
non-trivial (cf. [2]). What then about the real part of a uniformly closed subalgebra of Co(X), the space of all continuous complex-valued functions vanishing at infinity on X? Wermer's theorem,
[10], clearly applies to this situation. It turns out that the functional calculus for the real part of a uniformly closed subalgebra, which is not a C* -algebra, of Co(X) is trivial. This result,
which is due to O. Hatori, is to be found in [7]. To prove this one has to do more than just adapt the proofs for the compact case to the locally compact situation. The purpose of this paper is to
give a simple alternative proof using antisymmetric decomposition of uniform algebras. 2. Proofs and results Let X be a locally compact Hausdorff space and let Xl denote the one-point
compactification of X, with Xoo denoting the point at infinity. The functions in Co(X) have a natural extension to functions in C(Xl)' the space of all continuous complex-valued functions on Xl. If A
is a uniformly closed subalgebra of Co(X), that separates the points of X and does not vanish identically at any point of X, then Al = AEI1C is a uniformly closed subalgebra of C(Xd that separates
the points of Xl and contains the contant functions and A = {a E All a(x oo ) = O}. Thus, we can assume that there is a uniformly closed subalgebra Al of C(Xd containing the constant functions, where
Xl is compact, and a point Xoo in Xl, such that A is obtained by restricting the functions in the set {a E Al Ia(x co ) = O} to the set X = Xl \ {xoo}. We need some results from the theory of uniform
algebras. A subset E of Xl is a set of antisymmetry for Al if the only functions in Al that are real-valued on E are constant functions. Maximal sets of antisymmetry are closed generalized peak sets
(intersections of peak sets), they form a partition of Xl and a continuous complexvalued function f on Xl is in Al if and only if its restriction to each maximal set of antisymmetry E is in the space
AIlE of restrictions of functions in Al to E. Thus Al(Xd = C(Xd if and only if each maximal set of antisymmetry is a singleton. Also, AIlE is uniformly closed for each maximal set of antisymmetry,
there is actually for each a in AIlE an element a' in Al with a' = a on E and II a' 1100=11 a llco,(E). (The latter norm is the supnorm w.r.t E). This material can f. ex. be found in [3]. If Al is
the disc algebra on the closed unit disc in the complex plane and A is the algebra of functions vanishing at the origin, then any non-zero function in A takes real values of opposite signs in any
neighbourhood of O. Thus any real-valued function, defined on an interval which is not a neighbourhood of 0, operates trivially on Re A. We shall therefore always assume that the interval on which an
operating function is defined is a neighbourhood of O. We can now state the main result of this note. THEOREM 1. Let A be a uniformly closed subalgebra of Co(X) which separates the points of X and
does not vanish identically at any point in X and let Re A be the space of real parts of functions in A. If Re A has a non-affine operating function h: I ........ JR where I, an interval, is a
neighbourhood of 0, then Re A = Co(X, JR) and thus also A = Co(X).
The first step in the proof of Theorem 1 is to show that operating functions are continuous. To prove this we need the following lemma. LEMMA 1. Let E be a maximal set of antisymmetry for A1 and let
b E Re A. Then b(E) is either an interval or a singleton.
Proof. Suppose b(E) is not connected, and that b = Rea for some a in A 1. Then we can find two disjoint closed rectangles Ro and R1 in the complex plane such that a(E) is a subset of Ro u R1 but not
a subset of anyone of the two rectangles. By Runge's theorem there is sequence of polynomials (Pn) converging uniformly to o on Ro and to 1 on R 1. Since AdE is uniformly closed, the sequence (Pn 0
a) converges on E to an element of A1IE. But the sequence converges on E to a function which takes only the values 0 and 1, contradicting the fact that E is a set of antisymmetry. 0 LEMMA 2. Let E be
a maximal set of antisymmetry for A 1 such that E\ {xoo } =I- 0. Then there is an Xo E E such that for every to E I (resp. to in the interior of I), there exists bo E ReA with bo(xo) = to and bo(X) C
I (resp. bo(X) is a subset of the interior of I).
Proof. If to = 0, the existence of the desired function is easily proved. We consider the case where to > O. (The proof for the case to < 0 is similar.) Let Xo be a generalized peak point in E, other
than x oo , for A 1 • Simple calculations show that we can take 1 E A with I(xo) = 11/1100 = 1. Let c: be a positive number such that (-2c:, to] C I (resp. (-2c:, to] is a subset of the interior of
I), and R the closed rectangle with corners -c: ± i and to ± i. Let
be an operating function for Re A. Then, unless
Proof. If A =I- Co(X,~), there is a maximal set of antisymmetry E for A1 which contains more than one point. Thus E \ {xoo} =I- 0. Let t be an arbitrary point in I. Then by Lemma 2 there exists and x
E E and bEReA such that b(x) = t and b(X) C I. Since hob is in ReA, hob is continuous. It follows that h is continuous at t. We conclude that h is continuous on its domain of definition. If A = Co
(X), h need only be continuous at 0, f. ex., if A = Co. 0
In the compact case, where A contains the constant functions, Re A is dense in CR(X) (cf. Proposition 4 below). Here we obtain a similar result. PROPOSITION 2. Let h : I --+ IR be an operating
function for ReA which is non-affine in every neighbourhood of an interior point to for I. Let Xo E X and suppose that there is a function bo E ReA which maps X into the interior of I and satisfies
bo(xo) = to· Then there is an open neighbourhood Xo of Xo such that every f E Co (X, ~) which vanishes outside Xo can be approximated uniformly on X by elements of ReA.
Proof. We are assuming that X is locally compact and not compact so that
h(O) = o. We may assume that h is continuous by Proposition 1. Let us first look at the case where h is a polynomial of degree at most three in some neighbourhood of 0 but not affine there. Then bed
E Re A for all b, e, d E Re A and hence fb E cl(Re A) for all b E cl(Re A) and all f in the algebra generated by Re A. Here cl(Re A) denotes the uniform closure of Re A. It follows that
Co(X,JR)· cl(ReA)
and hence ReA is dense in Co(X,JR). Suppose first that h is affine in some neighbourhood ofO. Replacing h by h(t)-,t we may assume that h = 0 in a neighbourhood of o. Put e = tC; l bo. For bEReA, r,t
E JR and '{) E CO'(JR) the function
h 0 (rb o + tb - se)'{)(s)ds
is in cl(Re A) if r is close to 1, t is close to 0 and the support of'{) is contained in a small neighbourhood of 0, so that the expression above is defined. We differentiate twice w.r.t. 0, put t =
0 to obtain
b2 e- 2
h 0 (rbo - se)'{)"(s)ds E cl(ReA),
for all bEReA. (If le(x)1 is small the expression is 0.) Put
d= We have
d(xo) =
h 0 (rb o - se)'{)"(s)ds.
h 0 (rto - s)'{)"(s)ds = (h
* '{))"(rto).
Since h is not affine in any neighbourhood of to we can choose '{) and r, where the support of'{) is contained an arbitrarily small neighbourhood of 0 and r is arbitrarily close to 1 such that d(xo)
'" O. Let
Xld(x) '" O}.
Since b2 e- 2 d E cl(Re A) for all b E Re A it follows that
b1 ,b2 e- 2 d E cl(ReA) for all b1 , b2 E Re A. Put
M(ReA) = {J
Co(X,JR) If· cl(ReA)
be- 2 d
Above we have E M(ReA) for bEReA. Since these functions separate the points of Xo and since M(ReA) is an algebra we deduce that cl(ReA) contains every f E Co(X,JR) which vanishes outside Xo. Suppose
now that h is not affine in any neighbourhood of o. For e E Re A let
B(e) = {u E cl(ReA) 13>' E JR+ s.t. lui::::; >'Iel}. If b ERe A and b + ie E A then be and be(b2 - e2 ) belong to B(e). Let also
M(e) = {g
Co(X, JR) Ig. cl(B(e))
a subalgebra of Co(X, JR). We note that since h is continuous h also operates on
For bl ERe A, for u E B(e), r, t E lR and
h 0 (rb l
+ tu -
is in cl(Re A), if rand t are sufficiently small and the support of
uie- i
ho (rb l - se)
d= With i
= 2 and
h 0 (rb l - se)
instead of
u 2 e- 2 d E cl(Re A) for all u E B(e) and hence
(2) for all
Ul, U2
E B(e). With i
= 3 and u = be E B(e)
in (1) we get
b3 d E cl(ReA), and with i
1, u
= beW - e2 )
E B(e) and
e2 )d E cl(Re A)
and thus
If we put
be2 d E cl(ReA). = be2 d in (2) we find that ubd2 E cl(Re A)
for all u E B(e). Since u E B(e) implies ubd2 E B(e) we deduce that
bd 2 E M(e). Let us determine how well the functions bd2 separate the points of the set
Xo = {x
X Ib(x)e(x) i- a}.
Let x, y E Xo. Suppose that does not separate x and y for every d. Since A is an algebra we can choose bl E ReA such that bl(x) = 0 and bl(y) = 1. Then
bd 2
d2 (y)
= (b(x)/b(y))d 2 (x).
The right hand side is independant of r so that
h 0 (r - se(y))
a differentiable function of r, is constant in for r in some neighbourhood of O. It follows that h 0 (r - se(y))
for all r in a neighbourhood of O. Since this holds for all
dense in Co(X,JR). Otherwise, the functions bd2 separate the points of X o, and since these functions vanish outside X o, it follows that
{f E Co(X, JR) I I = 0 on Xo \ Xo}
Thus by a simple calculation we have
Co(X,JR) I I = 0 on Xo \ Xo}
o To proceed to the proof of Theorem 1 we need a local version of the so called Bernard's Lemma [1]. Let A be an infinite set and let C be a Banach space with norm I . II. Put
eOO(C) = {{c.d ICA E C and sup
I CA 11< oo},
where {cA} denotes a function from A into C such that {cA}(a) = a E A. Then eOO(C) is a Banach space in the norm
{cA }
E C for each
11= sup II CA I . AEA
The space Re A is a Banach space in the norm
lib 11= inf{11 a 1100 Ia E A and clearly lib 1100::;11 b II. We thus have eOO(ReA)
and b = Rea},
Bernard's lemma (see [1]) says that if eOO(ReA) is dense in eOO(Co(X,JR)) then ReA = Co(X,JR). To prove that eOO(ReA) is dense in eOO(Co(X,JR)), we represent eOO(Co(X,JR)) as a space of functions on
a compact Hausdorff space. We identify Co(X, JR) with a subspace of CIR(Xd in the obvious manner. Let A have the discrete topology and let (3(A x Xd) be the Stone-Cech compactification of Ax Xl' The
space eOO(C(xd) can be identified with C({3(A x Xd) in a natural way
FUll} where F{!lI}b,x) = 1'Y(x) for each b,x) in A x Xl' subspace of CIR ({3(A x Xl)) and we have the inclusions
Thus eOO(Co(X,JR)) is a
CIR({3(A x Xd).
The subsets of the Stone-Cech compactification may be complicated but the subsets we are interested in have a simple description. For each I in C(Xl ) let {f} denote the net {fA} where IA = I for
each A Let now Po be in {3(A x Xd. The map
{f }(Po)
is a homomorphism of C(X l ) and is thus given by point-evaluation at some point in Xl' Put
5: 0
= {p E (3(A x Xd I {f}(p)
VI E CIR(Xl )}.
The sets x form a partition of {3(A x Xt} into closed sets. Their significance stems from the following local version of Bernard's lemma, similar to the one given by O. Hatori [6].
Bernard's lemma (local version) Let A denote the unit ball of Co(X,JR.), let x be in X and suppose that the restriction space lOO(ReA)lx is dense in CIR(x). Then there is a compact neighbourhood Kx
of x such that ReAIKx = CIR(Kx )' Proof. For each A in A let 1>.. = A. By assumption there is an element {b A } in lOO(ReA) such that we have the inequality
I{fA} - {bA}1 < 1/2 on x and hence also on some open subset U of {3(A x Xt} containing X. By a simple calculation we have
Xo = {p E (3(A x Xl) I I{f}(p) - f(xo)1
:=; ~
Vf E CIR(Xl )}.
Thus there is a function fo in CIR(Xt} such that the set
{3(A x Xt}
I I{fo}(p) -
:=; 1/2}
is a subset of U. Put
Kx = {y E Xd Ifo(Y) - fo(x)1 :=; 1/2}, a compact neighbourhood of x. Then A x Kx is contained in U so that IfA - bA I :=; 1/2 on Kx for all A in A. Let M = SUPA II bA II, a finite quantity because (b
A ) is in lOO(Re A). Take any f in Co(X, JR.) with II f lloo:=; 1. By induction we construct a sequence (b n ) of elements from ReA, with II bn II:=; M for all n, such that n-l
12n( f - L
2- i
bi ) - bnl < ~
on Kx. The function b =
E:o 2- b is in Band b = f on Kx. 0 i i
PROPOSITION 3. Let Xo E X and suppose that there is an open neighbourhood Xo of Xo such that every function in Co(X,JR.), which vanishes outside Xo can be unifromly approximated on X by elements from
ReA. Then lOO(ReA) sepamtes the points of xo.
Proof. Suppose p,q are in xo. There is an element {fA} in lOO(Co(X,JR.», where each 1>.. vanishes outside X o, such that {fA}(P) = 0 and {fA}(q) = 1. By assumption we can for each A E A find a
function bA in ReA with II fA - bA lloo:=; ~. Thus we have {bA}(p) :=; ~ and {bA}(q) ~ ~ since II{fA} - {bA}lIoo,{3(Axxt} :=; ~. Let a A = bA + iC A be in A for each A. Now, the A-net {e a ... - 1}
is in lOO(A) and it separates p from q since {e an } does. Since
I{e a ... }(p)1 =
{e b... }(p)
:=; e l / 4
and I{e a ... }(q)1 = {e b... }(q) ~ e3 / 4
we see that lOO(A) separates p from q and thus lOO(ReA) does so as well. 0
We also need the aforementioned extension of the Stone-Weierstrass theorem due to de Leeuw and Katznelson (see [4], Theorem 4.21). We can prove a version of this result, Proposition 4, in the same
way as the original one, the proof is omitted. PROPOSITION 4. Let Y be a compact Hausdorff space and B a subspace ofCR(Y) which separates the points of Y and contains the constant functions. Suppose
that B is also a normed space with the norm II· liB which dominates the uniform norm. Let h be a continuous function defined on an interval I. Suppose that h is nonaffine in every neighbourhood of an
interior point to in 1. Suppose that there exists a positive real number 8 > 0 with (to - 8, to + 8) ~ I such that h 0 (to + u) E B for every u E B with Ilull < 8. Then B is uniformly dense in CIR
Proof of Theorem 1. We are going to use the local version of Bernard's Lemma. We thus have to show that the conditions stated there are satisfied. For this we use the result of de Leeuw and
Katznelson above. The main obstacle is that we can not conclude that if {b.>.} is in [00 (Re A) then h e {b.>.} is in [00 (Re A) although the composite function is defined, i.e. we can not conclude
that composition with h maps bounded nets to bounded nets. To overcome this difficulty we use a method of Sidney, [8], to obtain local boundedness for composition with h. Suppose A =I- Co(X) and thus
also Al =I- C(X 1 ). Let E be a maximal set of antisymmetry for Al containing more than one point. Let further to be an interior point of I such that h is not affine in any neighbourhood of to and
let bo be a function in Re A which maps X into the interior I for which to is an interior point of bo(E). By Lemma 2 such a function exists. We now choose E > 0 such that if b is in the E-ball,
(ReA)., of ReA then ho(bo+b) is defined and to is an interior point of (b o + b)(E). We write (ReA).
= U{b E (ReA). I I h 0 (bo + b) II::; n}. n
The Baire Category Theorem shows that the closure of one of the sets on the right hand side has an interior point and thus there is a function b1 E Re A and positive numbers 8, lvI, with I b1 I +8 <
E, such that
I he (bo +b 1 +c) II::; lvI for c in a dense subset of the 8-ball of Re A. Let b = bo + b1 , and let [00 (Re A) be as in the local version of Bernard's Lemma. We then have
h 0 {b + c.>.} E cl([OO(Re A)), the uniform closure of [OO(ReA), if {c.>.} E [OO(ReA) and such that b(xo) = to, an interior point of b(E). We restrict to Xo and deduce that
h 0 {to
11< 8.
Take Xo E E
+ c.>.} E cl([OO(ReA)lxo),
for every {c.>.} E [OO(ReA)lxo whose quotient norm satisfies II {c.>.} 11< 8. Since {c} is constant on xo for any c E ReA, the space [OO(ReA)lxo contains the constant functions, it also separates the
points of Xo by Proposition 2 and 3. Then Proposition 4 shows that [OO(ReA)lxo is dense in CIR(xo) and thus, by the local version of Bernard's Lemma, ReAIK = CIR(K) for some compact neighbourhood K
of Xo. The theorem of Sidney and Stout, [9], then shows that AIK = C(K). By Proposition 2, there exists an open neighbourhood Xo of Xo such that every f E Co(X, 1R) which vanishes outside Xo can be
approximated uniformly on X by
elements of ReA. We may assume K c Xo. Let Kl be a compact neighbourhood of Xo such that K 1 is in the interior of K. Since E is antisymmetric and contains more than one point, EnK l contains more
than one point. On the other hand, we will show that {f E Co(X, IR) II = 0 outside Kd CAl. It will follow that En Kl = {xo}, which will be a contradiction proving A = Co(X, IR). Let 1 E Co(X, IR)
with 1 = 0 outside K 1 . A function u E Co(X, IR) with lui::; 1 on X, u = 1 on K l , and u = 0 outside K can be uniformly approximated by functions in ReA. Thus, for every positive integer n, there
exists bn E ReA such that 1 - ,& ::; bn ::; 1 on K 1 and bn ::; ,& outside K. Without loss of generality we may assume that Ibnl ::; 1 on X. Take Cn E A with Recn = bn and put an = (eCn-l)n. Then an
E AI, e-1. ::; lanl ::; Ion Kl and lanl ::; e- n+1. outside K. Since AllK = C(K) and thus AllKl = C(Kd, there exists a positive real number M such that for every positive integer n there exists gn E
Al such that gnan = 1 on Kl with IIgnlloo ::; M. Since AllK = C(K), there is a function af E Al with af = 1 on K. Then afgnan E Al and by a simple calculation lIafgnan - 11100 --+ 0 as n --+ 00, that
is, 1 E AI' 0 References [lJ A. Bernard, Espaces des parties relies des Iments d'une algebre de Banach de fonctions, J. Funct. Anal. 10 (1972), 387-409. [2J E. Briem, Approximations from Subspaces of
Co(X), J. Approx. Theory 112 (2001), 279-294. [3J A. Browder, Introduction to Function Algebras, W. A. Benjamin, Inc. !969. [4J R.B. Burckel, Characterizations of C(X) among its subalgebras (Lecture
Notes in Pure and Appl. Math. 6). Marcel Dekker, New York 1972. [5J O. Hatori, Functions which operate on the real part of a uniform algebra, Proc. Amer. Math. Soc. 83 (1981), 565-568. [6J O. Hatori,
Separation properties and operating functions on a space of continuous functions, lntemat. J. Math. 4 (1993), 551-600. [7J O. hatori, Range transformations on a Banach function algebra. IV, Proc.
Amer. Math. Soc. 116 (1992), 149-156. [8J S. J. Sidney, Functions which operate on the real part of a uniform algebra, Pac. J. Math. 80 (1979), 265-272. [9J S. J. Sidney and E. L. Stout, A note on
interpolation, Proc. Amer. Math. Soc. 19 (1968), 380-382. [10J J. Wermer, The space of real parts of a function algebra, Pac. J. Math. 13 (1963), 1423-1426. SCIENCE INSTITUTE, UNIVERSITY OF ICELAND.
E-mail address:briemillhi.is
Contemporary Mathematics Volume 328, 2003
Some mapping properties of p-summing operators with Hilbertian domain Qingying Bu
ABSTRACT. We prove that if H is a Hilbert space, Y is a Banach space and u : H ---+ Y is absolutely p-summing for some p :::: 1, then for any 1 < q < 00, u takes absolutely q-summable sequences in H
into members of iq®Y, the projective tensor product of lq and Y.
Given a real or complex Banach space X and 1 ::; p < 00, we denote by and f;eak(X) the Banach spaces of sequences in X with norms II(xn)nlle;trong(X) = 11(llxnll)nllep and II(xn)nlle;eak(X) =
sUP",*eB x * II(x*xn)nllep, respectively (cf. [4, pp. 32-36]). For 1 < p < 00, let fp(X) denote the space of all (strongly p-summable) sequences in X such that E~=l Ix~(xn)1 < 00 for each (X~)n E
f't},eak(x*), normed by f~trong(x)
II(xn)nlltp(X) = sup
{I ~ x~(xn)1
II(x~)nlle~~eak(X.) ::; 1} ,
where pI is the conjugate of p, i.e., lip + 11pl = 1. With this norm fp(X) is a Banach space (cf. [1, 3]). Note: In [2] it was shown that fp(X) is exactly fp®X, the projective tensor product of fp
and X. In this note we use this identification of fp(X) with fp®X to deduce a surprising mapping property of absolutely p-summing operators that have a Hilbert space domain. While the main result of
this note can be derived from some by-now famous results of Kwapien, it was discovered because of the identification of fp(X) with fp®X, moreover, this identification leads itself to a proof that is
a clean and clear application of Khinchin's inequality, Kahane's inequality, and Pietsch's Domination theorem - all fundamental aspects of the theory of p-summing operators. From the definitions, we
have for 1 < p < 00, fp(X) ~ f;trong(x) ~ f;eak(x), and 11·lIe~..ak(X) ::; 1I'lIl~trong(X) ::; 11·llep(x)· Moreover, in case dimX = 00, all the containments are proper. For Banach spaces X and Y and
a continuous linear operator 'U : X ---+ Y, define ft. : XN ---+ yN by (xn)n 1---+ (uxn)n. Then ft. is a linear operator. Thanks 2000 Mathematics Subject Classification. 46B28. © 145
2003 American Mathematical Society
to the Closed Graph Theorem, each of
it: f;eak(x)
f~trong(x) ----> f~trong(y);
it: fp(X)
is a continuous operator with
Ilitll£;:,eak(x)_e;:,eak(Y) = Ilitll£~trong(x)_e~trong(y) =
Ilitllep(x)-£p(Y) = Ilull·
We should mention here Khinchin's inequality (cf. [4, p. 10]) and Kahane's inequality (cf. [4, p. 211]) each of which plays a critical role in this paper. Let rn(t) denote the Rademacher functions
(cf. [4, p. 10]), namely, rn : [O,IJ ----> ~, n E Pi! defined by rn(t) := sign(sin2n7rt).
< p < 00, there are positive constants Ap, Bp such that for any scalars a1, a2, ... ,an, we have
Khinchin's Inequality. For any 0
Kahane's Inequality. If 0 < p, q < which
([II t, r,('lx, I ,d')
1/, ,;
then there is a constant Kp,q > 0 for
K",' ([II
r,('lx,lI' dt )
regardless of the choice of a Banach space X and of finitely m.any vectors Xl, X2, ... , Xn from.X.
In 1967, A. Pietsch [5J introduced p-summing operators between Banach spaces, namely, a Banach space operator u : X ----> Y is called p-summing operator, 1 :::; p < 00, if it takes f~eak(x) into
f~trong(y). Let IIp(X, Y) denote the space of all psumming operators from a Banach space X to a Banach space Y. If u : X ----> Y is a p-summing operator then, thanks to the Closed Graph Theorem, it :
f~eak(x) ----> f~trong(y) is continuous. So we define the p-summing norm 7rp(u) on IIp (X, Y) to be 7rp (u) = Ilitll£;:,eak(x)_£~trong(y). With this norm IIp (X, Y) is a Banach space. There is an
equivalent definition of p-summing operators, namely, a Banach space operator u : X ----> Y is p-summing if and only if there is a constant c > 0 such that for any XI,X2,'" ,Xn E X, (1)
In this case, 7rp (u)
= inf{c
> 0: for all possible c in (I)}.
In 1973, J. S. Cohen [3J introduced strongly p-summing operators between Banach spaces, namely, a Banach space operator u : X ----> Y is called strongly p-summing operator, 1 < p < 00, if it takes
f~trong(x) into fp(Y). Let Dp(X, Y) denote the space of all strongly p-summing operators from a Banach space X to a Banach space Y. If u : X ----> Y is a strongly p-summing operator then, thanks to
the Closed Graph Theorem, it : f~trong(x) ----> fp(Y) is continuous. So we define a
strongly p-summing norm Dp(u) on Dp(X, Y) to be Dp(u) = lIulll~trong(X)-+lp(Y)' With this norm Dp(X, Y) is a Banach space. There is an equivalent definition of strongly p-summing operators, namely, a
Banach space operator U ': X ~ Y is strongly p-summing if and only if there is a constant c > 0 such that for any Xll X2,'" , Xn E X, and any yi, Y2,'" , y~ E Y*,
In this case, Dp(u) = inf{c > 0: for all possible c in (2)}.
Note: Actually, U E Dp(X, Y) means that u takes absolutely p-summable sequences in X into members of ip®Y.
Main Theorem. Let 1 < p, q < 00; and let H be a Hilbert space and Y be a Banach space. Then IIp(H, Y) ~ Dq(H, Y), i.e., if u : H ~ Y is absolutely p-summing, then u takes absolutely q-summable
sequences in H into members of
PROOF. First consider H = i2 for n E N. Let u E IIp(l2' Y). By Pietsch's Domination Theorem [4, p. 44], there is a regular probability measure J.L on Bl'.J: such that for any x E i 2, lip (
Iluxll ::; 1Tp (U)' Lt'.J: I(x, z)IP dJ.L(z) Now for
2, and Yi'Y2"" ,Y;" E Y*, we have n
Xk =
~Xk ·e· L.-J ,'&"',
k = 1,2,··· ,m.
Then m
L I(Lxk,iuei, yZ)1 k=l
< <
t. (t, lx,.,1 (t,1 t.IlX,II' 1" ' (1' 1t, 2) ,/, ,
(He;, y;)I' ) II'
r,(t)(ue" Yk)
I" II"~ dt)
by Khinchin's inequality,
L· (t." (t.[1 t, I"
x."') II, .
r,(t)(ue, , yk)
(X) • ( [
(4) Now by Kahane's inequality and (3),
([II t, r,(t)"" II" dt) 'I,' "K,.,.· ([II t, r,(t)ue,II'
(L'r I(t,
q ([
u) .
r,(t)e;, z)I' dp(z) )
r,(t)e;, z) dt) dP(Z)),,'
u) .
B, .
dp(z) )'/'
B, .
(5) Combining (4) and (5),
Dq(H, Y) with 1
Dq(u) :::; -A . Bp· Kp,ql· 7rp(u). ql Now consider a general Hilbert space H. Let
Xl, X2,··· , Xm E
H and
Yi,y2,··· ,y:n, E Y*. Then there is an n E N such that span{xk}r is isometrically isomorphic to i 2. Let P be the orthogonal projection from H onto span{xk}r.
Then by (6), m
L I(UPXk, Yk)1
. Bp' Kp,ql' ll"p(U) . II (PXk)i"lI egtron 9 (X)
. Bp' Kp,ql' ll"p(U) ·11(Xk)i"lIe~tr.on9(X) ql This shows us that U E Dq(H, Y) and again 1 Dq(u) :::; A . Bp' Kp,ql' ll"p(u). ql The proof is complete. :
ACKNOWLEDGEMENT. I am grateful to my advisor, Professor Joe Diestel, for his good suggestions for this paper.
References 1. H. Apiola, Duality between spaces of p-summable sequences, (p, q)-summing operators and characterization of nuclearity, Math. Ann. 219 (1976), 53-64. 2. Q. Bu and J. Diestel,
Observations about the projective tensor product of Banach spaces, I - ip®X, 1 < p < 00, Quaestiones Math. 24 (2001),519-533. 3. J. S. Cohen, Absolutely p-summing, p-nuclear operators, and their
conjugates, Math. Ann. 201 (1973), 177-200. 4. J. Diestel, H. Jarchow, and A. Tonge, Absolutely Summing Operators, Cambridge Univ. Press, Cambridge, 1995. 5. A. Pietsch, Absolut p-summierende
abbildungen in normierten riiumen, Studia Math. 28 (1967), 333-353. DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MISSISSIPPI, UNIVERSITY, MS
E-mail address: qbu(llolemiss. edu
Contemporary Ma.thema.tics Volume 328, 2003
The Unique Decomposition Property and the Banach-Stone Theorem Audrey Curnock, John Howroyd, and Ngai-Ching Wong* ABSTRACT. We show an affine version of the Banach-Stone theorem. Given compact convex
sets K and S with unique decomposition property, we show that every surjective linear isometry T between the affine function spaces A(K) and A(S) induces an affine homeomorphism between K and S.
Furthermore, T can be written as a weighted composition operator in this case.
1. Introduction
The celebrated Banach-Stone Theorem states that two compact Hausdorff spaces X and Y are homeomorphic if and only if the corresponding real continuous function spaces C(X) and C(Y) are linearly
isometric (see, for example, [3, Chapter 7]). As is well known, see for example [2, Theorem 1.4.9], C(X) can be identified with the Banach space A(K) of real continuous affine functions on K = {'P E
C(X)* : II'PII = 1 = 'P(l)}, the state space of C(X), where the norm in A(K) is the usual supremum norm. Here K is a Bauer simplex and consequently there is a natural reformulation of the
Banach-Stone theorem in the context of affine geometry as follows. Two Bauer simplexes K and S are affinely homeomorphic if and only if their corresponding affine function spaces A(K) and A(S) are
linearly isometric. Clearly an affine homeomorphism between any two compact convex sets K and S induces a linear isometry between A(K) and A(S). However, an example of J.T. Chan given in [9] shows
that the converse of this cannot hold for arbitrary compact convex sets in locally convex (Hausdorff) spaces, even in finite dimensions. Thus it is natural to ask what conditions on K and S imply
that this converse does hold. Lazar [11] showed that the converse holds if both K and S are Choquet simplexes. Ellis and So [9] extended this to the case when both K and S have the property that
every pair of complementary closed faces is split, which applies to the state spaces of function algebras. In this paper we use another geometric condition on K and S, namely the unique decomposition
property of Ellis [6, 7, 8], under which K and S are affinely homeomorphic whenever there is a linear isometry T between A(K) and A(S). 2000 Mathematics Subject Classification. Primary 46A55;
Secondary 46B04. Key words and phrases. Affine function space, isometry, unique decomposition property. *Partially supported by Taiwan National Science Council Grants: 89-2115-Mll0-009,37128F. 151
Moreover, we show that in this case T can be written as a weighted composition operator. This includes Lazar's result as a special case and, more generally, the state spaces of C*-algebras. Further
results by the same authors relating to skew-symmetries are to be found in [4]. We would like to express our deepest gratitude to Professor Cho-Ho Chu for many useful discussions. 2. Preliminaries We
note that every compact convex set K (in a locally convex space) can be embedded into A(K)* as the state space {cp E A(K)* : Ilcpll = 1 = cp(l)} of A(K) with the weak* topology, where x E K is
identified with the linear functional f
(j, x) = f(x)
for all f in A(K). If Y is a subset of K then we denote the convex hull of Y by co (Y). A point x in K is called an extreme point of K if K \ {x} is a convex set. The set of extreme points of K is
denoted by 8K. In the above setting the closed unit ball BA(K)' of A(K)* is the convex hull of K and -K; namely co (K U -K) where -K denotes {-k: k E K}. Thus 8B A(K)' is contained in 8KU8(-K). The
reverse inclusion follows easily from the fact that for cp E BA(K)" \ (K U -K) we have -1 < cp(l) < 1. Therefore, 8B A(K)" is equal to 8KU8(-K). A convex subset F of K is called a face of K if
whenever x E F with x = >..y + (1 - >..)z for some y, z E K and >.. E (0,1), then both y and z are in F. A pair of faces (FI, F2) is said to be complementary whenever FI n F2 = 0 and K = co (FI U F 2
). Thus, in this case, each x in K has a decomposition relative to (FI' F2); namely x = >..y + (1 - >..)z for some y E F I , Z E F2 and>" E [0,1]. If >.. is unique, for each x E K \ (FI U F 2 ) but
independent of y and z, then FI and F2 are said to be parallel faces; parallel faces are automatically norm closed. If in addition y and z are unique then FI and F2 are called split faces. See, for
example, [1, 2] for the general theory of compact convex sets and related topics. Let K be a compact convex set. Recall that the facial topology on 8K is given by defining
{F n 8K : F is a closed split face of K} to be the family of all closed sets. The facial topology is weaker than the relative topology on 8K. The centre Z(A(K)) of A(K) is the set of all those
functions in A(K) whose restriction to 8K is facially continuous. The central functions h E Z(A(K)) are characterised by the following property (see, for example, [1, Corollary 11.7.4] or [2, Theorem
3.1.4]): for all f E A(K), there exists 9 E A(K) such that g(x) = h(x)f(x) for all x in 8K. The uniqueness of the continuous affine function 9 is clear, since a continuous affine function on a
compact convex set is completely determined by its values on the extreme boundary, and consequently we may write 9 = h . f. In this way it is useful to think of the central functions as the
multipliers of A(K). A compact convex set K is a Choquet simplex whenever for all bounded (real) linear functionals cp in A(K)* and 0 > 0 the set K n (cp + oK) is either empty or of the form 'IjJ +
(3K for some '¢ in A(K)* and {3 ~ 0 (see, for example, [12]); note that {3 = 0 allows K n (cp + oK) to be a singleton. In a Choquet simplex every closed face is split (see [2, Theorem 2.7.2]).
In [6, 7, 8], Ellis defined the unique decomposition property of K by the condition that for every
0 the set Kn(
O. In particular, every Choquet simplex has the unique decomposition property. A result of Grothendieck [10], see also [5, p. 272], shows that the state space of a (unital) C*-algebra has the unique
decomposition property. Thus the results of this paper apply, giving an affine homeomorphism between the state spaces K and S of two C*-algebras whenever the associated (real) affine function spaces
A(K) and A(S) are linearly isometric. We give some examples below, the second and third of which show that the unique decomposition property and the condition of Ellis and So are independent
geometric properties of a compact convex set. EXAMPLE 2.1. Let K be the state space of the C*-algebra M 2 , of all 2 x 2 matrices over C. Then, by Grothendieck's result, K has the unique
decomposition property. Also, K is affinely homeomorphic to a closed ball in 1R3 (see [2, p. 241]) and satisfies the condition of Ellis and So since it has no proper complementary faces. EXAMPLE 2.2.
Let K be a triangular bi-simplex in 3-dimensional space as in the figure below, Figure 1. Then no proper face is (geometrically) parallel to any other and hence K n (
o. It follows, by the geometric characterisation of Ellis, that K has the unique decomposition property. However (F, F') is a pair of complementary faces of K, but not split, and hence K does not
satisfy the condition of Ellis and So. EXAMPLE 2.3. Let K be an icosahedron in 3-dimensional space. Then K satisfies the condition of Ellis and So because it has no proper complementary faces.
However it does not have t.he unique decomposition property because it has
FIGURE 1. The bi-simplex of Example 2.2
faces (geometrically) parallel to each other and hence K n (cp + aK) can have empty interior for some cp E A(K)* and a > O. 3. Results
Throughout this section K and 8 will denote compact convex sets of locally convex (Hausdorff) spaces. We will also let T: A(K) ~ A(8) denote a surjective linear isometry. Notice that if T1 = 1 then
for cp E A(8)* with Ilcpli = 1 = cp(1), we have IIT*cpll = licp 0 Til = licpli = 1 = cp(1) = cp(T1) = T*cp(1); here T*: A(8)* ~ A(K)* denotes the dual map of T. Consequently T*(8) = K and hence T*
induces an affine homeomorphism a: 8 ~ K such that Tf(s) = T*s(f) = f(a(s)) for all s E 8; that is, T is a composition operator f f-+ f 0 a. In Proposition 3.3 we see that T is a weighted composition
operator if and only if T1 is central. To do this we 'decompose' 8 by defining (3.1)
= {s E 8: (Tl)(s) = I}
82 = {s E 8 : (T1)(s)
= -I}.
It is clear that 8 1 and 8 2 are closed faces of 8. LEMMA 3.1. Let 8 1 and 8 2 be as in (3.1). Then 88 are closed parallel faces of 8.
8 1 U 8 2, and 8 1 and 8 2
PROOF. Observe that the dual map T* is a linear isometry from A(8)* onto A(K)*. Hence T* maps the extreme points of the closed unit ball of A(8)* onto the extreme points of the closed unit ball of A
(K)*. Thus
T*(88 U 8( -8)) = 8K U 8( -K). Consequently, for each s E 88 we have T* s is in 8K or 8( - K) and hence
Tl(s) = T*s(l) = ±1. Therefore 88 ~ 8 1 U 8 2 and thus by the Krein-Milman Theorem 8 = co (81 U 8 2), since 8, 8 1 and 82 are all (weak*) compact. Thus 8 1 and 8 2 are complementary faces since they
are clearly disjoint. For each s in 8 with s = >.x + (1 - >')y, where x E 8 1 , Y E 8 2, and>' E (0,1), we have T* s = >'T*x + (1 - >')T*y. Thus,
= T*s(1) = >'T*x(1) + (1- >')T*y(1) = >. - (1- >.) = 2>' - 1. This establishes the uniqueness of>. = ((Tl)(s) + 1)/2 and the result follows. Tl(s)
We now specialise to the case when K and 8 have the unique decomposition property. LEMMA 3.2. Let 8 1 and 8 2 be as in (3.1). 8uppose that K has the unique decomposition property. Then 8 1 and 8 2
are complementary split faces of 8. PROOF. By Lemma 3.1, for each s E 8\(81 U 8 2) we may write s = >.x + (1 - >.)y where x E 8 1 , Y E 8 2 and 0 < >. < 1, and>' is unique. We consider the
decomposition T* s = >'T*x - (>. -1)T*y. Since x E 8 1 we have T*x E K and hence >'T*x is positive. Similarly, T*y E -K and hence (>. - I)T*y is positive. Also
IIT*sll = 1 = >. + (1- >.) = Ii>'T*xli + 11(>' -
I)T*yli· Thus, by the unique decomposition property, >'T*x and (>. -1)T*y are unique. By the uniqueness of >., we have T*x and T*y are unique. Since T is surjective, T* is injective and the result
follows. 0
By replacing T by T- 1 in (3.1) we may 'decompose' K by defining (3.2) K1 = {k E K : T- 11(k) = I} and K2 = {k E K : T- 11(k) = -I}. Applying Lemmas 3.1 and 3.2 to T- 1 we see that K1 and K2 are
complementary split faces of K whenever S has the unique decomposition property. We say that T is a weighted composition operator whenever there exists a central function h in A(S) and a continuous
affine mapping (1: S -+ K such that Tf = h· f 0 (1 for all f E A(K); that is, Tf(s) = h(s)f((1(s)) for all sEaS. The following proposition asserts that the linear isometry T is a weighted composition
operator, with Tl(s) = ±1 on as, if and only if Tl is central. PROPOSITION 3.3. Let T: A(K) -+ A(S) be a linear mapping. Then the following are equivalent: a) T is an isometry and Tl is central; b) T
is a weighted composition operator of the form T f = h . f 0 (1 for all f E A(K) where (1 is an affine homeomorphism and h(s) = ±1 for all sEas. PROOF. See [4, Theorem 3.3] or [13].
We now apply the above decompositions of K and S to prove our main theorem. THEOREM 3.4. Suppose that K and S have the unique decomposition property. Then the real affine function spaces A(K) and A
(S) are linearly isometric if and only if K and S are affinely homeomorphic. Moreover, every linear isometry from A(K) onto A(S) may be written as a weighted composition operator. PROOF. It suffices
to show necessity. Suppose that T is a linear isometry from A(K) onto A(S). We 'decompose'S into the complementary split faces Sl and S2 of (3.1) and, similarly, K into the complementary split faces
K1 and K2 of (3.2). Since (T-1)* = (T*)-l, we have T*(Sd = K1 and T*(S2) = -K2' and hence we may define (1: S -+ K by (1(,xx + (1 - ,x)y) = ,xT*(x) - (1 - ,x)T*(y)
whenever x E Sl, Y E S2 and 0 :5 ,x :5 1. We see that (1 is an affine homeomorphism from S = co (Sl U S2) onto K = co (K1 U K 2). Moreover, to show that T is a weighted composition operator it
suffices, by Proposition 3.3, to show that h = Tl is central. Let f E A(S) then, since (1: S -+ K is an affine homeomorphism, we may write f = gO(1 for some g E A(K). Note that for x E Sl we have Tg
(x) = T*x(g) = g((1(x)) = h(x)f(x). Similarly for x E S2 we have Tg(x) = T*x(g) = -g((1(x)) = h(x)f(x). Therefore, for all x E as £;; Sl U S2 we have Tg(x) = h(x)f(x), and the result follows. 0
References [1] E.M. Alfsen, Compact convex sets and boundary Integmls, Ergebnisse der Mathematik, 57, (Springer-Verlag, Berlin-Heidelberg-New York, 1971). [21 L. Asimow and A.J. Ellis, Convexity
theory and its applications in functional analysis, London Math. Soc. Monograph, 16 (Academic Press, London, 1980). [3] E. Behrends, M -Structure and the Banach-Stone Theorem, Lecture Notes in
Mathematics 736, (Springer-Verlag, Berlin-Heidelberg-New York, 1979). [4] A. Curnock, J. Howroyd, and N.-C. Wong, Isometries of affine function spaces, preprint. [5] J. Dixmier, C*-Algebms
(North-Holland Publishing Co., Amsterdem-New York-Oxford, 1982).
[6] A.J. Ellis, An intersection property for state spaces, J. London Math. Soc., 43 (1968), 173176. [7] A.J. Ellis; Minimal decompositions in partially ordered normed spaces, Proc. Camb. Phil. Soc.,
64 (1968),989-1000. [8] A.J. Ellis, On partial orderings of normed spaces, Math. Scand., 23 (1968), 123-132. [9] A.J. Ellis and W.S. So, Isometries and the complex state spaces of uniform algebras,
Math. Z., 195 (1987), 119-125. [10] A. Grothendieck, Un result at sur Ie dual d'une C*-algebre, J. Math. Pures Appl., 36 (1957), 97-108. [11] A.J. Lazar, Affine products of simplexes, Math. Scand.,
22 (1968), 165-175. [12] R.R. Phelps, Lectures on Choquet's Theorem, Second Edition, Lecture notes in Mathematics 1757 (Springer-Verlag, Berlin, 2001). [13] T.S.R.K. Rao, Isometries of Ac(K), Proc.
Amer. Math. Soc., 85 (1982), 544-546. SCHOOL OF COMPUTING, INFORMATION SYSTEMS AND MATHEMATICS, SOUTH BANK UNIVERSITY, LONDON SE1 OAA, ENGLAND. E-mail address:curnocaOsbu.ac.uk DEPARTMENT OF
MATHEMATICAL SCIENCES, GOLDSMITHS COLLEGE, UNIVERSITY OF LONDON, LONDON SE14 6NW, ENGLAND. E-mail address:masOljdhOgold.ac.uk DEPARTMENT OF ApPLIED MATHEMATICS, NATIONAL SUN YAT-SEN UNIVERSITY,
KAOHSJUNG 80424, TAIWAN, R.O.C.
Contemporary Mathematics Volume 328, 2003
A Survey of Algebraic Extensions of Commutative, Unital Normed Algebras Thomas Dawson ABSTRACT. We describe the role of algebraic extensions in the theory of commutative, unital normed algebras, with
special attention to uniform algebras. We shall also compare these constructions and show how they are related to each other.
Introduction Algebraic extensions have had striking applications in the theory of uniform algebras ever since Cole used them (in [5]) to construct a counterexample to the peak-point conjecture. Apart
from this, their main use has been in (a) the construction of examples of general, normed algebras with special properties and (b) the Galois theory of Banach algebras. We shall not discuss (b) here;
a summary of some of this work is included in [29]. In the first section of this article we shall introduce the types of extensions and relate their applications. The section ends by giving the exact
relationship between the types of extensions. Section 2 contains a table summarising what is known about the extensions' properties. A theme lying behind all the work to be discussed is the following
(Q) Suppose the normed algebra B is related to a subalgebra A by some specific property or construction. (For example, B might be integral over A: every element b E B satisfies ao + ... +an_1bn-1 +bn
= 0 for some ao, ... ,an -1 EA.) What properties of A (for example, completeness or semisimplicity) must be shared by B? This is a natural question, and interesting in its own right. Many special
cases of it have been studied in the literature. We shall review the related body of work in which B is constructed from A by adjoining roots of monic polynomial equations. Throughout this article, A
denotes a commutative, unital normed algebra, and A its completion. The fundamental construction of [1] applies to this class of 1991 Mathematics Subject Classification. Primary 46J05, 46J10. This
research was supported by the EPSRC. © 2003 American Mathematical Society 157
algebras. Algebraic extensions of more general types of topological algebras have received limited attention in the literature (see [19], [21]). If E is a subset of a ring then (E) will stand for the
the ideal generated by E. 1. Types of Algebraic Extensions and their Applications 1.1. Arens-Hoffman Extensions. Let a(x) = ao + ... + an_lx n- l + xn be a monic polynomial over the algebra A. The
basic construction arising from A and a(x) is the Arens-Hoffman extension, Ao. This was introduced in [1]. Most of the obvious questions of the type (Q) for Arens-Hoffman extensions were dealt with
in this paper and in the subsequent work of Lindberg ([18]' [20], [13]). See columns two and three of Table 2.2. All the constructions we shall meet are built out of Arens-Hoffman extensions.
DEFINITION 1.1.1. A mapping 0: A - B between algebras A and B is called unital if it sends the identity of A to the identity of B. An extension of A is a commutative, unital normed algebra, B,
together with a unital, isometric monomorphism
The Arens-Hoffman extension of A with respect to a(x) is the algebra Ao := A[x]/(a(x)) under a certain norm; the embedding is given by the map v: a t-+ (a(x)) + a.
To simplify notation, we shall let x denote the coset of x and often omit the indeterminate when using a polynomial as an index. It is a purely algebraic fact that each element of Ao has a unique
representative of degree less than n, the degree of a(x). Arens and Hoffman proved that, provided the positive number t satisfies the inequality t n ~ ~~:~ lIakll tk, then
~ IIbkll t
(bo, ... , bn -
defines an algebra norm on Ao. The first proposition shows that Arens-Hoffman extensions satisfy a certain universal property which is very useful when investigating algebraic extensions. It is not
specially stated anywhere in the literature; it seems to be taken as obvious. 1.1.2. Let A(l) be a normed algebra and let 0: A(l) _ B(2) be a unital homomorphism of normed algebras. Let al (x) = ao +
... + an_lXn - l + xn E A(l) [x] and B(1) = A~l}. Let y E B(2) be a root of the polynomial a2(x) := O(ad(x) := O(ao) + ... +O(an_l)X n - l +xn. Then there is a unique homomorphism 1>: B(l) _ B(2)
such that PROPOSITION
is commutative and 1>(x) = y.
The map 1> is continuous if and only if 0 is continuous. PROOF.
This is elementary; see [7]
1.2. Incomplete Normed Algebras. A minor source of applications of Arens-Hoffman extensions fits in nicely with our thematic question (Q): these extensions are useful in constructing examples to show
that taking the completion of A need not preserve certain properties of A. The method uses the fact that the actions of forming completions and ArensHoffman extensions commute in a natural sense. A
special case of this is stated in [17]; the general case is proved in [7), Theorem 3.13, and follows easily from Proposition 1.1.2. It is convenient to introduce some more notation and terminology
here. Let O(A) denote the space of continuous epimorphisms A --+ Cj when n appears on its own it will refer to A. As discussed in [1], this space, with the weak *-topology relative to the topological
dual of A, generalises the notion of the maximal ideal space of a Banach algebra. In fact, it is easy to check that 0 is homeomorphic to 0(..4), the maximal ideal space of the completion of A. The
Gelfand transform of an element a E A is defined by
a: n --+ C; W
and the map sending a to a is a homomorphism, r, of A into the algebra, C(O), of all continuous, complex-valued functions on the compact, Hausdorff space O. We denote the image of r by A. A good
reference for Gelfand theory is Chapter three of [24]. DEFINITION
1. 2.1 ([1)). The algebra A is called topologically semisimple if
is injective. If A is a Banach algebra then this condition is equivalent to the usual notion of semisimplicity. The precise conditions under which Aa is topologically semisimple if A is are
determined in [1]. In [17] Lindberg shows that the completion of a topologically semisimple algebra need not be semisimple. In order to illustrate Lindberg's strategy we recall two standard
properties of normed algebras.
1.2.2. The normed algebra A is called regular if for each closed subset E ~ 0 and wE O-E there exists a E A such that a(E) ~ {O} and a(w) = 1. The algebra is called local if A contains every complex
function, f, on 0 such that every wE 0 has a neighbourhood, V, and an element a E A such that flv = alv. DEFINITION
It is a standard fact that regularity is stronger than localness; see Lemma 7.2.8 of [24). EXAMPLE 1.2.3. Let A be the algebra of all continuous, piecewise polynomial functions on the unit interval,
I, and a(x) = x 2 - id/ E A[x]. Let A have the supremum norm. By the Stone-Weierstrass theorem, A = C(I) and hence n is identifiable with I. Clearly A is regular. We leave it as an exercise for the
reader to find examples to show that Aa is not local. This is not hardj it may be helpful to know that in this example the space O(Aa) is homeomorphic to {(s, oX) E I xC: oX 2 = s}. This follows from
facts in [1]. In the present example, neither localness nor regularity is preserved by (incomplete) Arens-Hoffman extensions.
Finally we can explain the method for showing that some properties of normed algebras are not shared by their completions because, in the above, 'non-regularity' is not preserved by completion of Ao
(nor is 'non-localness'). To see this, note that Ais clearly regular if A is and so by a theorem of Lindberg (see Table 2.2) the ArensHoffman extension (A)o is regular. But, by a result of [17]
referred to above, this algebra is isometrically isomorphic to the completion of Aa. Of course Lindberg's original application was much more significant; there are simpler examples of the present
result: for example the algebra of polynomials on I.
1.3. Uniform Algebras. It is curious that the application of Arens-Hoffman extensions to the construction of integrally closed extensions of normed algebras did not appear in the literature for some
time after [1]. It was seventeen years later until a construction was given in [22]. Even then the author acknowledges that the constuction was prompted by the work of Cole, [5], in the theory of
uniform algebras. Cole invented a method of adjoining square roots of elements to uniform algebras. He used it to extend uniform algebras to ones which contain square roots for all of their elements.
Apart from feeding back into the general theory of commutative Banach algebras (mainly accomplished in [22] and [23]) his construction provided important examples in the theory of uniform algebras.
We shall describe these after recalling some basic definitions. DEFINITION 1.3.1. A uniform algebra, A, is a subalgebra of C(X) for some compact, Hausdorff space X such that A is closed with respect
to the supremum norm, separates the points of X, and contains the constant functions. We speak of 'the uniform algebra (A, X)'. The uniform algebra is natural if all of its homomorphisms wEn are
given by evaluation at points of X, and it is called trivial if A = C(X).
Introductions to uniform algebras can be found in [4], [11], [26], and [16]. An important question in this area is which properties of (A, X) force A to be trivial. For example it is sufficient that
A be self-adjoint, by the Stone-Weierstrass theorem. In [5] an example is given of a non-trivial uniform algebra, (B,X), which is natural and such that every point of X is a 'peak-point'. It had
previously been conjectured that no such algebra existed. We shall describe the use of Cole's construction in the next section, but now we reveal some of the detail. PROPOSITION 1.3.2 ([5],[7]). Let
U be a set of monic polynomials over the uniform algebra (A, X). There exists a uniform algebra (AU, XU) and a continuous, open surjection 7r: XU ---> X such that (i) the adjoint map 7r*: C(X) ---> C
(XU) induces an isometric, unital monomorphism A ---> AU, and (ii) for every a E U the polynomial 7r*(a)(x) E AU[x] has a root Po E AU. PROOF. We let XU be the subset of X x such that for all a E U f
cU consisting of the elements (K, >.)
+ ... + /(0) (K)>.n(0)-1 + >.n(o) n(a)-1 a 0
= 0
where o:(x) = f~Ol) + ... + f~(l)_lXn(Ol)-l + xn(Ol) E U. The reader can easily check that XU is a compact, Hausdorff space in the relative product topology and so the following functions are
continuous: 11":
XU -+ X; (~, >.) ...... ~
POI: XU -+ C; (~, >.) ...... >'01
E U).
The extension AU is defined to be the closed subalgebra of C(XU) generated by 1I"*(A) U {POI: 0: E U} where 11"* is the adjoint map C(X) -+ C(XU) ; g ...... go 11". It is not hard to check that AU is
a uniform algebra on XU with the required properties. 0 We shall call AU the Cole extension of A by U. Cole gave the construction for the case in which every element of U is of the form x 2 - f for
some f E A. It is remarked in [22] that similar methods can be used for the general case; these were independently, explicitly given in [7]. By repeating this construction, using transfinite
induction, one can generate uniform algebras which are integrally closed extensions of A. Full details of this, including references and the required facts on ordinal numbers and direct limits of
normed algebras, can be found in [7]. Again this closely follows [5]. Informally the construction is as follows. Let v be a non-zero ordinal number. Set (Ao,Xo) = (A,X). For ordinal numbers T with 0
< T ~ V we define (A~" , X!:" )
(A.,-, X T )
if T = a
= { l~ . (A u,Xu)u<.,-, (* 1I"p,u, 1I"p,u ) p~U
if T is a limit ordinal.
The construction requires sets of monic polynomials, Uu ~ Au [x], to be chosen inductively. The notation (Au, Xu )u<.,-, (1I";,u, 1I"p,u )P~U
0 be an ordinal number. Then (A.,-, X T)T
(A, X).
Thus (A1' Xl) is just a Cole extension of (Ao, Xo). When U1 is a singleton we call Al a simple extension of Ao; the same adjective can be applied to ArensHoffman extensions. An integrally closed
extension, (Av, Xv), is obtained by taking v to be the first uncountable ordinal. At the successor ordinals the whole set of monic polynomials is frequently used to extend the algebra, but this set
is larger than necessary. The same procedure is used to obtain the integrally closed extensions in other categories (to be discussed in Section 1.6).
1.4. Some Applications of Cole's Construction. Cole's method has been developed by others, including Karahanjan and Feinstein, to produce examples of non-trivial uniform algebras with interesting
combinations of properties. We cite the following example of Karahanjan. THEOREM 1.4.1 (from [15], Theorem 4). There is a non-trivial, antisymmetric uniform algebra, A, such that (1) A is integrally
closed, (2) A is regular, (3) n is hereditarily unicoherent, (4) G(A) is dense in A, and (5) the set of peak-points of A is equal to n. In the above, G(A) is our notation for the invertible group of
A. We refer the reader to [15] and the literature on uniform algebras for the definitions of other terms we have not defined here. A further example in [15] also strengthens Cole's original
counter-example. Both examples (of non-trivial, natural uniform algebras on compact, metriseable spaces, every point of which is a 'peak-point') are regular. Feinstein has varied the construction to
obtain such an example which is not regular in [10]. The same author also used Cole extensions in [9] to answer a question of Wilken by constructing a non-trivial, 'strongly regular', uniform algebra
on a compact, metriseable space. Returning to the sample theorem quoted above, note that some of these properties (for example the topological property of 'hereditary unicoherence') are consequences
of the combination of other properties of the final algebra. By contrast, (2) and (4) hold because they are true for the base algebra on which the example is constructed. It is therefore very useful
to know exactly when specific properties of a uniform algebra are transferred to those in a system of Cole extensions of it. The known results on this problem are summarised in the first column of
Table 2.2. Determining if an algebra's property is shared by its algebraic extensions has led to some interesting devices. We shall elaborate on this topic in the next section. We remark in passing
that the methods used in [15] to show that the final algebra has a dense invertible group have been simplified in [8]; in particular there is no need to develop the theory of 'dense thin systems' in
1.5. A Further Remark on Cole Extensions. The reader will notice from Table 2.2 that virtually all properties of uniform algebras are preserved by Cole extensions. The key to obtaining most of these
results is the following result, originating with Cole. PROPOSITION 1.5.1 ([5]'[23]). Let (Ar,Xrk:;v be a system of Cole extensions of (A, X). There exists a family of unital contractions (T.,.,r: C
(Xr) -+ C(X.,.)).,.::;r::;v such that for all a ~ T ~ v (i) T.,.,r(A r ) ~ A.,., and (ii) T.,.,r 07r;,r = idc(X u )'
PROOF. See [23]. For example, it is easy to see from the existence of T: C(Xu) Cole extension AU is non-trivial if A i= C(X).
C(X) that the
The operator T was constructed in [5] for extensions by square-roots. In the case of a simple Cole extension, (A{o}, X{o}), there are at most two points y±(x:) in the fibre 7r- l (x:) for each x: E X
and they correspond to the roots of the equation x 2 - f(x:) = 0 where o(x) = x 2 - f. The operator is then defined by
(g E C(X{o}), x: EX). For other sorts of monic polynomials it was not so obvious how to construct T. The basic techniques appeared in [22] (see the proof of Theorem 3.5) for simple extensions, and
were further developed in the proof of Theorem 4 of [15], but it was not until [23] that a comprehensive construction was given. We must also mention the role of E. A. Gorin: he appears to have paved
the way for [15] and [23].
1.6. Algebraic Extensions of Normed and Banach Algebras. As we have seen, algebraic extensions have had striking applications in the theory of uniform algebras. They have long been used as auxiliary
constructions in the general theory of Banach algebras. Notable examples of this are in [14] and [25]; the latter explicitly uses Arens-Hoffman extensions. However algebraic extensions for Hormed
algebras were apparently only studied in their own right in order to generalise the work of Cole and Karahanjan. We now turn to these generalisations. The basic extension generalising Arens-Hoffman
extensions is called a standard normed extension. It is defined in the following theorem of Lindberg. THEOREM 1.6.1 ([22]). Let A be a normed algebra and U a set of monic polynomials over A. Let ~ be
a well-ordering on U with least element 00' Then there exists a normed algebra, B u , with a family of subalgebras, (Bo)oEU, such that: (i) for all 0, (3 E U, Bo ~ Bf3 if 0 ~ (3, and, (ii) for all (3
E U, Bf3 is isometrically isomorphic to an Arens-HoffmBll extension of B
PROOF. See [22].
Lindberg shows how this leads to the construction of Banach algebras with interesting combinations of properties, one of which is integral closedness. Let the isometric isomorphism B
Let ~Ot be the standard root of a E U, witll associated norm parameter tOt, and suppose (T/Ot)OtEU ~ B(2) is such that £I(a)(T/Ot) = 0 for all a E U. Tllen there is a unique, unital homomorphism ¢: B
(l) --+ B(2) such that the following diagram is commutative B(1)
(Note added in proof: The map ¢ is continuous if and only if £I is continuous and
L OtEU
(n(a) -l)log+
(11~OtII) < +00 Ot
where log+ denotes the positive part of the logarithm, max(log,O).) PROOF. A simple application of transfinite methods and Proposition 1.1.2. 0 Purely algebraic standard extensions are defined in
[22] and the main content of Lemma 1.6.2 is a statement about these. Narmania gives ([23]) an alternative construction for integrally closed extensions of a commutative, unital Banach algebra, A. His
method is rather more conventional than the one used to define standard extensions. If U is a set of monic polynomials over A then the Narrnania extension of A by U is equal to the Banach-algebra
direct limit of (As: 8 is a finite subset of U) where each As is isometrically isomorphic to A extended finitely many times by the Arens-Hoffman construction. As this paper is not readily available
in English and we shall refer to the explicit construction of Narmania's extensions in the next result, we stop to report the precise details of this. If E is a set, the set of all finite subsets of
E will be written E<wo. Let 8 = {ai, ... , am} ~ U and let tOt (a E U) be a valid choice of Arens-Hoffman normparameters (see Section 1.1). It is important to insist that distinct elements a, /3 E U
are associated with distinct indeterminates XOt , x{3. Thus 8 is an abbreviation for {a1(xoJ, ... ,am(xo,J}. It is proved carefully in [23] that for q = Ls qsx~ll ... x~;;, E A[x U1 ' · •• ,xo: m ],
the algebra of polynomials in m commuting indeterminates over A (s is a mult.iindex in No where No = {O}UN), then (8) +q has a unique representative whose degree in 3:O: j is less than than n(aj),
the degree of aj(:ro: j ) (j = 1, ... ,m). For convenience we shall call such representatives minimal. Then if q is the minimal representative of (8) +q, 11(8) + qll := Ls IIqsll t~ll ••• t~: defines
an algebra norm on As. The index set, U<wo is a directed set, directed by ~. The connecting homomorphisms VS.T (for 8 ~ T E U<wo) are the natural maps; they are isometries. Thus (sec [24] Section
1.3) Au is the completion of the normed direct limit, D := UsEU<wo As / "', where '" is an equivalence relation given by (8) + q '" (T) + r if and only if q - r E (8 UT) for 8, T E U<wo. Furthermore,
the canonical map, Vs, which sends an element of As to its equivalence class in D, is an isometry. Note that A0 is defined to be A. We can now show how the types of extensions we have been
considering are related. Many of the idea.'3 behind Proposition 1.6.3 are due to Narmania but we take the step of linking them to Cole and standard extensions.
1.6.3. Let A be a commutative, unital Banach algebra and U a set of monic polynomials over A. Then, up to isometric isomorphism, Au = Bu. If A is a uniform algebra then we have PROPOSITION
= (Au)" = (Bur,
where the closures are taken with respect to the supremum norm. PROOF.
It is easily checked that if B is a normed algebra then the homeomor-
phism 12(B) ---t O(B) induces an isometric isomorphism B" ---t (B)". It is therefore sufficient to prove that Au = Bu and that AU = (Bu)". The last equality follows very quickly from the universal
property of standard extensions mentioned above and the simplicity of the definition of AU. We shall only prove the first identification; the second can be proved by a similar approach. Although what
follows is routine, we hope that it will help to clarify the details of standard and Narmania extensions. As before let to (0 E U) be a valid choice of Arens-Hoffman norm-parameters for the
respective extensions Ao. We shall show that there is then an isometric isomorphism between Bu and D (when defined by these parameters); the result then follows from the uniqueness of completions.
For each a E U let Yo be the equivalence cla..<;s [({o(:c o )}) + xo] E D. Since Yo is a root of V0(0)(X) in D there exists, by the universal property of standard extensions, a (unique) homomorphism
¢: Bu ---t D such that ¢I A = V0 and for all a E U, ¢(~o) = Yo' Here, ~o is the the element of Bu associated with x by the isometric isomorphism 1/)0 : B
L,j!!j-111¢(bj)11 tb.
Since the algebras vs(As) are directed there exists 8 E U<W(I such that ¢(bj ) E vs(As) (j = 0, ... ,n(,6) -1). We can assume that 8 = {al,'" ,am} and ol(X) = ,6(x). Let qo,·.· ,qn((3)-IE A[XQ1""
,XOm] be the minimal representatives such that ¢(bj ) = [(8) + qj] (j = 0, ... ,n(,6) -1). So Ilbjll = Ilqjll (j = 0, ... ,n(,6) - 1). A routine exercise in the transfinite induction theorem shows
that for all 'Y E U, ¢(B"() ~ UTE[O,"(]<w" vT(AT). It follows that the
degree of qj in
is zero. Hence n({3)-l
11¢(b)11 =
reS) + qjX~l]
j=O n(tJ)-l
(S) +
j=O n({3)-l
II t~l
Ilbll ,
from above. The penultimate equality above follows from noting that the representative of the coset is minimal and then expanding and collecting terms. By the transfinite induction theorem, .:J = U
as required. 0
2. A Survey of Properties Preserved by Algebraic Extensions 2.1. Introduction. We summarise in Table 2.2 what is currently known about the behaviour of certain properties of normed algebras with
respect to the types of extensions we have been considering. Some preliminary explanation of the entries is in order first. Extra information about the polynomial(s) generating an algebraic extension
can help to determine whether certain properties are preserved or not. For example if a(x) has degree nand factorises completely over A with distinct roots AI, ... ,An E A such that for all W E fl,
.x:(w) =I- :X;(w) if i =I- j then n(Ao<) decomposes into n disjoint homeomorphs of fl in which case very many properties of A, for example localness, are shared by Ao<' This property, referred to as
'complete solvability', is investigated in [12]. The condition on a(x) most frequently encountered in the literature is that it should be 'separable'. This means that its 'discriminant', which is a
certain polynomial in the coefficients of a(x), is invertible in A. It is interesting to compare columns two and three. Of course one can make additional assumptions on the algebra (for example that
A be regular and semisimple) but the resulting table would become too large and we have restricted it to three popular categories. References to the results follow the table. We should mention that
some of the entries have trivial explanations. For example Sheinberg's theorem, that a uniform algebra is amenable if and only if it is trivial, explains the entries for amenability in column one.
Also, applying the Arens-Hoffman construction to a uniform algebra need not result in a uniform algebra so not all the entries make sense. We have already met most of the properties listed in the
table. We end this section by discussing the ones which have not yet been specially mentioned. 1. Denseness of the invertible group. Although this property is self-explanatory it might not be obvious
why it is listed. However, the condition G(A) = A appears in the literature in various contexts; see for example [8].
2. The Banach algebra, A, is called sup-norm closed if A is uniformly closed in C(f2) (and therefore a uniform algebra). It is called symmetric if A is selfadjoint. 3. For the definitions of
'amenability' and 'weak amenability' we refer the reader to section 2.8 of [6]. All the properties in the table are preserved by forming the standard unitisation of a normed algebra. Most of these
results are standard facts or ea'iy exercises; some are true by definition. However this question does not fit into our scheme because the embedding is not unital in this case.
2.2. Table. Cole extensions have only been defined for uniform algebra'l; the algebra is therefore assumed to be a uniform algebra throughout column one. Colulllns two and three, a'l mentioned above,
refer to Arens-Hoffman extensions of a normed algebra, A, by a monic polynomial a(x); in column three it is given that a(:1:) is separable. Type of Extension: Property:
A.-H. standard Narmania a sep.
for normed algebras
complete topologically semisimple non-local local regular
2. 3. 4. 5.
• • • ?
• 0
• • •
? '?
• • • • •
• •
• • • •
for Banach algebras
6. 7. S. 9. 10.
local regular dense invertible group sup-norm closed symmetric 11. amenable 12. weakly amenable
for uniform algebms
13. non-trivial 14. trivial 15. natural
• • •
Key • property is always preserved o property is sometimes, but not always preserved
? ?
? ?
• •
? not yet determined - it doesn't always make sense to consider this property here References for the Entries. If we do not mention an entry here, it can be taken that the result is an immediate
consequence of the definition or was proved in the same paper in which the relevant extension was introduced (that is in [5], [1], [22], or [23]). The results of row three are not hard to obtain,
using appropriate versions of Proposition 1.5.1. Localness and regularity were discussed in Section 1.2. The main result about this is due to Lindberg in [18]; the same section of his paper also
deals with the results on the symmetry of Arens-Hoffman extensions. That regularity passes to direct limits of such extensions has been widely noted by many authors, for example in [15]. Results of
row eight follow from [8]; the case of Cole extensions was partially covered in [15], but the reasoning is not clear. The property of being sup-norm closed was investigated in [13]; this work was
generalised in [28]. Finally, examples of amenable Banach algebras which do not have even weakly amenable Arens-Hoffman extensions have been known for a long time. For example, the algebra C E9 C
under the multiplication (a, b) (e, d) = (ae, be + ad) is realisable as an Arens-Hoffman extension of C. Examples with both A and Ao semisimple have been found by the author. However the entries
marked "?' in rows eleven and twelve represent intriguing open problems. 3. Conclusion The table in Section 2.2 still has gaps, and there are many more rows which could be added. For example it would
be interesting to be able to estimate various types of 'stable ranks' (see [2]) of the extensions in terms of the stable ranks of the original algebras. (The condition G(A) = A is equivalent to the
'topological stable rank' of A not exceeding 1.) Remember too that there are many more questions which can be asked, of the form: 'if n has the topological property P, does n(Ao) have property P?' By
way of a conclusion we repeat that algebraic extensions have proved immensely useful in the construction of examples of uniform algebras. There is therefore great scope for and potential usefulness
in augmenting Table 2.2. It might also be valuable to reexamine the techniques used to obtain the entries to produce more general results (of the kind in [28] for example) in the context of question
References 1. Arens, R. and Hoffman, K., Algebraic Extension of Normed Algebras., Proc. Am. Math. Soc. 7 (1956), 203-210. 2. Badea, C., The Stable Rank of Topological Algebras and a Problem of R. G.
Swan., J. Funet. Anal. 160 (1998), 42-78. 3. Batikyan, B. T., Point Derivations on Algebraic Extension of Banach Algebra., Lobachevskii J. Math. 6 (2000), 3-37. 4. Browder, A., Introduction to
Function Algebras., W. A. Benjamin, Inc., New York, 1969.
5. Cole, B. J., One-Point Parts and the Peak-Point Conjecture., Ph.D. Thesis, Yale University, 1968. 6. Dales, H. G., Banach Algebms and Automatic Continuity., Oxford University Press Inc., New
York,2000. 7. Dawson, T. W .. Algebmic Extensions of Normed Algebms., M.Math. Dissertation, accessible from the web at: http://xxx.lanl.gov/abs/math.FA/0102131, University of Nottingham, 2000. 8.
Dawson, T. W., and Feinstein, J. F., On the Denseness of the Invertible Group in Banach Algebms., Proc. Am. Math. Soc. (to appear). 9. Feinstein, J. F., A Non-Trivial, Strongly Regular Unif01m
Algebm., J. Lond. Math. Soc. 45 (1992), 288-300. 10. Feinstein, J. F., Trivial Jensen Measures Without Regularity., Studia Math. 148 (2001).6774. 11. Gamelin, T. W., Uniform Algebms., Prentice-Hall
Inc., Engelwood Cliffs, N. J., 1969. 12. Gorin, E. A., and Lin, V ..J., Algebmic Equations with Cont'inuous Coefficients and Some Problems of the Algebmic Theory of Bmids., Math. USSR Sb. 7 (1969),
569-596. 13. Heuer, G. A., and Lindberg, J. A., Algebmic Extensions of Continuous Function Algebms., Proc. Am. Math. Soc. 14 (1963),337-342. 14. Johnson, B. E., Norming C(O) and Related Algebms.,
Trans. Am. Math. Soc. 220 (1976), 37-58. 15. Karahanjan, M. I., Some Algebmic Chamcterizations of the Algebm of All Continuous Functions on a Locally Connected Compactum., Math. USSR Sb. 35
(1979),681-696. 16. Leibowitz, G. M., Lectures on Complex Function Algebm,~., Scott, Foresman and Company, Glenview, Illinois, 1970. 17. Lindberg, J. A .• On the Completion of Tractable Normed
Algebms., Proc. Am. Math. Soc. 14 (1963),319-321. 18. Lindberg, J. A., Algebmic Extensions of Commutative Banach Algebms., Pacif. J. Math. 14 (1964), 559-583. 19. Lindberg, J. A., On Singly Genemted
Topological Algebras, Function Algebras (ed. Birtel, F. T.), Scott-Foresman, Chicago, 1966, pp. 334-340. 20. Lindberg, J. A., A Class of Commutative Banach Algebms with Unique Complete Norm Topology
and Continuous Derivations., Proc. Am. Math. Soc. 29 (1971), 516-520. 21. Lindberg, J. A .. Polynomials over Complete l.m.-c. Algebms and Simple Integml Extensions., Rev. Roumaine Math. Pures Appl.
17 (1972), 47-63. 22. Lindberg, J. A., lntegml Extensions of Commutative Banach Algebms., Can. ,/. Math. 25 (1973), 673-686. 23. Narmaniya, V. G., The Construction of Algebmically Closed Exten,~ions
of Commutative Banach Algebms., Trudy Tbiliss. Mat. Inst. Razmadze Akad. 69 (1982), 154-162. 24. Palmer, T. W., Banach Algebras and the Geneml Theory of *-Algebms. Vol. 1, Cambridge University Press,
Cambridge, 1994. 25. Read, C. J., Commutative, Radical Amenable Banach Algebms., Studia Math. 140 (2000), 199-212. 26. Stout, E. L., The Theory of Uniform Algebms., Bogden and Quigley Inc.,
Tarrytown-onHudson, New York, 1973. 27. Taylor, J. L., Banach Algebms and Topology, Algebras in Analysis. (ed. Williamson, J. H.), Academic Press Inc. (London) Ltd., Norwich:, 1975, pp. 118-186. 28.
Verdera, J., On Finitely Genemted and Projective Extensions of Banach Algebms., Proc. Am. Math. Soc. 80 (1980), 614-620. 29. Zame, W. R., Covering Spaces and the Galois Theory of Commutative Banach
Algebms., J. Funet. Anal. 27 (1984), 151-171.
Acknowledgements The author would like to thank the Division of Pure Mathematics and the Graduate School at the University of Nottingham for paying for his expenses in
order to attend the 4th Conference on FUnction Spaces (2002) at the Southern Illinois University at Edwardsville. The author is grateful to Mr. Brian Lockett who provided him with a translation of
the paper [23]. Special thanks are due to Dr. J. F. Feinstein who offered much valuable advice and encouragement and also proofread the article. DIVISION OF PURE MATHEMATICS, SCHOOL OF MATHEMATICAL
SCIENCES, UNIVERSITY OF NOTTINGHAM, UNIVERSITY PARK, NOTTINGHAM, NC7 2RD, UK.
E-mail address:pmxtwdlDnottingham.ac.uk
Contemporary Mat.hematics Volume 328, 200a
Some more examples of subsets of Co and Ll[O, 1] failing the fixed point property P.N. Dowling, C.J. Lennard, and B. Thrett We give examples of closed, bounded. convex, non-weakly compact subsets of
Co on which the right shift is expansive, and we construct two nonexpansive self-mappings (one affine and one non-affine) on these sets which fail to have a fixed point. We also prove that every
closed, bounded, convex subset of L1 [0,1] with a non-empty interior fails the fixed point property for nonexpansive mappings. Finally, we extend this result by showing that every closed, bounded,
convex subset of L1 [0,1] that contains a non-trivial order interval must fail the fixed point property. ABSTRACT.
1. Introduction
In [5], Llorens-Fuster and Sims construct examples of closed, bounded, convex subsets of Co that are not weakly compact but are compact in a topology that is slightly coarser than the weak topology,
and they nonetheless fail the fixed point property for nonexpansive mappings. These examples led Llorens-Fuster and Sims to conjecture that closed, bounded, convex non-empty subsets of Co have the
fixed point property if and only if they are weakly compact. This conjecture has been recently settled in the affirmative [2, 3]. All the examples constructed in [5] had a common feature - they all
support a nonexpansive right shift. In the first part of this short note we produce a collection of sets of the type considered by Llorens-Fuster and Sims, but which do not support a nonexpansive
right shift and yet they fail the fixed point property for nonexpansive mappings. In fact, we will produce two nonexpansive fixed point free mappings: one affine and the other non-affine. Variations
on the themes of these examples are important in the papers [2, 3]. Llorens-Fuster and Sims [5] also proved that a closed, bounded, convex subset of Co with non-empty interior fails the fixed point
property for nonexpansive mappings. In the second part of tllis note we prove an analogous statement in the setting of L1 [0, 1]. We also generalize this result to show that every closed, bounded,
convex 2000 Mathematics Subject Classification. Primary 47HI0, 47H09, 46E30. The authors wish to thank Professor Kaz Goebel for his helpful suggestions concerning the proof of Theorem 3.2. The second
author thanks the Department of Mathematics and Statistics at Miami University for their hospitality during part of the preparation of this paper. He also acknowledges the financial support of Miami
© 171
2003 American l\1athematical Society
P.N. DOWLING, C.J. LENNARD, AND B. TURETT
subset of L1 [0, 1J that contains a non-trivial order interval must fail the fixed point property. We refer the reader to the text of Goebel and Kirk [4J for any unexplained terminology. 2. The fixed
point property in Co We begin this section with the Llorens-F\lster and Sims examples. We have slightly modified their examples to simplify the computations. Let r denote the set of sequences 'Y = ("
(n)n in the interval (0,1). For all 'Y E r, define K., by
("(ntn)n E Co 112: h 2: t2 2: t3 2: ... 2: O} .
We define the right shift Ton K..., by
T( ("(1 t 1 , 'Y2t2, 'Y3 t3, 'Y4t4, ... ))
("(1, 'Y2t 1, 'Y3 t2, 'Y4t3, ... ).
Note that if x = ("(ntn)n and y = ("(nsn)n are elements of K..." then Ilx - yll sUPn 'Ynlt" - 8 n l and IIT(x) - T(y)11 = SUPn 'Yn+1Itn - 8 n l· Clearly, if the sequence ("(n)n is decreasing, then T
is a nonexpansive mapping on K..., - this is the case considered by Llorens-Fuster and Sims [5J. However, it is equally obvious that if the sequence ("(n)n is strictly increasing, then T is an
expansive mapping on K...,; that is, IIT(x) - T(y)11 > IIx - yll whenever x i= y. We will show that even though some of these sets do not support a nonexpansive right shift, they do support
nonexpansive fixed point free mappings. To simplify our computations we will only consider the sets K..., where the sequence 'Y = ("(n)n is in (0,1), is strictly increasing and satisfies 1 - 'Yn < 4-
n for all n E N. EXAMPLE 2.1. Let I be the identity mapping on K..." let T be the right shift defined above, T2 = ToT, T3 = ToT 0 T, and so on. Define R: K..., --+ K..., by
:= ~I
+ -b T + -1a T2 + 21
+... .
A simple calculation shows that if x = ("(1 t1, 'Y2t2, 'Y3t3, ... ) E K..." then
R(x) = ('Y1(~t1
+ ~),'Y2(~t2 + it1 + i),'Y3(~t3 + it2 + kt1 + ~), ... ). that if R(x) = x, then tn = 1 for all n E N, and thus
It is easily seen x is not an element of K...,; that is, R is fixed point free on K...,. To see that R is nonexpansive on K..." let x = ("(ntn)n and y = ("(nsn)n be elements of K...,. Then, 12 + - 4
1 + ... + 2,,1...,1 ) < 1 for each n E N. since 1 - 4- n < 'Yn < 1, we have 'Yn( -...,,, "",,-1 Consequently,
IIR(x) - R(y)11
sup hnl~(tn - sn) n
+ i(tn-1
- sn-d
+ ... + 2~ (t1
< sup hn(~ltn - snl + iltn-1 - 8 n -11 + ... + 2~' It 1 n
< sup {'Yn(_l+ _1_ + ... + +) max 'Yilti - Sil} n 2...,,, 4"",,-1 2""1 l::;'i::;n
< sUP'Ynltn n
Ilx-yll· Thus R is a nonexpansive mapping on K...,.
st)l} Sl
The mapping R, given in example 2.1, is an affine mapping on K"I' Our next example is non-affine on K"I' EXAMPLE 2.2. For an element x = (xn)n in K"I' we denote by bl,X) the sequence bl,Xl,X2,X3, ...
). We define a mapping S: K"I -+ Co by S(x):= X, for each x E K"I' where x = (Xl, X2, X2, ... ) is the decreasing rearrangement of the sequence bl' x); that is, X = bl, x)*. Note that
X= (Xl,X2,X3, ... ) =
(~~) ,12 (~:) ,13 (~:) , ... ).
Also Xl 2: X2 2: X3 2: ... 2: 0 and 0 < 11 < 12 < 13 < ... < 1. Therefore ~ > ~ > ~ > ... > O. Since ~ = max("("x) > 1 S(x) does not necessarily belong ~-n-n~ ~ -, to K"I' However, the mapping S is
nonexpansive on K"I because the operation of decreasing rearrangement is nonexpansive on Co, so for all x and y in K"I' we have IIS(X) - S(Y)II = IIx -
IIbl,X)* - bl,Y)*11 :::; Ilb1.x) - b1.y)11
Ilx - YII·
We now introduce a modification U of S that will be nonexpansive and fixed point free on K"I' Define U : K"I -+ Co by
11j AXj - 1j AYj I :::; IXj - Yj I for all j E N, it follows that IIU(x) - U(Y)II :::;
Ilx - yll = IIS(x) - S(Y)II :::; IIx - YII·
Thus U is a nonexpansive mapping on K"I' Furthermore, since 1j AXj =
1j (1 A ~) for all j E Nand 1 2: 1 A ~ 2: 1 A ~ 2:
1 A ~ 2: ... 2: 0, U maps K"I into K"I' To finish, we will show is fixed point free on K"I' Suppose, to get a contradiction, that there exists x E K"I such that x = U(x). Thus, for all j EN, Xj = 1j
AXj. A well-known fact about decreasing rearrangements that we will use is that for for each mEN, all W E
+ ... + Wm
wi + ... + w;;'.
Since 1 = (1n)n is strictly increasing with limit 1, while X is decreasing with 1 > Xl 2: 11, there exists a unique kEN such that Xk+1 < 1k+l and Xk 2: 1k. Thus, for all mEN with m > k, we have, k +
+ ... + Xm+l > (Xl + ... + Xk) + (Xl + ... + xm+d (Xl
+ ... + Xk) +
bl A Xl + ... + 1k A Xk + 1k+l A Xk+1 + ... + 1m+1 A Xm+1 = (Xl + ... + Xk) + bl + ... + 1k + Xk+l + ... + xm+d bl + ... + 1k) + (Xl + ... + Xk + Xk+1 + ... + Xm+1) bl + ... + 1k) + (Xl + ... + Xm+l)
> bl+"'+1k)+bl+Xl+"'+Xm),
P.N. DOWLING, C.J. LENNARD, AND B. TURETT
It follows that for all
E N with Tn
> k, k
bl+"'+'Yk)-k+'Yl='YI-L(1-'Yj) j=1
> 'Y1 -
L (1 -
'Yj) ~ 'Yl -
L 4-
= 'Yl -
This contradicts the fact that x E Co and so completes the proof that U is fixed point free on K-y.
3. The fixed point property in £1 [0, 1] One of the most notable works in metric fixed point theory is the construction of Alspach [1] of a non-empty weakly compact convex subset of Ll[O, 1] which
fails the fixed point property. We begin this section by recalling some of the details of Alspach's construction. Let C:= {f E Ll[O, 1] : 0::; f(t) ::; 1, for all t E [0, I]}. Now define T: C -+ C by
Tf(t) := {min{2f (2t), I} max{2f(2t - 1) - 1, O}
for 0 ::; t ::; ~ for ~ < t ::; 1.
for all f E C. Alspach showed that the mapping T is an isometry on C which has two fixed points; namely 0 and X[O,lj' Alspach also showed that T is an isometric self map of the closed convex subset
Co := {f E C : J~ f dm = 1/2} of C, such that T is fixed point free on Co. Here, m denotes Lebesgue measure. We now follow a modification of Alspach's example due to Sine [7]. Define S : C -+ C by S
(f) := X[O,lj - f, for all f E C. The mapping S is clearly an isometry of C onto C. Thus the mapping ST is a nonexpansive mapping on C. Sine proved that ST is fixed point free on C. In [5],
Llorens-FUster and Sims prove that a closed bounded convex subset of Co with non-empty interior fails the fixed point property. We will use the above construction of Alspach, and modification by
Sine, to prove a result analogous to the Llorens-FUster and Sims result in the setting of Ll [0, 1]. Specifically we prove the following result. THEOREM 3.1. Let K be a closed, bounded, convex subset
of Ll [0, 1] with nonempty interior. Then K fails the fixed point property for non expansive mappings. PROOF. By translating and scaling, we ca.n assume that K contains the unit ball of £1 [0, 1].
Consequently, the set C, constructed above, is a subset of K. Define the mapping R : K -+ K by
Rf(t) := min{lf(t)l, I}, for 0::;
t::; 1, for all f
It is easily seen that R is a. nonexpansive mapping on K and R(f) E C for all K. Now define U: K -+ K by
U(f) := ST(R(f)), for all
The mapping U is nonexpansive since all of the mappings, R, S, and Tare nonexpansive.
We now show that U is fixed point free. Suppose that f E K is a fixed point of U, that is, U(f) = f. Since f E K, R(f) E C, and since ST maps C into C, f = U(f) = ST(R(f)) E C. Note that the mapping
R restricted to C is the identity on C. Therefore, f = ST(R(f)) = ST(f) and so f is a fixed point of ST in C. This contradicts Sine's result that ST has no fixed point in C [7], and thus the proof is
complete. D THEOREM 3.2. Let K be a closed, bOllnded, convex sllbset of Ll[O, 1] that contains an order interval [h,g] := {f E L1[0, 1] : h ::; f ::; 9 a.e.}, for some h,g E Ll[O, 1] with h ::; 9
a.nd h ::f g. Then K fa'ils the fixed point property for nonexpansive mappings,
a a
PROOF. By translating by -h, we may assume that h = and 9 ~ a.e. with 9 non-trivial. Next, note that there exists a real number c > and a measurable set E with Lebesgue measure rn(E) > 0, such that 9
~ CXE. By rescaling K by lie, we may assume without loss that c = 1. Now, define the mapping R : K ~ [0, xel ~ K by R(f) := ifi /\ XE. Note that R is nonexpansive and R equals the identity on [0,
xel. At this stage, consider E. There exists to in the interval [0,1] such that rn(En [0, to]) = ~ rn(E), Let E1.1 := En [0, to] and E1.2 := En (to, 1]. Clearly E is the disjoint union of E1.1 and E
1,2 and rn(E 1,1) = rn(El,2) = ~ rn(E). Proceed iteratively from here. Similarly to above, there exist pairwise disjoint measurable subsets E2,1, E 2,2, E2,3 and E2,4 of [0, 1] such that El,l = E 2,1
U E2,2, E 1 ,2 = E 2,3 U E 2.4 , and each rn(E2,k) = ~ rn(E). Repeating this construction inductively, we produce a family of measurable subsets (EO,1 := E, En,k : n E N, k E {I, ... , 2n}) of [0,1]
such that (XE",k)n.k is a dyadic tree in Ll [0,1]. Moreover, letting the measure v be defined on the measurable subsets of E by v = (1/rn(E))rn, it follows that the Banach space L 1 ( E, v) is
isometrically isomorphic to L 1 ( [0, 1], m) = L 1 [0, 1] via the mapping Z defined as follows: Z(XEn, k):= X[k-l k), 2l'r'2"'t"
for each XE",k' Then Z is extended to L := the linear span of the functions XE",k in the usual way. Of course, Z is an isometry on L. Finally, since L is dense in Ll(E, v), with dense range in Ll
[0,1], Z extends to a linear isometry from L1(E, v) onto Ll[O, 1]. Let W := ST be Sine's variation on Alspach's example, as described above, and note that W maps the order interval C := [0, X[O,I]]
into C. Let's use W to define E : [0, xel ~ [0, XE], by E := Z-1 W Z. We have that E is a fixed point free Ll [0, 1]-isometry on [0, xel. Finally, we define U : K ~ [0, XE] ~ K via U := E R. In a
manner analogous to the argument in the proof of Theorem 3.1 above, we see that U is a fixed point free Ll[O, 1]-nonexpansive mapping on K. D REMARK 3.3. In [6], MatlI'ey proved that closed, bounded,
convex, non-empty subsets of reflexive subspaces of Ll [0, 1] have the fixed point property for nonexpansive mappings. Consequently, Maurey's result, in tandem with Theorem 3.2, shows that reflexive
subspaces of U [0,1] cannot contain a non-trivial order interval. In fact, as pointed out by the referee, the argument in the proof of Theorem 3.2 shows
P.N. DOWLING, C.J. LENNARD, AND B. TURETT
that infinite-dimensional subspaces of £1[0,1] which contain non-trivial order intervals actually contain isometric copies of £1 [0, 1] and thus are nonreflexive. The authors thank the referee for
his/her comments. References 1. D. Alspach, A fixed point free nonexpansive mapping, Proc. Amer. Math. Soc., 82 (1981), 423-424. 2. P.N. Dowling, C.J. Lennard and B. Thrett, Characterizations of
weakly compact sets and new fixed point free maps in co, to appear in Studia Math. 3. P.N. Dowling, C.J. Lennard and B. Thrett, Weak compactness is equivalent to the fixed point property in co,
preprint 4. Kazimierz Goebel and W.A. Kirk, Topics in metric fixed point theory, Cambridge University Press, Cambridge, 1990 5. Enrique Llorens-Fuster and Brailey Sims, The fixed point property in
co, Canad. Math. Bull. 41 (1998), no. 2, 413-422. 6. B. Maurey, Points fixes des contractions de certains faiblement compacts de Ll, Seminaire d'Analyse Fonctionelle, 1980-1981, Centre de
Mathematiques, Ecole Polytech., Palaiseau, 1981, pp. Exp. No. VIII, 19. 7. R. Sine, Remarks on an example of Alspach, Nonlinear Anal. and Appl., Marcel Dekker, (1981), 237-241. DEPARTMENT OF
E-mail address: dowlinpnGmuohio. edu DEPARTMENT OF MATHEMATICS, UNIVERSITY OF PITTSBURGH, PITTSBURGH, PA
E-mail address: lennard+lDpi tt. edu DEPARTMENT OF MATHEMATICS AND STATISTICS, OAKLAND UNIVERSITY, ROCHESTER,
48309 E-mail address: turettlDoakland. edu
Contemporary Mathelnatics Volume 328. 2003
Homotopic composition operators on Hoo (Bn) Pamela Gorkin, Raymond Mortini, and Daniel Suarez We characterize the path components of composition operators on Hoo(B n ), where Bn is the unit ball of
en. We give a geometrical equivalence for the compactness of the difference of two of such operators. For n = 1, we give a characterization of the path components of the algebra endomorphisms.
1. Introduction
Consider the Hardy space H2 on the unit disk D. Littlewood's subordination principle tells us that for an analytic self-map ¢ of D and a function f in H2, the function f 0 ¢ is once again in H2. Thus
one defines the composition operator C'" on H2 by C",(f) = f 0 ¢. The interplay of operator theory and function theory leads to several interesting results. One of these results is Berkson's theorem
on isolation of composition operators (see [1] and [14]): THEOREM 1 (Berkson). Let ¢ be an analytic self-map of D. If ¢ has mdial limits of modulus one on a set E of positive measure, then for every
other analytic self-map 'l/J of D, the following estimate holds:
IIC", - C",II2:
where C'" and C'" are the corresponding composition opemtors on H2.
Thus, Berkson's theorem tells us that every such operator is isolated in the set of composition operators in the operator norm topology. For example, the identity operator, C z , is at least a
distance of ..[f72 from every other composition operator on H2 (as is C"', where ¢ is any inner function). However, not every composition operator is isolated. If ¢ is analytic and ¢ : D ~ sD for
some s with 0 < s < 1, then it is easy to check that
Thus, C'" is not isolated. Inner functions induce highly noncompact operators, as well as isolated operators. The operators C'" for which ¢(D) is contained in sD for some s with 0 < s < 1 2000
Mathematics Subject Classification. Primary 47B33; Secondary 47B38. Key words and phrases. composition operator, path components, compact differences.
© 177
2003 American lvlathematical Society
are compact. As Shapiro and Sundberg [14J indicate in their paper, "compact composition operators are dramatically nonisolated." They show that the set of compact composition operators is path
connected, and therefore these operators are never isolated. It is interesting to ask which composition operators are, in fact, isolated. Shapiro and Sundberg studied this problem, and showed (among
other things) that if ¢ is au analytic self-map of D that is not an extreme point of the algebra HOC(D), then C> is not isolated; in their words, "isolated composition operators can only be induced
by extreme points." This allowed them to exhibit an example of a non-compact non-isolated operator. They also raised several questions at the end of their paper: (1) Characterize the components in
Comp(H2), the space of all composition operators on H2. (2) Which composition operators are isolated? (3) Characterize composition operators whose difference is compact. Before stating the final
question, we remind the reader that the essential norm of an operator T defined on a Banach space H is the distance to the compact operators; that is, liT lie = inf{IIT - KII : K compact on H}. It is
clear that IITII ~ IITlle, and therefore every essentially isolated operator is isolated. In fact, because of the abundance of weakly null sequences in H2, all the results on isolation appearing in
Shapiro and Sundberg's paper hold true if we replace the norm with the essential norm (see [14], p. 148). Thus they raised the following question. (4) Is every isolated operator essentially isolated?
Other papers of interest on this subject include [10J. Of course, one is not limited to the space H2, and the study of composition operators on various spaces has lead to a large body of literature.
In this paper, we are interested in the same problems for composition operators on HOC (Bn), where Bn denotes the open unit ball in en. While the problem 011 H2 seems to be difficult, MacCluer, Ohno
and Zhao [l1J were able to obtain partial results about operators on the algebra HOO(D). They showed that for two analytic self-maps of the disk, C> and C.p are in the same path component in the
space of composition operators, Comp(HOC(D)), if and only if IIC", - c.pll < 2. In particular, an operator C> is isolated in Comp(HOC(D)) if and only if IIC", - C",II = 2 for any other analytic
self-map 'IjJ of the disk. The authors show that this result can be rephra..<;ed in terms of the pseudohyperbolic metric, and they posed the question of whether or not every isolated composition
operator on HOC is, in fact, essentially isolated. The answer to this la..<;t question was given by Hosokawa, Izuchi and Zheng [8J. Their technique was to develop something called asymptotic
interpolating sequences, or a.i.s. for short. Essentially, this definition allowed them to interpolate sequences with a good bound on the norm (see [6J for more information about these sequences).
They then used these sequences and Blaschke products to obtain HOC(D) functions that provide good estimates for the essential norm of the difference of two composition operators.
HOMOTOPIC COMPOSITION OPERATORS ON Hoo(Bn)
In this paper, we give simpler proofs of the results obtained by Hosokawa, Izuchi and Zheng, and combine them with the proofs of MacCluer, Ohno and Zhao. Because our proofs are significantly simpler
and do not refer to asymptotic interpolating sequences or Blaschke products, we are able to obtain the results on the ball in en. While our results do not rely on interpolating sequence results, they
do rely on a construction of Gamelin and Garnett relating interpolating sequences to peak sets [5]. The proof we will provide is simple enough to be applicable to other algebras. We conclude the
paper with an example of such an application: Using these same techniques, we are able to "lift" these results to obtain a characterization of the path components of endomorphisms of HOO(D). After we
completed this paper, we learned that some of the results on composition operators on Hoo(Bn) were also obtained by Carl Toews [15]. Two other papers directly related to the results described here
are [3] and [9]. Finally, we mention that recent results on Shapiro and Sundberg's first question in the space H2 can be found in [12].
2. Preliminary results Our goal is to prove results about composition operators on Hoo(Bn) and endomorphisms of HOO(D). In this section, we present proofs of several lemmas that will be important in
obtaining estimates on norms and essential norms of operators. Our discussion begins with functions of n variables and Z = (Zl' Z2, .•. , zn), where each Zj is a complex number. As usual, the
associated norm of Z is given by
Izi =
(z, z)1/2,
and the unit ball Bn is the set of all Z E en for which Izi < 1. We let Hoo(Bn) denote the space of bounded holomorphic functions on Bn. If n = 1 our situation reduces to the familiar space of
functions on the unit disk: HOO(D). We need some background on general uniform algebras, some information specific to Hoo(Bn) and some deeper results for HOO(D). Everything we need is presented in
this paper. Let A be a uniform algebra. The maximal ideal space of A, denoted M(A), is the set of complex-valued, linear, multiplicative maps of A that map the identity of A to the value 1 E C. Since
evaluations at points of Bn are linear multiplicative functionals on Hoo(Bn), we may think of Bn as a subset of M(Hoo(Bn)). It is well known that M(A) is a compact Hausdorff space when endowed with
the weak-* topology induced by A * . We will always consider this topology for M(A). For an element a E A, the Gelfand transform of a is a complex-valued map defined on M(A) by a(x) = x(a). This map
establishes an isometric isomorphism between A and a closed subalgebra of C(M(A)). It is usual to identify the function with its Gelfand transform, since the meaning is generally clear from the
context. For x, y E M(A), the pseudohyperbolic and hyperbolic metrics are defined, respectively, by
p(x, y) = sup{l/(y)1 : I E A, 11/11 ::; 1, and I(x) = O} and y) h( x, y ) -- Iog 11 + p(x, ( ). -p x,y
It is well-known that P is a [0, 1]-valued metric and that h is a [0, +oo]-valued metric on M(A). The triangle inequality for h immediately implies that the condition p(x, y) < 1 (i.e. h(x, y) < 00)
is an equivalence relation on M(A). If A is a uniform algebra and ¢ : M(A)-+M(A) is a continuous map such that a 0 ¢ E A for all a E A, then the map C'" defined by
C",a = a 0 ¢ is an endomorphism of A. We will write C'" E End (A). Conversely, if E E End (A), we may define ¢ : M(A)-+M(A) by ¢(x)(a) = x(E(a)), obtaining E = C"'. The Shilov boundary of A is the
smallest closed subset of M(A) on which every function in A attains its maximum. We denote the Shilov boundary by 8A. LEMMA 2. Let A be a uniform algebra and let C"', C1/J E End (A). Let V M(A) be a
set whose closure contains 8A. Then,
sup xEV
2p(¢(x),1/I(x)) J :::; IIC", - C1/J1I :::; 2 sup p(¢(x), 1/I(X)). 1 + 1 - p(¢(x), 1/I(x))2 xEV
First we show that if x, y
(2.2 )
M(A), then
II 2p(x, y) + Jl - p(x,y )2:::; X
II 2 ( ) Y A*:::; p x, y .
The proof uses the techniques of [4, p. 144]. Let I E A with Ilfll < 1. It is clear that 1 = U - f(y))/(1 - I(y)f) E A, l(y) = and 11111 :::; 1. By definition of p then lj(x)1 :::; p(x, y), and
If(x) - f(y)1 :::;
11 -
f(y)f(x)1 p(x, y) :::; 2p(x, y).
Taking the supremum over Ilfll < 1 we get the upper inequality. Now we turn to the lower inequality. For simplicity, we write p = p(x, y). Choose fn E A with I!fnl! < 1, In(x) = and fn(y) > p-l/n.
Let Pn = fn(Y) and Ln(z) = (tn - z)/(I- tnz), where tn = (1- Jl- P~)/Pn' Therefore Ln 0 fn E A, IILn 0 Inll :::; 1 and
IIx -
yl!A* ~ I(Ln
fn)(x) - (Ln
fn)(Y) I = Itn - L .. (Pn)l·
A simple computation shows that
Itn - Ln(Pn)1 = 1Pn(1 - t;) 1-+ 2p . I-tnPn 1+~ The lemma will follow immediately from (2.2) and the following chain of identities: IIC",-C1/J1I
sup sup IU 0 ¢)(x) - U 0 1/I)(x) I 11/11=1 xE8A sup sup IU 0 ¢)(x) - U 0 1/I)(x) I 1I/1I=lxEV
sup sup 11(¢(x)) - f(1/I(x))1 11/11=1 sup 11¢(x) -1/I(x)IIA*.
o LEMMA
IIC", - C1/J1I
3. Let A be a uniform algebra and let C"', C,,}
< 2 is an equivalence relation.
End (A). The condition
HOMOTOPIC COMPOSITION OPERATORS ON Hoo(Bn)
PROOF. It is obvious that the relation is reflexive and symmetric. So, suppose that ¢,'l/J and t.p define endomorphisms of A such that IICeI> - Cvlli < 2 and IICob C
sup p(¢(x),'l/J(x)) < 1 and
sup p('l/J(x),t.p(x)) < 1. xEM(A)
Using the hyperbolic metric h on M(A) we obtain
1 + 0'1 h(¢(x), t.p(x)) :::; log - XEM(A) 1 - 0'1 sup
1 + 0'2 + log - = /3, 1-
and consequently sUPXEM(A) p(¢(x), t.p(x)) :::; (ei3 -1)/(ei3 + 1) tion of (2.1) yields the desired result.
< 1. A new applica0
LEMMA 4. Let A be a uniform algebra. If {fn} is a sequence of functions in the unit ball of A tending pointwise to zero on 8A, then {fn} tends to zero weakly in A.
PROOF. As indicated in the introduction, using the Gelfand transform, we
may think of A ~ C(8A). Let x be any element of the dual space of A. By the Hahn-Banach theorem, x has a continuous norm preserving extension to the space of continuous functions on the Shilov
boundary. Therefore, there exists a finite measure ILx on 8A such that x(f) =
f dJtx •
But Ilfnll :::; 1 for all n, and fn ...... 0 pointwise on 8A, so we may apply the Lebesgue dominated convergence theorem to conclude that 3;(fn) ...... O. Therefore, the sequence {fn} converges to
zero weakly. 0 We will need another estimate, but this will depend on the pseudohyperbolic metric particular to the ball, Bn. For a, z E Bn, let Sa = lal 2 , and the
Jl -
((z' a)) a and Qa = 1- Pa. Relevant a,a computations can be found in [13, p. 25]. On Bn, the pseudohyperbolic metric induced by Hoo(Bn) is given by projections Pa and Qa be given by Pa(z)
p a, z
la - Pa(z) - saQa(z) I 1 _ (z, a) ,
In what follows, for points a and z in the ball and numbers sand t in the closed interval [0,1]' we let as = a + s(z - a) and asH = a + (s + t)(z - a). LEMMA
5. Let a, z
Bn. For s, t
[0,1] .satisfying t :::; 1 - s we have
tp(a,z) p(a + s(z - a), a + (s + t)(z - a)) :::; 1 _ (1 _ t)p(a, z)·
PROOF. We can assume that a =I z, because otherwise there is nothing to prove. We consider first the case s = O. Since Pa and Qa are linear operators satisfying Pa(a) = a and Qa(a) = 0, the nmnerator
of p(a, a + t(z - a)) satisfies
a - Pa(a + t(z - a)) - saQa(a + t(z - a)) = ta - tPa(z) - tsaQa(z).
We have assumed that a =I- z, and therefore a - Pa(z) - saQa(z) =I- O. A simple computation gives
Ia -
p(a,a+t(z-a)) =
Pa{a + t(z - a)) - satQa(z) 1 - (a + t(z - a), a)
t(a - Pa(z) - saQa(z)) + (z, a) - (a, a) - t(z - a, a)
1 - (z, a)
< 11/p(Z,a) -1(1- t)(z - a,ta)I/la - Pa(z) - saQa(z)ll· But
I(z -
a,a)1 = I(a - Pa(z) - saQa(z),a)1 ::; la - Pa(z) - saQa(z)llal. Therefore t tp(a,z) p(a. at) < = . . - (1/ p(a, z)) - (1 - t) 1 - (1 - t)p(a, z)
This proves (2.3) for s = O. For the general case we can assume s =I- 1 since otherwise t = 0 and there is nothing to prove. From (2.4) we obtain
p(a s, as + t(z - a)) p(as,a s + (t/(I- s))(z - as)) t/(1 - s) (l/p(a s,z)) - (1- (t/(I- s)))'
p(a + s(z - a), z) p(z + (1 - s)(a - z), z) p(z, z + (1 - s)(a - z)). So we may apply (2.4) again to conclude that
I-s p(as,z)::; (l/p(a,z)) -s Combining this with our estimate on p(a s , as+t) above, we see that
t p(a s , as+t) ::; (1/ p(a, z)) - (1 - t)
Simplifying, we obtain the desired conclusion. 3. Composition operators on Hoc (Bn)
In this section we study composition operators on Hoc(Bn); that is, given an analytic self-map ¢> of the unit ball, we look at maps C'" : Hoc(Bn) ~ Hoc(Bn) defined by C",(f) = f 0 ¢>. These maps are
all endomorphisms of the algebra HOC (Bn). For the special case of n = 1 we will say more in the final section of the paper. We are interested here in estimates on the essential norm of the
difference of two composition operators. If T is a bounded operator, we denote its essential norm by IITlle. THEOREM
6. Let ¢> and '!/J be holomorphic self-maps of Bn such that
max{II¢>II, II'!/JII} =
HOMOTOPIC COMPOSITION OPERATORS ON H"""(Bn)
max {lim sup p(¢(Z), 'lj!(Z)) , lim sup p(¢(Z), '¢(Z))} . 1'I/J(z)l-l
Then (3.1) PROOF. By hypothesis there is a sequence of points {Zj} in Bn, such that p(¢(Zj),-¢(Zj)) ~ e, and one of them, say {¢(Zj)}, converges to a point ( on the boundary of Bn. Without loss of
generality we may assume that p( ¢( Zj ), 'lj!( Zj)) > e-1fj. Let kj E Hoo(Bn) be a function of norm one, whose existence is guaranteed by the pseudohyperbolic distance definition, satisfying kj(¢
(zj)) > e - 1fj and kj('lj!(zj)) = O. Consider the functions
f(z) = (1 + (z, () )/2 and g(z) = (1 - (z, () )/2. Then f,g E HOO(B n ), f(() = 1, If(7])1 < 1 for all 7] E aBn satisfying 7] -I- (, and g(() = O. We will now produce a sequence of functions, {hj},
tending to zero weakly for which Ihj(¢(zj)) - hj (1/I(Zj)) I ~ e. We proceed as follows. Let j E N. Since f(¢(zj)) ~ 1, we may choose Zmj so that If(¢(zmj))lj > 1 - 1 fj. Now since 9 -I- 0 on Bn,
there exists an integer Ij such that
Ig(¢(zmj))1 1/ 1j 2: J1- 1fj. Consider the functions hj = (gl/1 j )(J3)kmr Then Ilhjll ~ 1, hj (1/I(zmj)) = 0, and Ihj(¢(zmJ)1 > (1 - 1fj)(e - 1/mj). We note that hj ~ 0 on the Shilov boundary and,
by Lemma 4, hj ~ 0 weakly. Thus for any compact operator K we have IIC", - C,p
+ KII > 2:
IIC",hj - C,phj + Khjll Ihj(¢(zmj)) - hj('lj!(zmj))
+ (Khj)(zmJI ~ e·
= Ihj(¢(zmJ) + (Khj)(zmj)1
This proves the lower inequality in (3.1). For the upper inequality, let I: > 0 and choose 8 with 0 to 1 so that
p(¢(Z), 'lj!(z)) ~
e + I:
on the set {1¢(z)1
< 8 < 1 close enough
> 8} U {1'lj!(z)1 > 8}.
Now choose a = a(I:,8) E (0,1) close enough to one so that p(¢(z),a¢(z)) < I: and p('lj!(Z) , ml'{z)) < I: on the set {1¢(z)1 ~ 8}n{I1/I(z)1 ~ 8}. Since max{lla¢lI, Ila'lj!ll} ~ a < 1, the operator K
~f Co", - Co,p is compact. If Z and ware any two points of B'\ we may view Z and w as elements of the dual space of Hoo(Bn) in the obvious way. Therefore, for any function f in the unit ball of Hoo
(Bn) we may apply (2.2) to conclude that If(z) - f(w)1 ~ 2p(z,w). Henceforth, applying (2.2) to the functions f and fo(z) = f(az), we have
I(C",f)(z) - (C,pf)(z) - (Kf)(z)1
If(¢(z)) - f(a¢(z))1 + If('lj!(z)) - f(a'¢(z))1 < 2p(¢(z), a¢(z)) + 2p('lj!(z), a'lj!(z)) < 4c
when z E {I¢I ~ 8} n Hl/)I ~ 8}, while
I(C",f)(z) - (C,pf)(z) - (Kf)(z)1
< If(¢(z)) - f('lj!(z)) I + If(a¢(z)) - f(a1/l(z))1 < 2p(¢(z), 'lj!(z)) + 2p(¢(z), 'lj!(z)) < 4e + 41:
184 when
> 8} U {I'!/JI > 8}. Since the function f is arbitrary, IIG> - G", - KII
4e + 4c, and since c is arbitrary we obtain (3.1).
COROLLARY 7. Let cf> and '!/J be two holomorphic self-maps of the unit ball. Then G> - G", is compact if and only if either max {11<;b11, II'!/JII} < 1, or lim sup p(cf>(z), '!/J(z))
lim sup p(cf>(z), '!/J(z))
= O.
PROOF. It is clear that G> and G", are compact, if max {11cf>11, 11'!/J11l < 1. On the other hand, if max {11cf>11, II'!/JII} = 1 and e is the parameter of Theorem 6, then (3.1) says that Gel> - G",
is compact if and only if e = o. D Our next goal is to characterize the path components of composition operators on Hoo(Bn). We write G> '" G", to indicate that there is a norm-continuous homotopy of
composition operators joining Gel> with G",. Also, if K denotes the ideal of compact operators, we write G> "'e G", to indicate that there is an essential norm-continuous homotopy of classes {G", +
K: cp: Bn -4 B n holomorphic} joining Gel> + K with G", + K. Let cf> be a holomorphic self-map of Bn. For x E M (HOO (Bn)) we can define
cf>(x) E M(Hoo(Bn)) by the rule <;b(x)(f) ~f x(f 0 cf». Thus we can extend cf> : B n -4Bn to a self-map of M(Hoo(Bn)), which we also denote by cf>. The continuity of this extension is immediate. We
now have everything we need to prove the main theorem of this paper. As indicated in the introduction, this theorem unifies and extends many of the results appearing in [11], as well as [8]. THEOREM
8. Let cf> and '!/J be holomorphic self-maps of the unit ball in en. Then the following are equivalent.
(a) (b) (c) (d)
G> '" G",. G> "'e G",. IIGeI> - G",II < 2. SUPzEBn p(cf>(z), '!/J(z)) < 1.
PROOF. (a) => (b) is obvious. (c) ¢:} (d). A boundary for HOCJ (Bn) is a closed set F c M (HOCJ (Bn)) such that Ilfll = SUPxEF If(x)1 for all f E HOCJ(Bn). It is clear that the closure B n of B n in
M(HOCJ(Bn)) is a boundary for HOCJ(B n ), and since oHOCJ(Bn) is the intersection of all the boundaries [4, p. 10], then oHOCJ(Bn) c F. The equivalence then follows from (2.1). (b) => (c). By
hypothesis there is a family {cf>t}, with t E [0, 1], of holomorphic self-maps of Bn such that cf>o = cf>, cf>1 = '!/J and for every c > 0 there is some 8 > 0 satisfying
< c: if It - sl < 8. Then we can take finitely many points ty = 0 < ... < tm = 1 in [0,1] such that IIG>t. - GeI>tHl lie < 1/2 for every i = 1, . .. , m - 1. We claim that IIG>t - Gel>. lie
sup p(cf>t. (z), cf>t'+l (z)) < 1
for every i. In fact, if r = max{llcf>d,lIcf>tHlll} < 1, then both functions map B n into the closure of r Bn, and since the pseudohyperbolic diameter of this ball
is smaller than 1, we are done. If some of the maps have norm 1, then the first inequality of (3.1) tells us that there is some 0 < 8 < 1 close enough to 1 such that sup p(¢t;{Z), ¢tHl (Z)) < 3/4,
{I>., 1~t5}U{I>"+11~6} while the set {I¢t, I < 8} n {I¢t'+ll < 8} is mapped by both functions into the ball 8Bn, whose pseudohyperbolic diameter is smaller than 1. Our claim follows. Since the
closure of Bn in M(Hoo(Bn)) contains the Shilov boundary, (3.2) and (2.1) imply that IIC>" - C>'<+1 11 < 2 for i = 1, ... , Tn - 1. Lemma 3 now says that (c) holds. (d) =} (a). By (d) there exists 0:
< 1 such that sup p(¢(z), 'l/J(z)) -:; 0:. zEBn
We define a map ¢t = ¢
+ t('l/J -
IIC>, - C>. II
¢) for t E [0,1]. Now, if t < s < 1
< 2 sup p(¢t(z), ¢s(z)) zEBn
< <
2(s - t)p(¢(z), 'l/J(z)) 1 - (1 - (s - t)) p(¢(z), 'l/J(z)) 20: (s - t) 1 _ 0:'
where the first inequality holds by (2.1), since 8Hoo(Bn) C 13", and the second inequality from (2.3). From this, we see that t 1--+ C>t is a continuous mapping. 0 Therefore C'" lies in the same path
component as C>. As a corollary, we obtain the following generalization of the work on isolated points in [11]. COROLLARY 9. Let ¢ be a holomorphic self-map of the ball. Then C> is isolated in the
set of composition operators if and only if C> is essentially isolated. PROOF. Since IIC> - C",II ~ IIC> - C",lIe, it is clear that if C> is essentially isolated, then it is isolated. If C> is
isolated and 'l/J =I- ¢, Theorem 8 implies that SUPzEBn p(¢(z),'¢(z)) = 1. This can only happen if there are points z E Bn such that 1¢(z)I--+1 or I'l/J(z) 1--+1, and p(¢(z), 'l/J(z))--+1. Hence,
Theorem 6 says that IIC> - C'" lie ~ 1, and C> is essentially isolated. 0
4. Examples So what are some examples of isolated operators? If ¢ : Bn --+ Bn has radial limits of Euclidean norm 1 on a set of positive measure, we claim that C> is isolated. If 'l/J =I- ¢ there
must exist a set of positive measure in 8Bn on which ¢ has radial limits of norm 1 and 'l/J does not equal ¢ (see [13, Ch. 5]). Thus, there exists a sequence {zd c Bn for which ¢(Zk) --+ ( E 8Bn and
'l/J(Zk) --+ 'Tf, with 'Tf =I- (. Therefore p(¢(Zk), 'l/J(Zk)) --+ 1 and Lemma 2 tells us that IIC> - C",II = 2. In particular, the automorphisms of B n induce isolated composition operators. It is
clear from Theorem 8 that if ¢, 'l/J are holomorphic self-maps of Bn and O> - C'" is compact, then C> '" C"'. It is not completely clear, though, that the converse fails. In [8] Hosokawa, Izuchi and
Zheng constructed an example that shows this for n = 1. By eliminating variables, every example that works for n = 1 can be made to work for general n. Here we construct a simpler example of
holomorphic self-maps of the ball, 4> and '1/), such that C'" '" C'" but C - C,,) is not compact. Let n = {UJ ED: ~1-=-!:1 > ~} be a nontangential region in D at the point Z = 1. We want to estimate
p(w, (w + 1)/2) for wEn. We recall that p(z, w) = Iz - wi/II - zwl for z, wED. By straightforward calculation,
( ( + 1)/2) -1
p w, w
11 -lwl 2+ 1- wi Il-w I
(1 -lwl)(1 + Iwl) 1 3 Il-w I + ~
- Iwl21 11+ 1l-w
p(w, (w + 1)/2)-1
> ~ (1 + 1- Iw12) 1- w
1-lwl 2
1+ 11_wI2(1-~w)
1+ 1+ Iwl > ~
- 2
when wEn. That is,
3 ~ p(w, (w + 1)/2) ~ 3
for all wEn. Let c.p : D---+n be a one-to-one and onto holomorphic function and define 4>, 'I/J : Bn---+Bn by
= (c.p(Z1), 0, ... ,0) and that 114>11 = 11'1/)11 = 1. For z
4>(Z1,"" zn) It is clear that
'I/J(Z1, ... , zn) E
= ((c.p(zt) + 1)/2,0, ... ,0).
Bn, a straightforward calculation shows
p(¢(z), 'I/J(z)) = p(c.p(zd, (c.p(zd
+ 1)/2).
Since c.p(zd E n, the inequalities in (4.1) show that p(4)(z), 'I/J(z)) E [1/3,2/3]. Therefore Theorem 8 says that C'" '" C"', while Corollary 7 says that C - C'" is not compact..
5. Endomorphisms of HOO(D) In this section we investigate the path components of elldomorphisms of H OO (D). For x E M(HOO(D)), the Gleason part of x is P(x) = {y E M(HOO(D)) : p(x,y) < I}. Since the
condition p(:r, y) < 1 is an equivalence relation, the Gleason parts form a partition of M(HOO(D)). In [7] Hoffman produced a continuous and onto map Lx: D---+P(x) such that Lx(O) = x and foLx E HOO
(D) for every x E M(HOO(D)) and f E HOO(D). There are two possibilities: either Lx(z) = x for all zED (so P(x) = {x}) or Lx is one-to-one. We write G = {x E M(HOO) : Lx is one-to-one} and
= {x E M(HOO) : Lx = {x}}.
It is well-known that every endomorphism T of HOO(D) can be factored as T = C",CL", , where 4> is a holomorphic self-map of D and x E M(HOO(D)). Although it is clear that this factorization is not
unique, two different factorizations of the same endomorphism are related in the following way (see [2]): if p(x, y) < 1, then there is a biholomorphic map r of D (depending on x and y) such that Ly
(z) = Lx(r(z))
Hoo(B n )
for every zED. This means that every endomorphism of the form T also be factored as
= Cq,CLy can
T = Cq,CLy = Cq,CTCL", = CToq,CLx ' Of course, if x E rand p(x, y) < 1, then x = y and T = C Lr . LEMMA 10. Let x E G and A = Ej=1 )..,jCq,j' where composition operators on H'XJ(D). Then IIACLJI =
E C and Cq,j are
PROOF. Since IIACLxll ::; IICLxllliAIl ::; IIAII, one direction is easy. For the other direction, if 0 < f < 1, there exists a function f in the ball of HOO(D) such that IIA(f)11 > (1 - f)IIAIi. By
the definition of the norm, there exists r with o < r < 1 such that n
L )..,jf(cPj(z)) ZETD j=l
sup IA(f)(z)1 = sup
> (1 - f)2I1AII·
By a result of Hoffman [7, p. 91]' there exist Blaschke products bk such that (bk 0 Lx)(z) ~ z uniformly on compact subsets of D. But rD is a precompact subset of D, and therefore cPj (r D) is
precompact for each j. That is, there is 0 < 0: < 1 such that Uj=l cP j(rD) C o:D. Fix {J with 0: < {J < 1. Since f is analytic, there is 8> 0 such that for z, wE (JD with Iz - wi < 8 we have If(z) -
f(w)1 < f. Clearly we can also require 8 < (J - 0:. Therefore we may choose k sufficiently large so that I(b k 0 Lx)(cPj(z)) - cPj(z)1 < 8 for all z E rD. Thus, for k that large, z E rD and f as
above, (b k o Lx)(cPj(z)) E (3D and consequently If(bk(Lx(cPj(z))) - f(cPj(z))1 < f. Therefore there exists a constant At depending only on nand )..,l"",)..,n such that II ACLx II
sup I(ACLx(f
zErD j=l
)..,j(f 0 bk 0 Lx)(cPj(z))1
> Letting
f ~
)..,jf(cPj(z))I- Mf
(1 - f)2I1AII- Mf.
0 yields the desired result.
THEOREM 11. Let T 1, T2 E End(HOO(D)). Then the following ar'e equivalent.
(a) T1 rv T2 in End(HOO(D)). (b) IIT1 - T211 < 2. (c) There exist x E M(HOO(D)) and holomorphic self-maps cP. 'ljJ of D such that T1 = Cq,C Lx • T2 = CtfJCLx and IICq, - CtfJlI < 2. PROOF. Suppose
that (a) holds. Then there is a homotopy
G: [0, 1] ~End(HOO(D)) with G(O) = T1 and G(I) = T 2. We can find finitely many points 0 = it < ... < tn = 1 such that IIG(tj) - G(tj+d II < 2 for j = 1, ... , n - 1. Lemma 3 then says that IIG(O) -
G(I)1I < 2. Suppose that (b) holds and write T1 = CLxoq, and T2 = CLyo"" where x,y E M(HOO(D)) and
p(Lx(¢(O)), Ly(
- CLyo'!'lI = 2. Thus (b) implies that p(x,y) < 1. If x E r, then Tl = T2 = C Lx ' If x E G and we write T2 = CLxo.p, where 1/J is a holomorphic self-map of D. Lemma 10 says that 2> IITI - T211 =
IICt/> - C,pll, so (c) holds. If (c) holds Theorem 8 says that there is a homotopy of composition operators F(t), with t E [0,1] such that F(O) = Ct/> and F(l) = C,p. By Lemma 10, G(t) ~f F(t)CLx is
a homotopy of endomorphisms connecting Tl with T 2, which proves (a). 0 Acknowledgement. The last author thanks Bucknell University for its hospitality and peaceful environment during the preparation
of this paper. References [1] E. Berkson, Composition opemtors isolated in the uniform opemtor topology, Proc. Amer. Math. Soc. 81 (1981),230-232. [2] P. Budde, Support sets and Gleason parts,
Michigan Math. J. 37 (1990), 367-383. [3] P. Galindo and M. Lindstrom, Factorization of homomorphisms through HOO(D), preprint. [4] T. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood
Cliffs, N. J., 1969. [5] T. Gamelin and J. Garnett, Distinguished homomorphisms and fiber algebras Amer. J. Math. 92 (1970), 455-474. [6] P. Gorkin and R. Mortini, Asymptotically interpolating
sequences in uniform algebms, J. London Math. Soc., to appear. [7] K. Hoffman, Bounded analytic functions and Gleason parts, Ann. of Math. (2) 86 (1967), 74-111 . [8] T. Hosokawa, K. Izuchi, and D.
Zheng, Isolated points and essential components of composition opemtors on HOC, Proc. Amer. Math. Soc. 130 (2002), no. 6, 1765-1773. [9] U. Klein, Kompakte multiplikative Opemtoren auf uniformen
Algebren, Mitt. Math. Sem. Giessen 232 (1997), 1-120. [10] B. MacCluer, Components in the space of composition opemtors, Integr. Equ. Oper. Theory 12 (1989), 725-738. [11] B. MacCluer, S. Ohno, and
R. Zhao, Topological structure of the space of composition oper'ators on HOC, Integr. Equ. Oper. Theory 40 (2001), 481-494. [12] J. Moorhouse and C. Toews, Differences of composition opemtors,
preprint. [13] W. Rudin, Function theory in the unit ball of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], 241. Springer-Verlag, New York-Berlin,
1980. [14] J. Shapiro and C. Sundberg, Isolation amongst the composition opemtors, Pacific J. Math., 145 (1990), 117-151. [15] C. Toews, Topological components of the set of composition opemtors on
HOC (B N ), preprint.
E-mail address: pgorkinlDbucknell. edu DEPARTEMENT DE MATHEl\fATIQUES, UNIVERSITE DE METZ, ILE DU SAULCY,
F-57045 METZ,
E-mail address:mortinilDponcelet.univ-metz.fr DEPARTA~IENTO DE MATEMATICA, UNIVERSIDAD DE BUENOS AIRES, PAB. I, CIUDAD UNIVER-
(1428) NUNEZ, CAPITAL FEDERAL, ARGENTINA E-mail address:dsuarezlDdm.uba.ar
Contemporary Mathematics Volume 328, 2003
Characterization of conditional expectation in terms of positive projections J.J. Grobler and M. De Kock ABSTRACT. A description of positive, order continuous projections in ideals of measurable
functions is given in terms of conditional expectation-type operators. The dual of such an operator can also be represented as a conditional expectation operator. We use this result to characterize
conditional expectation in terms of a positive, order continuous projection that preserves one, and such that an extension of the dual also preserves one.
1. Introduction
Dodds, Huijsmans and De Pagter (see [2]) give a complete description of positive projections in ideals of measurable functions in terms of conditional expectationtype operators. Let E be an ideal of
measurable functions such that Loo(O, E, Jl) C E C L1 (0, E, Jl). They characterize a positive, order continuous projection T : E -+ E with the property that T and the operator dual T' preserves one,
in terms of conditional expectation. We extend this result to an ideal of measurable functions L with the property that Loo(O, E, Jl) C L, but we omit the assumption L C Ll (0, E, Jl). In this case
we prove that a positive, order continuous projection S : L -+ L, which is onepreserving and with the property that an extension of its dual is also one-preserving, can be characterized in terms of
the conditional expectation. In order to prove this extension, we need the fact that if an operator can be characterized in terms of conditional expectation, then its dual can also be characterized
in terms of conditional expectation.
2. Preliminaries Let (0, E, Jl) be a a-finite measure space. The vector lattice of all Jl-a.e. finite E-measurable functions on 0, with the usual identification of Jl-a.e. equal functions, is denoted
by Lo(O, E, Jl). M+(O, E, Jl) denotes the set of all (equivalence classes of) real, positive, Jl-measurable functions into [0,00]. We define x Vy := sup{x, y}. A linear subspace G of a vector lattice
E is called a sublattice if x Vy belongs to G for all x, y E G. A linear subspace A in which Ixl :::; IYI, with yEA implies x E A, is called an ideal. A subset AcE is order bounded if A is contained
in an order interval, i.e. A is bounded from above and below. We denote the set of order bounded linear operators © lRO
2003 American Mathematical Society
J.J. GROBLER AND M. DE KOCK
from the vector lattice E into itself by Cb(E). Cb(E) is a Dedekind complete vector lattice (see [3), p3). All vector lattices considered will be Dedekind complete. A net (xQ)QEf in E is called order
convergent if there exists a net (YQ)QEr satisfying (YQ) 1 0 and Ix - x", I ~ y", for all Q E r, where r is an index set. We write X ---> x in order. If Eo is a sub-a-algebra of E, then we denote the
restriction of p, to Eo by IL again. Et denotes the collection of subsets of Eo of positive measure. The characte1"istic function of a set A E E is denoted by lA. We write 1 rather than In. For any
measurable function Jon 0, the support of f is denoted by supp(J) = {w EO: f(w) =1= o}. Our attention will, to a great extent, be focused on ideals of measurable functions on (0, E, I.L), i.e., on
ideals L in the vector lattice Lo(O, E, p,). The set Z E E is called an L-zero set if every .f E L vanishes p,-a.e. on Z. There exists (modulo IL-llUll sets) a maximal L-zero set ZI in E and the set
0 1 = 0 \ ZI is called the carrier of the ideal L. There exists a sequence An i 0 1 in E such that 1.L(An) < 00 and IAn E L for all n E N, (see [5], p143). Clearly, the carrier of L is equal to n, if
and only if L is order dense in Lo(n, E, p,). Let L c Lo(O, E, p,) be an order dense ideal with order continuous dual L~. We identi(y L~ with an ideal L' of functions in Lo(n,E,p,), and we will
assume that L' is again an order dense ideal (which is always the case if L is a Banach function space; (see [5), Theorem 112.1). Equivalent to this assumption is that L ~ separates f gd,.L the
points of L. The duality relation between Land L' is given by (J, g) = for f ELand gEL' (see [5), Section 86). Let 8 E Cb(L) with L an ideal of functions in Lo(O, E, p,). We define its order
continuous adjoint 8' : L' ---> L' by (g, 8' J) = (8g, J) for all f E L' and gEL (see [5], Section 97). Then 8' E Cb(L'). If there is no reason for confusion, we will denote (n, E, p,) by (E) only. Q
DEFINITION 1. Let E and F be vector lattices and let T : E ---> F be a linear operator. Then (i) T is positive (denoted by T ~ 0) whenever Tx ~ 0 for all x ~ 0; T is called strictly positive (denoted
by T »0) if Tx > 0 for all x > O. (ii) T is order continuous whenever Txc< -+ 0 in order for every net (x satisfying X -+ 0 in order. (iii) T is order bounded if it maps order bounded subsets into
order bounded subsets. Q )
For a Banach function space (E, II . liE) defined on some finite measure space ~ E ~ Ll (E, I.L), we define the following (see [2), p69).
(n, E, p,) for which Loo(E, p,)
DEFINITION 2. (i) The linear map T : E -+ E is called averaging if for all f E Loo(E) and all gEE we have that T(JTg) = Tf· Tg. (ii) T: E -+ E is called contractive if II T II ~ 1. DEFINITION 3. Let
(n, E, p,) be a probability space (i.e. 1.L(n) = 1) and let Eo be a sub-a-algebra of E. For fELl (E), we denote by lFP(J I Eo) the IL-a.e. unique Eo-measurable function with the property that
lFf'(J I Eo)dl.L
for all A E Eo. The function lFP(J I Eo) is called the conditional expectation of f with respect to Eo. If there is no reason for confusion, we will denote the p,-a.e. Eo-measurable function lElL('IE
o) by lE('IE o) only. The existence of lE(J I Eo) is a consequence of
the Radon-Nikodym theorem. The conditional expectationlE('IEo) can be extended from a mapping from Ll (E) into itself, to a mapping from M+(E) into itself. If f E M+(E,), then 1E(f I Eo) E .l\J+(E)
is defined by 1E(f I Eo) = suplE(fn I Eo), where 0 :::; fn E Ll (E) (71. = 1,2, ... ) satisfy 0 :::; fn i f J,L-a.e. The conditional expectation operator has the following properties. For a proof of
properties (i) to (vi) we refer to [4], p7; for property (vii) we refer to [3], p7. 1. (i) lE(o:f + /1g I Eo) = 0:1E(f I Eo) + /11E(g I Eo) for all f,g E M+(E) and for all 0:::; 0:,/1 E R (ii) 0 :::;
f :::; 9 in M+(E) implies that 0 :::; 1E(f I Eo) :::; lE(g I Eo) and if 1E(lfll Eo) = 0, then it follows that f = O. By virtue of positivity we have 11E(f I Eo)1 :::; 1E(lfll Eo). (iii) 0:::; fn i f
IJ,-a.e. implies that 0:::; lE(fn I Eo) i 1E(f I Eo) J,L-a.e. (iv) lE(gf I Eo) = glE(f I Eo) for all f E M+(E) and all 9 E M+(Eo). (v) If 9 E M+(Eo) and f E M+(E), then fA gdJ,L = J~ fdJ,L for all A
E Eo if and only if 9 = 1E(f I Eo) IJ,-a.e. (vi) If Eo c Ao are sub-a-algebras of E, then 1E(f I Eo) = 1E(1E(f I Ao) I Eo) for all 0:::; f E M+(E). (vii) If f E M+(E) is such that 1E(f I Eo) E Lo(E),
then we also have that f E Lo(E). PROPOSITION
4. The domain domlE('IE o) of 1E('IEo) is defined by
domlE('IE o) := {f E Lo(E) : 1E(lfll Eo) E Lo(Eo)}. Clearly, domlE(·IE o) is an ideal in Lo(E) which contains L 1 (E). For f E dom 1E(·1 Eo), we define: 1E(f I Eo) := 1E(f+ I Eo) -1E(r I Eo). This
defines a positive linear operator 1E('IEo) : domlE('IE o)
Lo(Eo) C Lo(E).
Let (n, E, l.l) be a probability space and let L carrier n. Set
M(L) = {m E Lo(E) : 1E(lmfll Eo) E L
Lo(n, E, J,L) be an ideal with
V f E L}.
Since L C Lo(E), we have that mf E domlE('IE o) for all m E M(L) and f E L. For m E M(L) we define Smf : L --+ L by
Smf := lE(mf I Eo)
V f E L.
Sm is order bounded and ISml :::; Simi' Sm is also order continuous. The following proposition will be applied in the sequel. A proof can be found in [3], (p8). PROPOSITION
2. Let (n, E, J,L) be a probability space and Eo C E a sub-a-
algebra. (i) If f E domlE('IE o) and 9 E Lo(Eo), then it follows that gf E domlE('IE o) and lE(gf I Eo) = glE(f I Eo). (ii) If f E Lo(E), then f E domlE('IE o) if and only if there exists a sequence
{A n }:'=1 in Eo such that An in and .
IfldJ,L <
= 1,2, ....
J.J. GROBLER AND M. DE KOCK
Moreover, if f E domlE(·IE o), then, for all A E Eo with
lEU I Eo)dft =
fA Ifldft < 00,
The following lemma will be applied in t.he sequel. LEMMA
3. For a linear subspace N of Loo(E) the following statements are
equivalent. (i) There exists a sub-a-algebra Eo such that N = Loo(Eo). (ii) N is a subalgebra of Loo(E) containing the constants such that fn N, Ifni::; u E Loo(E) (11. = 1,2, ... ) and fn - t f a.e.
imply that fEN.
The proofs of the following propositions and corollaries rely mainly on t.he proofs by Dodds, Huijsmans and De Pagter (see [2]). Let L c Lo(E) be an ideal of measurable functions such that Loo C L.
We then have the following. PROPOSITION
4. Let S : L
L be an or'der continuous, positive linear opemtor
for which (i) Sf E Loo(E) whenever f E Loo(E),
(ii) SUSg) = Sf· Sg for' all f E Loo(E) and all gEL. Then there exists a sub-a-algebra Eo of E and there exists a 0 ::; m E M(L) such that Sf = lE(mf I Eo) for all f E L. We use the following
proposition in the proof of the main result. PROPOSITION
5. For a linear operator S : L
L, the following statements are
(i) S is positive and or'der continuous, S2 = S, SI = 1 and the range R(S) of S is a sublattice.
(ii) There exist a sub-a-algebra Eo of E and a function 0 ::; m lE(mIEo) = 1 such that Sf = lE(mf I Eo) for all f E L.
M(L) with
Because the range of a strictly positive projection is a sublattice, we obtain the following result. COROLLARY
6. For a linear operator S : L
L the following statements are
(i) S is a strictly positive, order continuous project'ion with SI = 1. (ii) There exists a sub-a-algebra Eo of E and a strictly positive function m M(L) with lE(ml Eo) = 1 such that Sf = lE(mfl Eo)
for all f E L.
In the following proposition we consider the case where the operator no more preserves onc, but where the image of the indicator function is strictly positive. We derive a similar result as in
Corollary 6 for S strictly positive. PROPOSITION 7. Let S : L ments are equivalent.
L be a linear operator, then the following state-
(i) S is a positive or'der continuous projection onto a sublattice such that SI is strictly posit'ive. (ii) There exist a s'nb-a-algebra Eo ofE, 0 ::; m E Lo(E) and a strictly positive function k E
Lo(E) with lE(mk I Eo) = 1, such that Sf = klE(mf I Eo) for all f E L.
We have the following basic characterization of conditional expectation on L1 (E). A proof can be found in [2], p71. PROPOSITION 8. (Douglas R.G. and Seever) If T is a continuous linear map on L 1
(E), then the following statements are equivalent.
(i) There exists a sub-a-algebra Eo of E such that for all f E L1 (E) we have that Tf = lEU I Eo). (ii) T is a contractive projection which preserves 1. 3. Main characterization of conditional
expectation. We prove that if an operator can be characterized in terms of the conditional expectation, then its dual can also be characterized in terms of the conditional expectation. As before, we
let L c Lo(E) be an ideal of measurable functions which contains Loo(E). LEMMA 9. If S : L --+ L is a linear operator such that Sf = lEU IEo) for all f E L, then S' : L' --+ L' satisfies S' 09 = lE(g
I Eo) for all E-measurable 09 E L'.
ProoF. Let f ELand gEL'. Then (j, S'g)
(Sf,g) l
l l
lE iL U I Eo)gdJ.l
lEiL(glEiLU I Eo) I Eo)dJ.l
llEiLU I Eo) ·1E1L(g I Eo)dJ.l l
f S'lE iL (g I Eo )dJ.l
Thus, we have proved that (3.1)
(j,S'IE(gIE o )).
For any Eo-measurable g, we have that (3.2) It follows from (3.2) that for Eo-measurable 9 we have that S'g = 9 and from (3.1) 0 and (3.2), for arbitrary gEL' that S'g = S'lE(g I Eo) = lE(g I Eo).
Now we are able to prove the main result, where we characterize conditional expectation in terms of a positive, order continuous projection and an extension of its dual. PROPOSITION
10. If S is a linear map, then the following statements are equi-
(i) Ther'e exists a sub-a-algebra Eo of E such that Sf = lEU I Eo) for' all f E L. (ii) S: L --+ L is a positive order continuous projection such that SI = 1 and S' has an extention S' : L' + Loo --+
L' + Loo satisfying S'I = 1.
J.J. GROBLER AND M. DE KOCK
Proof. (i) =} (ii) It follows from Proposition 5 that S is a positive, order continuous projection such that SI = 1. Since the conditional expectation operator is defined on L1 (E) into L1 (E), it
follows that
S : L n L 1(E)
Denote the restriction of S to L n L1 (E) by
8' : L' + Loo(E) and 8' is an extension of S' : L' have (3.3)
L n L 1(E).
L' + Loo(E)
L'. Because for
L' and gEL n L1 (E), we
(g, 8' J) = (8g, J) = (Sg, J) = (g, s' J).
Since L n L1 (E) is dense in L, (3.3) holds for all gEL, and so 8' f = S' f for all f E L'. For all gEL' + Loo(E), it follows from Lemma 9 that 8'g = lE(g 1 Eo), so 8'1 = 1, by the properties of
conditional expectation. (ii) =} (i) We first note that since Loo (E) c L we have L' C L1 (E) and also that L' is dense in L 1 (E). An argument of Ando (see [1], (p401)) shows that S' is contractive
for the L1 (E)-norm. In fact, if gEL', it follows from the assumption SI = 1 that,
In IS'
In In IgIIS(1
S' 9 sgn S' gd{t
< InlgIS(lsgnS'gl)dJL <
In In
IglSld{t Igld{t.
Since L' is dense in L1 (E) for the L1 (E)-norm, we can extend S' to a contraction on LdE). Since S is a projection, the same holds for S'. Thus, by Proposition 8, there exists a sub-a-algebra Eo of
E such that S' 9 = lE(g 1 Eo) for every 9 E Ll (E) and therefore also for all gEL'. It follows from Lemma 9 that S" f = lEU 1Eo) for all f E L". By restricting S" to L, we therefore have that Sf =
lEU 1 Eo) for all f E L. 0
References [1] ANDO, T., 1966, Contmctive projections in Lp-spaces, Pacific J. Math., 17,391-405. [2] DODDS, P.G., HUIJSMANS, C.B., and DE PAGTER, B., 1990, Chamcterizations of Conditional
Expectation type-opemtors, Pacific J. Math., 141,55-76. [3] GROBLER, J.J. and DE PAGTER, B., 1999, Opemtors representable as Multiplication Conditional Expectation opemtors, To appear in J. of
Operator Theory. [4] NEVEU, J., 1975, Discrete-pammeter martingales, North Holland/American Elsevier, Amsterdam Oxford New York. [5] ZAANEN, A.C., 1982, Riesz spaces II, North Holland, Amsterdam, New
York. SCHOOL FOR BUSINESS MATHEMATICS, POTCHEFSTROOM UNIVERSITY FOR CHE, POTCHEFSTROOM 2520, SOUTH AFRICA" MATHEMATICS DEPARTMENT, KENT STATE UNIVERSITY, KENT, OH 44240 E-mail
address:srsjjgClpuknet.puk.ac.za • mdekockClmath. kent. edu
Contemporary Mathematics Volume 328, 2003
The Krull nature of locally C* -algebras Marina Haralampidou ABSTRACT. Any complete locally m-convex algebra, whose normed factors in its Arens-Michael decomposition are Krull algebras is also Krull.
In particular, any locally CO-algebra is a Krull algebra. Considering perfect projective systems, we give another proof of the fact that any Frechet locally CO-algebra is a Krull algebra.
Furthermore, a proper complete locally m-convex H* -algebra with continuous involution and a normal unit is a locally C* -algebra, hence Krull. The class of Krull (topological) algebras is closed
with respect to cartesian products, topological algebra isomorphic images, and perfect projective limits.
1. Introduction and preliminaries
Every closed (left) ideal of a CO-algebra E is the intersection of the (closed) maximal regular (left) ideals containing it (see, for instance, [2: p. 56, Theorem 2.9.5]. Thus, E is a Krull algebra
in the sense of Definition 1.1. A natural question arises here whether, in general, any locally CO-algebra is Krull. In that direction, using the Arens-Michael decomposition, we get that a complete
locally m-convex algebra (E,(Po)oEA) is a Krull algebra, if each factor Eo. = E/ker(po.)' Q E A is a Krull algebra (Proposition 2.1). As a consequence, we get that any locally CO-algebra is Krull
(Corollary 2.2). Besides, a proper complete locally m-convex H*-algebra (E, (Po.)o.EA) with continuous involution and a unit element e, so that po.(e) = 1 for every Q E A, is a Krull algebra
(Corollary 2.6). Based on the notion of a perfect projective system (Definition 2.7), we provide another proof of the fact that any Fn3chet locally C* -algebra is Krull (Theorem 2.10). By the term
topological algebm we mean an algebra, which is a topological linear space such that the ring multiplication is separately continuous (see [11: p. 4, Definition 1.1] and/or [12: p. 6]). A topological
algebra E is called a Q'-algebm, if every maximal regular left or right ideal in E is closed (see [5: p. 148, Definition
1.1]). A locally m-convex algebm is a topological algebra E whose topology is defined by a family (Po.)o.EA of submultiplicative seminorms, i.e. Po.(xy) :::; Po.(x)Po.(y) for 1991 Mathematics Subject
Classification. Primary 46H05, 46H10, 46H20. Key words and phrases. Krull algebra, Q'-algebra, Arens-Michael decomposition, locally C*algebra, perfect projective system of topological algebras,
perfect projective limit algebra, Frechet locally CO-algebra, proper algebra, locally m-convex H*-algebra. © 2003 American Mathematical Society 195
all x, y E E, 0: E A (see for instance [11] and/or [12]). Such a topological algebra is denoted by (E, (PoJ"'EA)' A complete metrizable locally m-convex algebra E is called a Prechet locally m-convex
algebra. In this case, the topology of E is defined by a countable family (Pn)nEN of submultiplicative seminorms. A C*-seminorm is a seminorm P on an involutive algebra E, satisfying the
C*-condition, namely, p(x*x) = p(x)2 for every x E E [13: p. 1, Definition 1]. Such a seminorm is submultiplicative and *-preserving [ibid. p. 2, Theorem 2]. A locally pre-C* -algebra is an
involutive locally (m-) convex algebra (E, (P",)",EA), such that each p"" 0: E A is a C* -seminorm, while a complete algebra, as before, is called a locally C* -algebra [8: p. 198, Definition 2.2]. A
Frechet locally C* -algebra is an involutive Frechet locally (m-) convex algebra (E, (Pn)nEN) where each Pn is a C* -seminorm. A locally m-convex H* -algebra is an algebra E equipped with a family
(P"')"'EA of Ambrose seminorms in the sense that P"" 0: E A arises from a positive pseudoinner product <, >"" such that the induced topology makes E into a locally mconvex topological algebra.
Moreover, the following conditions are satisfied: For any x E E, there is an x* E E, such that
< xy,z >",=< y,x*z >", < yx, z >",=< y, zx* >", for any y, z E E and 0: E A. x* is not necessarily unique. In case, E is proper (viz. Ex = (0), implies x = 0), then x* is unique and * : E -+ E : x
f-+ x* is an involution (see [4: p. 451, Definition 1.1 and p. 452, Theorem 1.3]). Throughout of this work the considered algebras are over the field of complexes. To fix notation we recall the
following. Let (E, (P"')"'EA) be a complete locally m-convex algebra and
P'" : E
E/ker(p",) == E", : x
p"'(x) :=x + ker(p",)
the respective quotient maps. Then Ilx",ll", := p"'(x), x E E, 0: E A defines on E", an algebra norm, so that E", is a normed algebra and the morphisms P"" 0: E A are continuous. E"" 0: E A denotes
the completion of E", (with respect to II . II",). A is endowed with a partial order by putting 0: :::; /3 if and only if p"'(x) :::; P(3(x) for every x E E. Thus, ker(p(3) <;;;; ker(p",) and hence
the continuous (onto) morphism
j",(3 : E(3
E", : x(3
j",(3(x(3) := x""
is defined. Moreover, j ",(3 is extended to a continuous morphism !",(3 : E(3
Thus, (E"" j",(3), (E",'/",(3), 0:, /3 E A with (resp. Banach) algebras, so that
0: :::;
/3 are projective systems of normed
- -
lim E", 2:! lim E", (Arens-Michael decomposition)
within topological algebra isomorphisms (cf., for instance, [11: p. 88, Theorem 3.1 and p. 90, Definition 3.1] and/or [12: p. 20, Theorem 5.1]). Concerning the following notion see [7]. DEFINITION
1.1. A topological algebra is called a Krull algebra, if every proper closed left (resp. right) ideal is contained in a closed maximal regular left (resp. right) ideal.
For the statements (i) and (iii) in the next proposition see [7: Lemma 3.8). PROPOSITION 1.2. Let E, F be topological algebms and ¢ : E ---7 F a continuous epimorphism. Then the following hold true:
(i) If E is a Krull algebm and ¢ closed, then F is a Krull algebm. (ii) If E is a Krull algebm and F a Q'-algebm, then F is a Krull algebm. (iii) If F is a Krull algebm and ¢ closed with ker(¢) ~ I
for every proper closed left or right ideal in E, then E is a Krull algebm. (iv) If F is a Krull algebm and ¢ a closed injection, then E is a Krull algebm. PROOF. (ii) For a proper closed left ideal
J in F, ¢-l(J) is a proper closed left ideal in E with ker(¢) ~ ¢-l(J) and J = ¢(¢-l(J)) (see also [3: p. 316, Proposition B.5.4)). Thus, ¢-l(J) ~ M for some closed maximal regular left ideal Min E.
Hence J ~ ¢(M), so that ¢(M) is a maximal regular left ideal in F (ibid.), closed by Q'. Similarly, for proper closed right ideals. (iv) Immediate from (iii). 0 COROLLARY 1.3. A topological algebm is
a Krull algebm if and only if a topological algebm isomorphic image of it is so. PROPOSITION 104. Let (E"')"'EA be a family of topological algebms and F = II"'EA E", the respective cartesian product
topological algebm. Then F is a Krull algebm, if each E"" Q E A is a Krull algebm. The converse is true in case the factors are Q' -algebms. PROOF. Consider the canonical continuous epimorphisms
11"", :
E", : x = (X"')"'EA
E A.
Let I be a proper closed left ideal in F. Since the multiplication is separately continuous, it follows that the closure 11"",(1) of the left ideal 11"", (I), is a closed left ideal in E",. Moreover,
11"",(1) i- E", for some Q E A. Otherwise, II"'EA 11"",(1) = F. It is easily seen that n"'EA 11";;1 (E",) = ILEA E",. Besides, I = I = n"'EA 11";;1(11"",(1)). Hence I = II"'EA E", = F, a
contradiction. Now, since E", is a Krull algebra, 11"",(1) ~ M, for some closed maximal regular left ideal M. Hence I ~ 1I";;1(M) with 1I";;1(M) a closed maximal regular left ideal in F. Similarly,
for proper closed right ideals. The above argument shows that F is a Krull algebra. 0 For the rest of the assertion apply (ii) of Proposition 1.2. 2. The Krull property for locally C*-algebras
We provide first the following result akin to that of Proposition 104. PROPOSITION 2.1. Let (E, (P"')"'EA) be a complete locally Tn-convex algebm, such that the norrned algebms E"" Q E A in its
Arens-Michael decomposition, are Krull algebms. Then E is a Krull algebm, as well. On the other hand, if E is a Krull algebm, then a factor E", is a Krull algebm if it is also a Q' -algebm. PROOF. E
~ lim E", within a topological algebra isomorphism, say ¢ (see f-(1.3)). Consider the continuous epimorphic image p",(I) (see (1.1)) of a proper closed left ideal I in E. Claim that the closed left
ideal p",(I) is proper in E", for
some Q E A. Suppose the contrary. Then based on M. Exarchakos, concerning the first equality of the next rels, we get ¢(E)
= ll!!!Ea = ll!!!Pa(I) = ll!!!(fa(¢(I))) = ¢(I),
here fa denotes the restriction to ll!!! Eo of the projection map 7ra : TIaEA Eo -+ Eo, Q E A (cf. also [11: p. 87, Lemma 3.2 and p. 89, (3.24)]; we note that ¢(I) is a closed left ideal in lim Eo).
Thus, E = I, which is a contradiction. So, since tEa, Q E A is a Krull algebra, it follows that Pa(I) ~ M for some closed maximal regular left ideal M, and hence I ~ p~l(Pa(I)) ~ p~l(M). Besides, p~l
(M) is a maximal regular left ideal in E, closed by the continuity of Po. An analogous result holds for proper closed right ideals. The last part of the assertion follows from (ii) of Proposition
1.2. 0
By [1: p. 32, Theorem 2.4], the factor normed algebras, in the Arens-Michael decomposition of a locally C*-algebra, are C*-algebras and hence Krull (see, for instance [2: p. 56, Theorem 2.9.5]).
Thus, Proposition 2.1 implies the next. COROLLARY 2.2. Every locally C*-algebm is a Krull algebm. By Proposition 1.4 and Corollary 2.2, we get the next. COROLLARY 2.3. The cartesian product of
locally C* -algebras is a Krull (locally C* -) algebm. In view of Corollary 2.2, Theorem 4.7 in [6: p. 3732] is improved as follows: THEOREM 2.4. A locally C* -algebm is dual if and only if it is
complemented. Let (E, (Pa)aEA) be a proper complete locally m-convex H* -algebra with continuous involution. Then E can be made into a locally pre-C* -algebra, via a family (qa)aEA of C*-seminorms
given by (2.1)
so that, (2.2)
qa(X) ~ Po(x) for every x E E,
E A.
(Namely, the respective topology on E is weaker than the given one). Moreover,
Po(xy) ~ qa(x)Pa(Y) for every x, y E E,
E A.
(See [9: p. 265, Proposition 2.3]). In that framework we get the next two results. PROPOSITION 2.5. Let (E, (Po)oEA) be a proper complete locally m-convex H*algebm with continuous involution and a
unit element e. Then the following are equivalent: 1) po(e) = 1 for every Q E A (: normal unit). 2) (E, (Pa)aEA) is a locally C*-algebm. PROOF. 1) ==> 2): Let qa, Q E A be the seminorms given by
(2.1). By (2.3), p",(x) ~ qo(x) for every x E E, Q E A. Hence (see also (2.2)) Po = qa for every Q E A. Namely, (E, (Pa)aEA) is a locally C*-algebra. 2) ==> 1): C*-property implies Pa(e)(l-po(e)) = 0
for every Q E A. If po(e) = 0 for some Q E A, then Ileoli o = 0, where eo = e + ker(Pa) is the respective unit
element in the factor algebra EOl == EOl (see aslo [1: p. 32, Theorem 2.4] and [11: p. 91, Theorem 4.1]). Thus eOl = 0, which is a contradiction. Thus, pOl(e) =I- 0 for every Q E A, hence POl (e) = 1
for every Q E A. 0 As a consequence of Corollary 2.2 and Proposition 2.5 we have the next. COROLLARY 2.6. Every proper complete locally m-convex H* -algebm with continuous involution and a normal
unit is a Krull algebm. Our next aim is to provide another proof to the fact that a F'rechet locally C*-algebra is Krull (see Corollary 2.2). To do this, we use the notion involved in the next.
DEFINITION 2.7. A projective system {(EOl , fOl,8)}OlEA of topological algebras is called perfect, if the restrictions to the projective limit algebra
= ~EOl = {(x Ol ) E
EOl: fOl,8(X,8) = XOl , if Q:S; (3 in A} OlEA of the canonical projections 7r0l : I10lEA EOl -+ E Ol , Q E A, namely, the (continuous algebra) morphisms
fOl = 7rOl IE =limE", : E
EOl ,
E A,
are onto maps. The resulted projective limit algebra E = lim EOl is called perfect (topological) algebm. ~
LEMMA 2.8. Every Frechet locally m-convex algebm (E, (Pn)nEN) gives a perfect projective system of normed algebms. PROOF. For any n
:s; m in N, the connecting maps
fnm(X + ker(Pm)) = x + ker(Pn) are onto algebra morphisms (see, for instance, [11: p. 86, (3.6) and (3.7)]). So, since {(En, fnm)}nEN is a denumerable projective system of normed algebras, it follows
that fn, n E N (see (2.5)) are onto, as well (see [10: p. 229, Theorem 8]). 0 The proof in the next result is an adaptation of that in Proposition 2.1. PROPOSITION 2.9. Any perfect projective limit
of Krull algebms is a Krull algebm. PROOF. Let {(EOl,fOl,B)}OlEA be a perfect projective system of Krull algebras. Consider the projective limit algebra E = limEOl (see (2.4)), which is a closed
subalgebra of the cartesian product topological algebra I10l EA EOl (see, for instance, [11: p. 84, Lemma 2.1]). For a proper closed left ideal I in E, fOl(I) is a (closed) left ideal in E Ol , Q E
A. If f Ol (I) = EOl for every Q E A, then ~
1= limfOl(I) ~
= limfOl(I) = limEOl = E, ~
(see also [ibid. p. 87, Lemma 3.2]), which is a contradiction. Thus, fOl(I) =I- EOl for some Q E A. Since E Ol , Q E A is a Krull algebra, there exists a closed maximal regular left ideal, say M,
with fOl(I) ~ M and hence I ~ f;;l(1Ol(1)) ~ f;;l(M),
where J;;l(M) is a closed maximal regular left ideal in E and this terminates the proof for closed left ideals. Similarly, for closed right ideals. D THEOREM
2.10. Any Prichet locally C*-algebm is a Krull algebm.
PROOF. Let (E, (Pn)nEN) be an algebra as in the statement. By [1: p. 32, Theorem 2.4], the respective normed algebras En, n E N in the Arens-Michael decomposition of E, are C* -algebras and hence
Krull (see, for instance, [2: p. 56, Theorem 2.4.5]. In particular, {(En' Jnm)}nEN is a perfect system of normed algebras (see Lemma 2.8 and relation (2.6)). Proposition 2.9 assures that the
projective limit algebra lim En is a Krull algebra and hence E is a Krull algebra, as it fol+-lows from Corollary 1.3 and the fact that E ~ lim En within a topological algebra +-isomorphism (see
(1.3)). D References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
C. Apostol, b*-algebms and their representation, J. London Math. Soc. 3(1971), 30-38. MR 44:2040. J. Dixmier, C*-Algebms, North-Holland, Amsterdam, 1977. MR 56:16388. R.S. Doran and V.A. Belfi,
Chamcterizations of CO-Algebras. The Gel'fand-Na'tmark Theorems, Marcel-Dekker, 1986. MR 87k:46115. M. Haralampidou, On locally convex H*-algebms, Math. Japon. 38(1993), 451-460. MR 94h:46088. M.
Haralampidou, Annihilator topological algebras, Portug. Math. 51(1994), 147-162. MR 95f:46076. M. Haralampidou, On complementing topological algebms, J. Math. Sci. 96(1999), 3722-3734. MR
2000j:46085. M. Haralampidou, On the Krull property in topological algebms (to appear). A. Inoue, Locally C* -algebras, Mem. Faculty Sci. Kyushu Univ. (SerA) 25(1971), 197-235. MR 46:4219. A. EI
Kinani, On locally pre-C*-algebm structures in locally m-convex H*-algebms, Thrk. J. Math. 26(2002), 263-271. G. Kothe, Topological Vector Spaces, I, Springer-Verlag, Berlin, 1969. MR 40:1750. A.
Mallios, Topological Algebms. Selected Topics, North-Holland, Amsterdam, 1986. MR 87m:46099. E.A. Michael, Locally multiplicatively-convex topological algebms, Mem. Amer. Math. Soc. 11(1952).
(Reprinted 1968). MR 14,482a. Z. Sebestyen, Every C*-seminorm is automatically 8ubmultiplicative, Period. Math. Hung. 10(1979), 1-8. MR 80c:46065.
DEPARTMENT OF MATHEMATICS, UNIVERSITY OF ATHENS, PANEPISTIMIOPOLlS, ATHENS 15784, GREECE E-mail address:mharalamOcc.uoa.gr
Contemporary Mathematics Volume 328, 2003
Characterizations and automatic linearity for ring homomorphisms on algebras of functions Osamu Hatori, Takashi Ishii, Takeshi Miura, and Sin-Ei Takahasi ABSTRACT. Automatic linearity results for
certain ring homomorphisms between two algebras, in particular, semi-simple commutative Banach algebras with units are proved. For this purpose a representation by using the induced continuous
mapping between the maximal ideal spaces and ring homomorphisms on the field of complex numbers is given. Ring homomorphisms on certain non-complete metrizable algebras into the algebras of analytic
functions are also considered. A characterization of the kernel of complex-valued ring homomorphism on a commutative algebra is given. As a corollary of the results a complete description of ring
homomorphisms on the disk algebra into itself is given in terms of prime ideals.
Introduction A ring homomorphism between two algebras is a mapping which preserves addition and multiplication. If we assume that the mapping is linear, then it is an ordinary homomorphism. In the
case where the two algebras are just the field C of complex numbers, the assumption cannot be avoided; there are ring homomorphisms of C into C which are not linear nor conjugate linear (cf. [9]).
The history of ring homomorphisms on C probably dates back to the investigation of Segre [19] in the nineteenth century and that of Lebesgue [12]. A similar remark applies to finite-dimensional
Banach algebras. But this is not the case for several infinitedimensional ones; for instance, Arnold [1] proved that a ring isomorphism between the two Banach algebras of all bounded operators on two
infinite-dimensional Banach spaces is linear or conjugate linear (cf. [5]). Kaplansky [8] proved that if p is a ring isomorphism from one semi-simple Banach algebra A onto another, then A is a direct
sum Al EB A2 EB A3 with A3 finite-dimensional, p linear on All and p conjugate linear on A 2 . It follows that a ring isomorphism from a semi-simple commutative Banach algebra onto another with
infinite and connected maximal ideal space is linear or conjugate linear. 2000 Mathematics Subject Classification. Primary 46JlO, 46E25; Secondary 46J40. The first, the second, and the fourth author
were partialy supported by the Grants-in-Aid for Scientific Research, The Ministry of Education, Science, Sports and Culture, Japan.
© 201
2003 AJnerican Mathematical Society
O. HATORI, T. ISHII, T. MIURA, AND S.-E. TAKAHASI
It is interesting to study ring homomorphisms on Banach algebras which are not necessarily injective or surjective. We may expect that a number of ring homomorphisms on infinite-dimensional Banach
algebras are automatically linear or conjugate linear. By a routine work we see that a a ring homomorphism is real-linear if it is continuous. On the other hand, we can also arrive at automatic
linearity for several ring homomorphisms by results in [14, 15, 20, 21, 13]; they studied and characterized *-ring homomorphisms between commutative Banach algebras with involutions and ring
homomorphisms on regular commutative Banach algebras with additional assumptions. The heart of this paper is automatic linearity results for certain ring homomorphisms of a much more general nature.
Throughout the paper A and B denote semi-simple commutative Banach algebras with units eA and eB respectively. The maximal ideal space for A is denoted by MA. In this paper, we denote the Gelfand
transform of a E A also by a. For a ring homomorphism of C into C, we simply say a ring homomorphism on C. Let T be a ring homomorphism on C and x E MA. Then the complex-valued mapping p on A defined
p(a) = T(a(x)),
is a typical example of a ring homomorphism. Semrl [20, Example 5.4] showed that there exists a complex-valued ring homomorphism other than this type. In section 2 we show that if a ring homomorphism
of A into B satisfies a certain condition, say (m), then it is represented by a modified version of the above. Many ring homomorphisms satisfy this weak and rather natural condition (m): *-ring
homomorphisms on involutive algebras; p{A)(y) = C for every y E M B ; p(A) contains a subalgebra of B. Thus our result generalizes the previous ones in [14, 20, 21, 13]. In section 3, by using
results in section 2, we deduce some automatic linearity results for ring homomorphisms: p with (m) is real-linear on a closed ideal of finite co dimension in A; if p(CeA) = CeB and p(A) contains an
element with an infinite spectrum, then p is linear or conjugate linear. It is a natural question: under the two hypotheses (1) p(CeA) c CeB and (2) p(A) contains an element with an infinite
spectrum, does it follow that p is linear or conjugate linear? We give an affirmative answer under stronger hypotheses: (1) and (2)' p(A) contains an element whose spectrum contains a non-empty open
subset. Problems in the same vein are also considered not only for Banach algebras but also for algebras of analytic functions. Bers [3] proved that if U and V are plane domains and H(U), H(V) are
the rings of analytic functions on U, V respectively, then any ring isomorphism of H(U) onto H(V) is induced by a conformal (or anti-conformal) equivalence of V with U, thus the ring isomorphism is
linear (or conjugate linear). Nakai [17] and Rudin [18] have shown this also holds for open Riemann surfaces. Ring homomorphisms which are not necessarily injective or surjective are also considered
by many mathematicians (cf. [7, 10]). Among them, Becker and Zame [2] have proved automatic continuity and linearity for ring homomorphisms from certain complete metrizable topological algebras into
the algebra of analytic functions on connected, reduced analytic spaces. In section 4 we also consider ring homomorphisms into the algebras of analytic functions. In particular, we consider the case
of a ring homomorphism p from the algebra Rs of rational functions on C with poles off a subset SeC into an algebra of analytic funtions. Here Rs is a metrizable topological algebra, but it cannot be
a Banach algebra by the Baire category theorem. We show that, for certain subsets S, p is automatically linear or conjugate linear, if the range of p contains a 11011constant function. We also give
an example of S such that a ring hommomorphism that is neither linear nor conjugate linear, and whose range contains non-constant functions, is possible. In the final section we study ring
homomorphisms into C: ring homomorphisms whose ranges contain only constant functions. We characterize the kernels of ring homomorphisms from a unital commutative algebra into C, which is compared
with the one to one correspondence between maximal ideals and complex (linear) homomorphisms on commutative Banach algebras. As a corollary we show that there exists an injective ring homomorphism
from an algebra which consists of analytic functions into C. We also give a complete description of the ring homomorphisms on the disk algebra in terms of prime ideals. We say that a ring
homomorphism 7 on C is trivial if 7 = 0 or 7(Z) = Z (resp. z) for every Z E C. Other ring homomorphisms on C are said to be non-trivial. We note some properties of non-trivial ring homomorphisms on
C, which are used later in this paper. For a proof of the existence of non-trivial ring homomorphisms, historical comments, and further properties, see [9]. It is easy to see that every non-zero ring
homomorphism 7 on C fixes rational real numbers and 7(i) = i or -i. If 7 is non-trivial, then 7 does not preserve complex conjugation. (This is a standard fact. Here is a proof. Suppose 7 does
preserve complex conjugation: 7(Z) = 7(Z) for every Z E C. Then 7(JR) C JR, that is, 7 is a ring homomorphism on the set of all real numbers R If x> 0, then 7(X) = (7( JX))2 > O. It follows that 7 is
order preserving on R Since 7(r) = r for every rational real number r, we have 7(X) = x for every real number x. Thus 7(Z) = Z (resp. 7(Z) = z) for every Z E C if 7(i) = i (resp. 7(i) = -i), which is
a contradiction.) It is easy to see that T is non-trivial if and only if 7 is discontinuous at every (resp. one) point in C. Thus, if T is non-trivial, then it is unbounded on every neighborhood of
zero. It follows that there exists a sequence {w n } of complex numbers which converges to 0 such that IT(Wn)1 tends to infinity as n -> 00 if 7 is non-trivial. If the ring homomorphism on C is onto,
then it is said to be a ring automorphism on C. Note that there is a non-zero ring homomorphism on C which is not a ring automorphism. We also note that there is a non-trivial ring automorphism on C
(cf. [9, 11]).
1. Partial representation If ¢ is a non-zero complex homomorphism on A, then there exists a unique x E MA such that ¢(a) = a(x) for every a E A. By this fact a well-known representation of a (linear)
homomorphism VJ from A into B follows: There exists a continuous mapping defined on {y E MB : VJ(a)(y) :I 0 for some a E A} into MA such that
= a((y)),
a E A,
y E {y E MB : VJ(a)(y)
:I 0 for
some a E A}.
On the other hand, if is a continuous mapping of MB into AfA and Ty is a ring homomorphism on C for every y E AfB , then
p(a)(y) = Ty(a((y)),
a E A,
y E MB
defines a ring homomorphism from A into the algebra of all complex-valued functions on M B . Thus it defines a ring homomorphism from A into B under the condition that T. (a ( (. )) is in B for every
a E A, and this is the case when M B
O. HATORI, T. ISHII, T. MIURA, AND S.-E. TAKAHASI
is finite. A problem is the converse: Is every ring homomorphism represented as above? A negative answer is known even in the case where B = C by the example due to Semd [20, Example 5.4].
Nevertheless, we show that a partial representation is still possible in this section. DEFINITION 1.1. Let p be a ring homomorphism of A into Band y a point in M B . The induced ring homomorphism py
of A into C is defined by
py(a) = p(a)(y), Let Ie : C
A be defined by IdA)
= AeA
for every A E C. We denote Ty
= pyole.
For every y E M B , the induced mapping Ty is a ring homomorphism on C. DEFINITION 1.2. Let p be a ring homomorphism of A into B. We denote: Mo = {y E MB : Ty = a}; !vlt = {y E MB : Ty(Z) = Z for
every Z E C}; ALl = {y E MB : Ty(Z) = Z for every Z E C}; Md,l = {y E MB : Ty is non-trivial and Ty(i) = i}; Md,-l = {y E !vIB : Ty is non-trivial and Ty(i) = -i}. LEMMA 1.3. Let p be a ring
homomorphism of A into B. Then M o, Ml U Md,l and M-1 UMd,-l are clopen (closed and open) subsets of MB. The subsets M1 and M-l are closed in M B . PROOF. By the definitions it is easy to see that Mo
= {y E MB : p(ieA)(y) = a}, M1 U Md,l = {y E MB : p(ieA)(y) = i}, and ALl U Md,-l = {y E MB : p(ieA)(y) = -i}, so they are clopen since p(ieA) is continuous on M B . Next we show that Jl,ft is a
closed subset of MB. Let y E Md,l' Since Ty is non-trivial, there exists a complex number A such that Ty(A) =I- A. Put
Then G is an open neighborhood of y. We also see that G n M1 = 0. It follows that ]\,{1 is a closed subset of MB since M1 U M d,l is clopen. In the same way, we see that M-1 is a closed subset of M B
. 0 Suppose that p is a ring homomorphism of A into B. If y E M 1 , then it is easy to see that Py is a non-zero complex homomorphism on A. Thus there exists a unique cp(y) in MA with
p(a)(y) = a(CP(y)),
a E
In a way similar to the above we arrive at a partial representation as follows:
p(a)(y) =
a, { a(CP(y)), a(CP(y)),
yEMo, yE M 1 , y E M_ 1 .
If y E Md,l U Md.-1, then the situation is complicated, in particular, ring homomorphisms with large Md,l U Md,-l are possible (cf. [20, Examples 5.3 and 5.4]).
2. Ring homomorphisms which satisfy the condition (m) In general the kernel of a non-zero ring homomorphism of A into C is a prime ideal and need not be a maximal ideal. (See section 5 in this
paper.) In this section we consider ring homomorphisms P of A into B which satisfy the condition that the kernel of the induced ring homomorphism Py for each y E MB defined by
Py(f) = p(f)(y),
f EA
is a maximal ideal. DEFINITION 2.1. Let P be a ring homomorphism of A into B. We say that P satisfies the condition (m) if Py is zero or ker Py is a maximal ideal of A for every yEMB. By the
following Lemma 2.2, if py(A) = C for every y E M B , in particular, if p(A) :J CeB, then (m) is satisfied. A *-ring homomorphism also satisfies the condition (m). (See the proof of Corollary 2.5.)
LEMMA 2.2. Let Po be a non-zero ring homomorphism of A into C. Then the following are equivalent. (1) The kernel ker Po of Po is a maximal ideal of A. (2) The equation po(A) = PO(CeA) holds. (3)
There exist a non-zero ring homomorphism 7 on C and an x E MA such that the equation po(a) = 7(a(x)) holds for every a E A. In this case 7 = Po 0 Ie. Such a 7 and x are unique. (4) The mnge Po(A) is
a subfield ofC which contains a non-zero complex number. PROOF. First we show that (1) implies (2). Suppose that ker Po is a maximal ideal. Then there exists a non-zero complex homomorphism
p(a)(y) = {7y(a(
O. HATORI, T. ISHII, T. MIURA, AND S.-E. TAKAHASI
In particular, p(a)(y) = a(q>(y)) for every y E Ml and p(a)(y) = a(q>(y)) for every y EM_I' The set q>(Md,l U Md,-d is a (possibly empty) finite subset of MA and q>-l(X) n (Md.l U Md,-d is an open
subset of MB for each x E q>(Md,l U Md,-d. PROOF. Let y E MB \ Mo. Then by the condition (m) ker PY is a maximal ideal of A, so by Lemma 2.2 there exists a unique q>(y) E MA such that
py(a) = PY o Ic(a(q>(y))) holds for every a E A. By the definition Ty = py 0 Ie and since py(a) = p(a)(y) we see that p(a)(y) = Ty(a(q>(y))) holds for every a E A. If y E M l , then Ty(A) = A for
every A E C, so p(a)(y) = a(q>(y)). If y EM_I, then Ty(A) =). for every A E C, so p(a)(y) = a(q>(y)). If y E M o, then p(a)(y) = O. Put Md = Md,l UMd,-l. We show that q>(Md) is a finite subset of MA.
Suppose not. Then there is a countable subset {Xn}~=l of q>(Md). For each n choose a point Yn E Md with q>(Yn) = Xn · Since TYI is unbounded near zero, there exists an al E A such that lIalli < 2- 1
and hI (al(xd)1 > 2. By induction we can find, for every n, an E A such that an(xd = ... = an(xn-d = 0, lIanll < 2- n , and
ITYn(an(xn))1 > 2n + ITYn(al(Xn) + ... + an-l(xn))l· (Choose b2 E A with b2 (xd = 0 and b2(X2) = 1. Since TY2 is unbounded near zero, there is a non-zero complex number 02 such that 1021 < Ilb211- l
2- 2 and ITY2(0)1 > 22+ITy2 (al(x2))I· Then put a2 = 02b2. We have a2(xl) = 0, IIa211 < 2- 2, and ITy2 (a n (x2))1 > 22+ITy2 (al(x2))1. Suppose that al, .. ' ,an-l E A are choosenso that the
conditions are satisfied. Choose bn E A with bn(xd = ... = bn(xn-d = 0 and bn(xn) = 1. Since TYn is unbounded near zero, there is a non-zero complex number On such that 10nl < Ilbn ll- 1 2- n and
ITYn (on)1 > 2n + ITYn (al(xn) + ... + an-l(xn))l· Then a2 = onbn is a desired function for n.) Then E::'=l an converges in A, say to a. Then a(xn) = al(x n ) + ... + an(xn) since the Banach norm on
A dominates the uniform norm on AlA. On the other hand
so that p(a) is unbounded, which is a contradiction proving that q>(Md) is a finite set. Let q>(Md) = {Xl, ... , xn} and Yj = q>-l(Xj) n Md for each j = 1,2, ... , n. Choose an a E A such that a(xl)
= 1, a(x2) = ... = a(xn) = O. Then p(a)(y) = 1 if y E Y l while p(a)(y) = 0 if y E Md \ Y l . Because p(a) is continuous, Y1 is clopen in Md; but Md is open in M B , so Y l is open in MB. In the same
way we see that Yj is an open subset of MB for each j = 2,3, ... n. Finally we prove that q> is continuous. Since Yj is open and q>(Yj) = Xj, we only need to prove that q> is continuous at each point
in Ml U M_ l . Let y E Ml and {y>.hEA be a net which converges to y. Without loss of generality we may assume that {y>.} c Ml U Md,l since Ml U Md,l is clopen. Suppose that {q>(y>.)} does not
converge to q>(y), that is, there is an open neighborhood G of q>(y) such that for every A E A there exists a A' 2:: A with q>(YN) rf. G. There exist a finite number of points aI, ... , am in A and a
positive real number e such that
MA : laj(x) - aj(q>(y))1 < e,j = 1,2, ... , m}
Since {p(aj)(y,x)} converges to p(aj)(Y) for each j, there exists a Ao E A such that
JTy>. (aj(iP(y,x))) - aj(iP(y))J < e holds for every A ~ AO and j = 1,2, ... , m. Suppose that A ~ Ao, then there exists a A' ~ A such that iP(y,x/) f/. G, so that Jajl(iP(y,x/)) - ajl(iP(y))J ~ e
for some j'. lt follows that YN E Md,l. We also see that iP(y,x/) E {Xl, ... ,Xn } \ {iP(y)}. There exists an a E A such that a(iP(y)) = 1 and a = 0 on {Xl, ... ,Xn } \ {iP(y)}. We conclude that for
every A with A ~ Ao there exists a A' ~ A such that p(a)(y,x/) = 0 and p(a)(y) = 1, which is a contradiction since p(a) is continuous on M B . Thus we have that {iP(y,x)} converges to iP(y), so iP is
continuous at y. In the same way we see that iP is continuous at each point in M_ I . We have proved that iP is continuous on MB \Mo. 0 Note that the set Md,l U Md,-l need not be a finite set or even
a closed subset of AlB (cf. [20, Example 5.3]). In [21] the authors proved the following corollary in the case where A is regular and satisfies a certain additional condition. Now we can remove these
conditions. COROLLARY 2.4. Let p be a ring homomorphism from A into B. Suppose that py(A) = C for every y E M B . Then there exists a continuous mapping iP of MB into MA and a non-trivial ring
automorphism Ty on C for every y E Md,l U Md,-l s1Lch that Y E MI , a(iP(Y))' { p(a)(y) = a(iP(y)), Y EM_I, Ty(a(iP(y))), Y E Md,l U Md,-l.
Moreover iP(Md,1 U Md,-d is a finite subset of MA. PROOF. By Lemma 2.2 we see that ker py is a maximal ideal, so the condition (m) is satisfied. The conclusion follows by Theorem 2.3. In particular,
Ty = Py 0 Ie is onto, thus it is a non-trivial automorphism on C for y E Md,l U Md,-l. 0
Theorem 2.1 in [13] for the case of unital and semi-simple commutative Banach algebras is also deduced from Theorem 2.3 COROLLARY 2.5. Suppose that A is involutive and B is symmetrically involutive.
Let p be a *-ring homomorphism. Then MB = Mo U MI U M-I and there exists a continuous function iP from MB \ Mo into AlA such that
a(iP(Y))' p(a)(y) = { 0, --:a('-=-iP..,...( y77")) ,
MI ,
yE Mo,
y EM_I·
PROOF. Since p is a *-ring homomorphism, it it easy to see that Ty(Z) = Ty(Z) for every Z E C and for every y E MB \ Mo. It follows that Ty is 0 or linear or conjugate linear. Thus the conclusion
follows. 0
3. Automatic linearity One of the reasons for ring homomorphisms between infinite-dimensional Banach algebras to be linear or conjugate linear is that the range contains an element with large
spectrum. In this section we show evidence of this.
O. HATORI, T. ISHII, T. MIURA, AND S.-E. TAKA HAS I
COROLLARY 3.1. Let p be a ring homomorphism of A onto B. If MB contains no isolated point, then p is real-linear. If MB is infinite and connected, then p is linear or conjugate linear. PROOF. Since p
is a surjection, py(A) = C for every y E M B , so the condition (m) is satisfied by Lemma 2.2. We also have that the induced mapping is injective. Thus Md,l U Md,-l is a (possibly empty) finite set.
Because Md,l U Md,-l is open (by Lemma 1.3), each point of Md,l U Md,-l is isolated in M B . If MB contains no isolated point, then Md,l U Md,-l = 0. Thus p is real-linear. If lvIB is infinite and
connected, then MB contains no isolated point, so Md,l U Md,-l = 0. It follows by Lemma 1.3 that MB = lvh or MB = M_ l . Thus P is linear or conjugate linear. 0 COROLLARY 3.2. Let p be a ring
homomorphism of A into B. Suppose that p satisfies the condition (m). Then there exists a (possibly empty) finite subset {Xl, ... , xn} of M A such that p is real-linear on the finite-codimensional
closed ideal {a E A: a(xj) = O,j = 1,2, ... ,n} of A. PROOF. Put {Xl,""X n } = (Md,l UMd,-d. (The set is finite by Theorem 2.3.) Then for every a E {a E A: a(xj) = O,j = 1,2, ... ,n} p(a)(y)
{Ty(a((Y))), 0,
y E Ml U M_l' Y E Mo U Md,l U Md,-l.
Since Ty is real-linear for every y E Ml U M_l' the conclusion follows.
COROLLARY 3.3. Let p be a ring homomorphism from A into B such that p(CeA) = CeB. Then we have that Mo = 0, and there exists a continuous mapping from MB into MA such that one of the following three
occurs. (1) P is linear: p(a)(y)
= a((y)),
a E A,
y E MB .
a E A,
y E MB .
(2) p is conjugate linear: p(a)(y)
= a((y)),
(3) There exists a non-trivial ring automorphism T on C such that p(a)(y)
= T(a((y))),
a E A,
y E M B.
In particular, if there exists an a E A such that the spectrum of p( a) is an infinite set, then p is linear or conjugate linear.
PROOF. For every y E MB, we have py(CeA) = C, so py(A) = py(CeA) = c. Thus ker Py is a maximal ideal of A by Lemma 2.2, so that the condition (m) is satisfied. Since p(ieA) E CeB, MB = Ml U Md,l or
MB = M_l U Md,-l. Suppose that Md,l U Md,-l = 0. Then (1) or (2) occurs. Suppose that there exists some Yd E Md,l and some Yl E M l · Then there is a complex number>' with Tyd (>') =I=- >., so that p
(>.eA) is not a constant function, which contradicts our hypothesis. Thus Md,l =I=- 0 implies that MB = Md,l. It is also easy to see that Ty is identical for every y E Md,l since p(CeA) consists of
constant functions. Thus (3) follows. In the same way we see that (3) follows if Md,-l =I=- 0. Suppose that there exists an a E A such that p(a)(MB) is infinite, then (3) does not occur since ( M B)
is a finite set in this case. It follows that p is linear or conjugate linear. 0
Corresponding results for ring homomorphisms on rings of analytic functions are proved by Kra [10, Theorem I]. Suppose that p is a ring homomorphism from A into B which satisfies two conditions: p
(CeA) C CeB; there exists an a E A such that p(a)(MB) is infinite. Does it follow that p is linear or conjugate linear? Although the authors do not know the answer, we can provide a positive answer
under a stronger condition. THEOREM 3.4. Let p be a ring homomorphism from A into B. Suppose that the following two conditions are satisfied: (i) p(CeA) C CeB; (ii) there exists an a E A such that p
(a)(MB) contains a non-empty open subset ofC. Then p is linear or conjugate linear. PROOF. Since p(CeA) C CeB we may suppose that pole is a non-zero ring homomorphism on C. We have two possibilities:
po Ie(i) = i; po Ie(i) = -i. We show that, in the first case, po Ie(z) = z for every complex number z, so it will follow that p is linear on A. (In the same way we see that p is conjugate linear if
pole( i) = -i.) Suppose that pole( i) = i. We show that pole is continuous on C. For this it is enough to show that pole is continuous at O. Suppose not. Then there is a sequence {w n } of non-zero
complex numbers which converges to 0 such that {poIe(w n )} does not converge to O. Without loss of generality we may assume that Ip 0 Ie(w n ) I ~ 00 as n ~ 00. Let a be in A such that p(a)(MB)
contains a non-empty open subset G of the complex plane. Let s be a complex number in G such that the real part and the imaginary part of s are both rational numbers. Put Zn = S + 1/ po Ie(w n ).
Then there is a positive integer mo such that Zm E G for every m ~ mo since Ip 0 Ie(w n )I ~ 00 as n ~ 00, so ZmeB - p(a) ~ B- 1 . Thus we have (8 + l/wm)eA - a ~ A-I. Then 8 + l/wm is in the
spectrum of a for every m ~ mo, which is a contradiction since Is + l/wnl ~ 00 as n ~ 00. It follows that pole is continuous at 0, thus on C, so pole( w) = w for every complex number w since p(ieA) =
i. Then we see that p is linear on A. D Note that either of the two conditions (i) and (ii) in the above theorem itself does not suffice for p to be linear or conjugate linear. Let T be a non-trivial
ring automorphism on C. Suppose that x E MA and tp from A into C is defined by tp(a) = a(x) for every a E A. Put P = TO tp. Then p is a ring fomomorphism with (i) since p(CeA) = C, but P is neither
linear nor conjugate linear; p is not even real-linear. Let D be the closed unit disk in the complex plane. Let D + 3 = {z E C: Iz - 31::; I} and X = D U (D + 3). Define p(f)(z) = {f(Z)' f(z - 3),
zED zED+3
for every f E C(D). Then p is a ring homomorphism from C(D) into C(X) with the condition (ii). But p is neither linear nor conjugate linear. Even more is true. There is a ring homomorphism with the
condition (ii) which is not real-linear. Recall that the disk algebra A(D) is the algebra of all complex-valued continuous functions on D which are analytic on the interior D of D. Suppose that
K = {O} U {l/n : n is a positive integer}, X
= {2} U D
and Y
= K U {z E C : Iz - 31::; I}.
O. HATORI, T. ISHII, T. MIURA, AND S.-E. TAKAHASI
A = {f
E C(X) :
and B = C(Y), where CO denotes the algebra of all complex-valued continuous functions on '. Let ¢ be the ring homomorphism from C into C(K) defined in [20, Example 5.3]. For f E A, put
f(2), y=O, p(f)(y) = { ¢(f(2))(1/n), y = lin, f(y - 3), Iy - 31 :::; 1. Then p is a ring homomorphism and satisfies the condition (ii). But (i) is not satisfied and p is not real-linear on A.
4. Ring homomorphisms into algebras of analytic functions Suppose that A is a completely metrizable topological algebra with an identity and r(X, Ox) is the algebra of global sections of a connected
reduced complexanalytic space (X, Ox). Becker and Zame [2] proved among other things that if p is a ring homomorphism from A into r(X, Ox) such that the range of p contains a non-constant section,
then p is linear or conjugate linear. This is not the case for ring homomorphisms on non-complete algebras. (Suppose that P is the algebra of polynomials on C and H(CC) is the algebra of entire
functions. Let T be a nontrivial ring homomorphism on C and define p on P by p(Eanz n ) = ET(an)Zn for every polynomial E anz n . Then p is a ring homomorphism, but it is neither linear nor conjugate
linear. By Theorem 5.1 there also exists an injective ring homomorphism from Pinto C since {O} is a prime ideal in P.) Nevertheless we show automatic linearity results for ring homomorphisms on
certain non-complete metrizable algebras. THEOREM 4.1. Suppose that A is a complex (commutative or non-commutative) algebra with unit e. Suppose that Y is a non-empty set and B is a complex algebra
of complex-valued functions on Y which contains the constant functions. Suppose that for every non-constant function b E B the range of b contains a non-empty open subset of c. Let p be a ring
homomorphism from A into B. If there exists an element a in A such that the resolvent set of a contains a non-empty open subset G of C and p( a) is non-constant, then p is linear or conjugate linear.
PROOF. It is easy to see that p( e) = 0 or 1 since the range of a non-constant function in B contains a non-empty open set. If p( e) = 0, then p is 0 on A and so p is linear. Suppose that p(e) = 1.
In the same way as above, we see that p(ie) = i or -i. We show that p is linear if p(ie) = i. (If p(ie) = -i, then 15 defined by 15(a) = p(a) will be linear by what we will show, so p will be
conjugate linear.) We will show that p(,Xe) = ,X for every complex number 'x. First we show that p(Ce) C cc. Suppose not. Then there is a complex number ,x such that p(,Xe) is a non-constant
function. Note that Re'x or Im'x is irrational since p(xe) = x for every rational real number x by a simple calculation. Since p('xe)(Y) contains a non-empty open set, there exists r E p('xe)(Y) with
Rer and Imr both rational. Then p(,Xe) - r is not an invertible element in B. Therefore (,x - r)e is not invertible in A, that is, ,x = r, which is a contradiction. Thus we have proved that p(Ce) C
C, or p induces a ring homomorphism on cc. We denote the induced ring homomorphism also by p.
Suppose that p is non-trivial on C. Since p(a) is not a constant, the interior of p( a) (Y) contains a complex number s whose real and imaginary parts are both rational, by our assumption for B.
Since p is non-trivial, there exists a sequence {w n } of non-zero complex number such that Ip(wn)1 tends to infinity as n --+ 00 and s+ l/w n E G for every n. Since p(i) = i we see that p(s+ l/wn )
= s+ 1/ p(w n ) and we may assume that
s + l/p(w n ) E p(a)(Y) for every n. Thus p(a) - (s + 1/ p(wn )) is not invertible in B. It follows that a - (s + l/w n )e is not invertible in A, which is a contradiction, proving that p is trivial.
Since p(i) = i, we have that p(A) = A for every complex number A. We conclude that p is linear on A. 0 The spectrum of each element in a Banach algebra is compact, so the conditions for A in Theorem
4.1 are satisfied by every Banach algebra with unit. Since the range of non-constant analytic function is a non-empty open subset of C, algebras of global sections on connected, reduced
complex-analytic spaces satisfy the condition for B in Theorem 4.1. Thus we have the following, which is a version of a more general result of Becker and Zame [2, Theorem 3.1]. But our proof is
considerably simpler. COROLLARY 4.2. Let Ao be a Banach algebra with unit. Suppose that p is a ring homomorphism from Ao into r(X, Ox), the algebra of global sections on a connected, reduced
complex-analytic space (X , Ox). If p( Ao) contains a nonconstant section, then p is linear or conjugate linear.
Let S be a subset of C. We denote by Rs the algebra of all rational functions on C with poles off S. Although Rs is a wlital algebra, it cannot be a Banach algebra by the Baire category theorem. If S
= C, then Rs = P, and so there is a ring homomorphism p on Rc into r( X , Ox) for (X, Ox) = C such that p(Rc) contains a non-constant function, while p is neither linear nor conjugate linear. In the
case where C \ S contains an interior point, the situation is different; in this case we prove an automatic linearity result. COROLLARY 4.3. Let S be a subset of C whose complement contains interior
points. Suppose that (X, Ox) is a connected, reduced complex-analytic space and r(X, Ox) is the algebra of global sections. Suppose that p is a ring homomorphism from Rs into r (X , Ox). If the range
of p contains a non-constant section, then p is linear or conjugate linear. PROOF. In the same way as in the proof of Theorem 4.1 we see that p(C) C C. Suppose that z denotes the identity function: z
(w) = w for every complex number w. Then we have that p(z) is non-constant. (Suppose not. Then p(f) is a constant section for every f E Rs.) On the other hand z - A is invertible for every A E C \ S,
which contains a non-empty open set. Thus the conditions in Theorem 4.1 are satisfied. It follows by Theorem 4.1 that p is linear or conjugate linear. 0
Note that every ring homomorphism p of R0 into r(X, Ox) is constant-valued for the empty set 0. (We see that p(C) C C as before. Suppose that p(f) is not a constant section for some non-constant
rational function f. Then there is a complex number r in p(f)(X) with rational real and imaginary parts. It follows that f - r is not invertible in R0, which is a contradiction.)
O. HATORI, T. ISHII, T. MIURA. AND S.-E. TAKAHASI
Let A be one of the algebras P, njj or the disk algebra A(D), where jj denotes the closed unit disk in C, and H(D) the algebra of analytic functions on the open unit disk. Although both P and no are
dense in the disk algebra, automatic linearity results for ring homomorphisms on these algebras are different from each other. Suppose that p is a ring homomorphism from A into H(D) such that the
range of p contains a non-constant function. If A = njj (resp. A(D), then p is linear or conjugate linear by Corollary 4.3 (resp. Corollary 4.2). But that is not the case for A = P. The ring
homomorphisms defined by p(2:a n z n ) = 2:1'(a n )zn for polynomials 2: anz n are neither linear nor conjugate linear for non-trivial ring homomorphisms l' on C.
5. Complex-valued ring homomorphisms In this section we consider ring homomorphisms into the complex number field C. Suppose that A is a complex algebra and p is a non-zero ring homomorphism from A
into C. Then the kernel ker p of p is a prime ideal. Recall that a proper ideal I of A is said to be a prime ideal of A if fg E I implies that f E I or gEl. By using well-known results of algebra, we
see the converse is also valid; for every prime ideal such that the cardinal number of the quotient algebra of the algebra by the ideal is equal to that of the continuum, there exists a ring
homomorphism into C whose kernel coincides with the ideal. Let K be an extension field of a field k. (Here and after a field means a commutative field.) We recall a subset S of K is said to be
algebraically independent over k if the set of all finite products of elements in S is linearly independent over k. A subset T of K which is algebraically independent over k and is maximal with
respect to the inclusion ordering is said to be a transcendence base of Kover k. By definition, for every transcendence base T of Kover k, K is algebraic over the quotient field k(T) of the
polynomial ring of T over k. There exists a transcendence base of Kover k (cf. [11, Theorem X.l.I]). Using the same argument as in [9] we can prove the following (cf. [11, 20]). (This might be a
standard fact. But we present here with a proof for the convenient of the readers.) THEOREM 5.l. Let A be a commutative complex algebra with unit e. Suppose that I is a prime ideal of A such that the
cardinal number of AI I is that of the continuum c. Then there exists a ring homomorphism p from A into C such that kerp=I. PROOF. The quotient algebra AI I has no non-zero divisor of zero, for I is
a prime ideal. We denote by K the field of fractions over AI I. Let Q be the field of complex numbers whose real and imaginary parts are both rational. Let TK be a transcendence base for Kover Q and
T a transcendence base for Cover Q. Then the cardinal number of TK (resp. T) is c since that of AI I (resp. q is c. There exists au injection a defined from TK onto T. Since TK is algebraically
independent, there is a unique extension from Q(TK ) onto Q(T), which is also denoted by a, and a is a ring homomorphism. Since C is algebraically closed and K is an algebraic extension of Q(TK),
there exists an extension of a which defines a ring homomorphism of K into C by Theorem VII.2.8 in [11]. We also denote it bya. Let h be the natural homomorphism of A onto AI I. Put p = a 0 h. Then p
is the desired ring homomorphism. D
Note that the corresponding ring homomorphism p is not unique. Let T be any non-zero ring homomorphism on C. Then TOp is a ring homomorphism on A with ker p = ker TOp. As a corollary of Theorem 5.1
we display a pathological feature of ring homomorphisms on algebras of analytic functions into C; even injection can be possible. COROLLARY 5.2. Let A be a unital algebra which consists of
holomorphic functions on a domain in Then there exists an injective ring homomorphism of A into Co
PROOF. Since the ideal containing only zero is a prime ideal and the cardinality of A is the same as that of the continuum, there exists a ring homomorphism p of A into C whose kernel consists only
of zero, by Theorem 5.1. Then p is an injective ring homomorphism. 0 Note that the injective ring homomorphism in Corollary 5.2 can never be surjective if A contains non-constant functions since A is
not a field. Note also that every ring homomorphism from a unital commutative C* -algebra into C cannot be injective if the dimension of the algebra is greater than one since {O} is not a prime ideal
in this case. Together with the results in the previous sections we give a complete description of ring homomorphisms on the disk algebra A(D). COROLLARY 5.3. Let p be a non-zero ring homomorphism on
the disk algebra into itself. Then ker p is a prime ideal. If the range of p contains a non-constant function, then p is linear or conjugate linear; there exists 'P E A(D) with 'P(D) c fJ such that
zED, f E A(D) p(f)(z) = f 0 'P(z), or
p(f)(z) = f
z E fJ,
f E A(fJ).
On the other hand, suppose that'P E A(D) with 'P(D) a(f)(z) = f
D. Then
E A(fJ)
E A(D)
defines a linear ring homomorphism and a(f)(z) = f
defines a conjugate linear ring homomorphism.
PROOF. A(D) has no non-zero divisors of zero, so the kernel of any ring homomorphism from complex algebra with unit element into A(D) must be a prime (algebra) ideal. If p(A(D) contains a
non-constant function, then by Theorem 4.1 we see that p is linear or conjugate linear. Suppose that p is linear. Then it is well known and easy to prove, since the maximal ideal space of A(D) is the
closed unit disk D, that there exists 'P E A(fJ) with 'P(D) c D such that p(f)(z) =
holds for every f E A(D) and zED. Suppose that p is conjugate linear. Let h : A(fJ) -+ A(fJ) be defined as h(f)(z) = f(2),
E A(D),
O. HATORI, T. ISHII, T. MIURA, AND S.-E. TAKAHASI
Then hop is a linear ring homomorphism on the disk algebra. It follows that there exists r.p E A(D) with r.p(D) c D such that
h 0 p(f)(z)
1 0 r.p(z) ,
Thus we see that
p(f)(z) = 1 0 r.p(z), holds for every I E A(D) and zED. Conversely, suppose that r.p E A(D) with r.p(D) c D. Then it is easy to see that u(f)(z)
1 0 r.p(z),
IE A(D)
IE A(D)
defines a linear ring homomorphism and
u(f)(z) = 1 0 r.p(z),
defines a conjugate linear ring homomorphism.
Let n be a positive integer and An(D) the subalgebra of those I in A(D) whose n-th derivative I(n) on D is continuously extended up to D. An(D) is a unital commutative Banach algebra with the norm
IIIlIn = L~=o III(k)lIoo/k! for I E An(D), where II . 1100 is the supremum norm on D. Then Corollary 5.3 is also valid for An(D). Prime ideals in A(D) and An(D) are studied in [16]. (See also [4] for
the case of A(D).) Mortini proved that every non-zero prime ideal is contained in a unique maximal ideal. He in fact showed that a non-zero and nonmaximal prime ideal in An(D) (resp. A(D)) is dense
in exactly one of the ideals {J E An(D) : 1(>") = 1'(>..) = ... = I(j)(>..) = O} for some 0 ~ j ~ n (resp. {I E A(D) : 1(>") = O}), >.. E aD. We also see by a theorem of Dietrich [4] that the
cardinal number of the set of all prime ideals of A(D) which is contained in a maximal ideal {J E A(D) : 1(>") = O}, >.. E aD is 2', the cardinal number of the set of all the subsets of the
continuum. Thus we see that there are 2' ring homomorphisms on the disk algebra. Acknowlegement. The authers would like to thank Professor Ken-Ichiroh Kawasaki for his valuable comments. They also
would like to thank the referees for their careful reading of the paper and their valuable comments.
References [1] B. H. Arnold, Rings of opemtors on vector spaces, Ann. of Math. 45(1944), 24-49 [2] J. A. Becker and W. R. Zame, Homomorphisms into analytic rings, Amer. Jour. Math. 101(1979),
1103-1122 [3] L. Bers, On rings of analytic junctions, Bull. Amer. Math. Soc. 54(1948), 311-315 [4] W. E. Dietrich, Jr., Prime ideals in uniform algebms, Proc. Amer. Math. Soc. 42(1974), 171-174 [5]
M. Eidelheit, On isomorphisms of rings of linear opemtors, Studia Math. 9(1940),97-105 [6] J. B. Garnett, Bounded analytic functions, Academic Press, New York, 1981 [7] H. Iss'sa, On the meromorphic
junction field of a Stein variety, Ann. Math. 83(1966), 34-46 [8] 1. Kaplansky, Ring isomorphisms of Banach algebms, Canadian J. Math. 6(1954),374-381 [9] H. Kestelman, Automorphisms of the field of
complex numbers, Proc. London Math. Soc. 53(1951), 1-12 [10] 1. Kra, On the ring of holomorphic functions on an open Riemann surface, Trans. Amer. Math. Soc. 132(1968),231-244 [11] S. Lang, Algebm
(second edition), Addison-Wesley, California, 1984. [12] M. H. Lebesgue, Sur les tmnsformations ponctuelles, tmnsformaant les plans en plans, qu'on peut definir par des procedes analytiques, Atti
della R. Acc. delle Scienze di Torino 42(1907), 532-539
[13J T. Miura, Star ring homomorphisms between commutative Banach algebrns, Proc. Amer. Math. Soc. 129(2001), 2005-2010 [14J L. Molnar, The rnnge of a ring homomorphism from a commutative C'
-algebrn, Proc. Amer. Math. Soc. 124(1996), 1789-1794 [15J L. Molnar, Automatic surjectivity of ring homomorphisms on H* -algebrns and algebrnic differences among some group algebrns of compact
groups, Proc. Amer. Math. Soc. 128(2000), 125-134 [16J R. Mortini, Prime ideals in the algebrn An(D), Complex Variables Theory AppJ. 6(1986), 337-345 [17J M. Nakai, On rings of analytic and
meromorphic functions, Proc. Japan Acad. 39(1963), 79-84 [18J W. Rudin, An algebrnic charncterization of conformal equivalence, Bull. Amer. Math. Soc. 61(1955), 543 [19J S. de Corrado Segre, Un nuovo
campo di ricerche geometriche, Atti della R. Acc. delle Scienze di Torino 25(1889), 276-301 [20J P. Semrl, Non-linear perturbations of homomorphisms on C(X), Quart. J. Math. Oxford (2) 50
(1999),87-109 [21J S.-E. Takahasi and O. Hatori, A structure of ring homomorphisms on commutative Banach algebrns, Proc. Amer. Math. Soc. 127(1999),2283-2288 DEPARTMENT OF MATHEMATICAL SCIENCE,
950-2181 JAPAN E-mail address:hatorilDmath.se.niigata-u.ae.jp
951-8126 JAPAN
992-8510 JAPAN E-mail address:miura«lyz.yamagata-u.ae.jp
992-8510 JAPAN E-mail address:sin-eiOemperor.yz.yamagata-u.ae.jp
Contemporary :r...1athematics Volume 328. 2003
Carleson Embeddings for Weighted Bergman Spaces Hans Jarchow and Urs Kollbrunner ABSTRACT. We are going to discuss Carleson measures for the standard weighted Bergman spaces A~ (-1 < a < 00, 0 < p <
(0). These are finite, positive Borel measures J.L on the unit disk in IC such that, given 0 < q < 00, A~ embeds, as a set, continuously into Lq(J.L). Such measures have been closely investigated by
V.L. Oleinikov and B.S. Pavlov [15], W.W. Hastings [6] and D.H. Luecking [11], [12]. We complement their results, in particular by characterizing compactness, order boundedness and related
(absolutely) summing properties of the canonical embedding A~ <-t Lq(J.L).
1. Introduction
The Carleson measures under investigation are finite, positive Borel measures Jl. on the open unit disk l[J in the complex plane such that, given -1
and 0 < p, q < 00, the (classical) weighted Bergman space A~ is a subset of Lq(J.l) and the embedding is bounded. Such measures have been characterized, even in a more general setting, by V.L.
Oleinikov and B.S. Pavlov [15J, W.W. Hastings [6J and D.H. Luecking [11], [12J. Our first main topic is to complement their results by characterizing when the canonical embedding I : A~ ~ Lq(J.l) is
compact. We will see that this is always the case if p > q. Our second main topic is to characterize when I is order bounded, that is, the unit ball of A~ is a subset of some order interval in Lq
(J.l). As a consequence, we obtain necessary and sufficient conditions for I to have specific (absolutely) summing properties. Our results extend corresponding ones for composition operators which
have been obtained e.g. in [3]' [4], [20]' [21J. In fact, they can also be viewed as results on composition operators which may have rather unusual range spaces. They apply, for example, to pointwise
multipliers. Many of the results to be presented remain valid for measures J.l on D for which f f-> f induces just a bounded linear map A~ ~ Lq(J.l) (not necessarily injective). This is rather
straightforward; precise formulations, however, require somewhat 1991 Mathematics Subject Classification. Primary 46 E 15,47 B 38, 47 B 10; Secondary 46 B 25, 30 H 05, 32 H 10. Key words and phrases.
Weighted Bergman spaces, Carleson measures, composition operators, compactness, order boundedness, absolutely summing operators. The results of this paper are part of the dissertation of the second
named author written at the University of Ziirich under the supervision of the first.
© 217
2003 Alnerican Mathematical Society
clumsy notation. Also, there is an immediate extension to complex Borel measures on 1lJ whose variation is (a, p, q) - Carleson. We are indebted to the referee for providing Example 8 and for
bringing to our attention the paper [15] by V.L. Oleinikov and B.S. Pavlov. 2. Weighted Bergman spaces Throughout the paper, we will use standard results and notation from (quasi-) Banach space
theory. We will work on the open unit disk 1lJ = {z E C: Izl < I} in the complex plane. The space 1i(1lJ) of all analytic functions 1lJ -> C is a F'rechet space with respect to the topology of
uniform convergence on compact subsets of 1lJ. Let da be normalized area measure on 1lJ. For each a > -1,
dao(z) :=
(a + 1) (1 -lzI 2 )O da(z)
is a probability measure on 1lJ. For each 0 < p < Bergman space is defined to be A~ := A~
the corresponding weighted
1i(1lJ) n P(ao ).
is closed in LP (a 0); it is a Banach space if p 2:: 1 and a p - Banach space if Its (p- ) norm will be denoted by II . Ilo,p. A~ is a Hilbert space and has a reproducing kernel:
o < p < 1.
Ko(z, w) = K(z, w)o+2; here K(z,w) = (1 - ZW)-l is the reproducing kernel for the Hardy space H2. For reasons like this, the scale of Hardy spaces is often considered as the scale of weighted Bergman
spaces which corresponds to a = -1. Some of the results below actually remain true for this case, and some can even be extended to analytic Besov spaces B~ (f E B~ {::} f' E A~+p). Nevertheless, in
this paper we will only deal with the case -1 < a < 00. 3. Carleson measures All measures on 1lJ will be finite, positive Borel measures. Let -1 < a < 00 and 00 be given. We say that a measure /L on
1lJ is an (a, p, q) - Carleson measure if A~ c Lq(/L) and the embedding A~ '---+ Lq(/L) is continuous: there is a constant C> 0 such that IIfIILq(~) ::; C ·lIfIIA~ "If E A~. Given an (a, p, q) -
Carleson measure, the canonical embedding I : A~ -> Lq(/L) will be referred to as a Carleson embedding. As mentioned in the introduction, a number of the results to follow remain true if we just
require that f 1-+ f induces a bounded linear map A~ -> Lq(/L). Also, complex measures whose variation is (a, p, q) - Carleson can be incorporated. Moreover, there are extensions to analytic
functions of several variables. However, we are not going to discuss such generalizations in this paper. We say that /L is a compact (a,p, q) - Carleson measure if the embedding A~ '---+ Lq(/L)
exists and is compact. For example, an a.e. positive function h E Lq(a/3) defines the bounded multiplier Mh : A~ -> Lq(a/3) : f 1-+ fh iff the measure h q da/3 is (a,p, q) - Carleson. Moreover,
discrete (a, p, q) - Carleson measures on 1lJ can be defined using appropriate versions of 'sampling sequences', etc.
o < p, q <
An important example is obtained by looking at the composition operator C", : J 0 'P induced by a non-constant analytic function 'P : 10 ---> 10. Clearly, C",: A~ ---> A~ exists iff a o o'P- 1 is
(o:,p,q)-Carleson. More generally, an arbitrary measure J.l on 10 is (0:, p, q) - Carleson if and only if, for every analytic map 'P : 10 ---> 10, CI{J maps A~ boundedly into Aq(J.l) := 1i(1O) n Lq
(J.l). In fact, the condition applied to the identity of 10 shows that J.l is (o:,p,q)Carleson. On the other hand, if J.l is (0:, p, q) - Carleson and 'P : 10 ---> 10 is analytic, then CI{J : A~ --->
Aq(J.l) is well-defined and bounded. For non-constant functions 'P, the condition is further equivalent to J.l 0 'P -1 being (0:, p, q) - Carleson. This allows an interpretation of Carleson
embeddings, and in particular of multipliers as above, as composition operators. However, in such a general setting the range space of a composition operator might be unpleasent, and desirable
properties may not be available. For example, Aq(J.l) embeds continuously into 1i(1O) if and only if Aq(J.l) is a closed subspace of Lq(J.l) and all point evaluations Aq(J.l) ---> C : J f-+ J(z), z E
10, are continuous. (0:, p, q) - Carleson measures have been characterized, even in a more general setting, by V.L. Oleinikov and B.S. Pavlov [15], W.W. Hastings [6] and D.H. Luecking
[11],[12]. The hyperbolic metric on 10 is given by
. lJd(J "I 1 _ J(J
e(z, w) := l~f
where the infimum extends over all smooth curves 'Y in 10 joining z and w. For w E 10 and r > 0, let Br (w ) = {z E 10 : e( z, w) < r} be the corresponding hyperbolic disk. Actually, the particular
choice of r > 0 doesn't really matter in our context. Let us agree to write A~ '---+ Lq(J.l) if A~ is a subset of U(J.l) and the embedding is continuous. Similar for other function spaces. The
Carleson measures under consideration can be characterized in terms of the function
3.1. Let -1 < 0: <
and 0 < p, q <
be given.
(a) IJp::; q, then A~
Lq(J.l) iJ and only iJ Ho,p,q(w) is bounded on 10.
(b) IJp> q, then A~
Lq(J.l) if and only if Ho,p,q
Here dA(z) = da(z)(I-JzJ2)-2 is the Mobius invariant measure on 10. In [12] Luecking presents an interesting proof of (b) which is based on the inequalities of Khinchin and Kahane for Rademacher
functions (see e.g. [2]). It is well-known (compare K. Zhu [22]) that there is a constant C = C(o:, r) > 0 such that 1 C . ao(Br(w)) ::; (I-JwJ2)o+2 ::; C· ao(Br(w)) 'Vw E 10 . Therefore we may also
say that Theorem 3.1 refers to properties of the function w f-+ J.l(Br (W))I/ q(1 _JWJ2)-(o+2)/p .
It also follows that Ha,p,q E L~(A) if and only if J-t(B r (·))/C7 a (B r (·)) is in LP/(p-q) (C7 a ). This will be used in the proof of Theorem 4.3 below. If p ~ q; then the relevant parameter in
Theorem 3.1 is q (0: + 2)/p, whereas for p > q and fixed 0:, dependence is on p / q. As a first immediate consequence we may state: COROLLARY
if and only if A~
3.2. For any -1 < 0: < "---> Ltq(J-t).
< p,q < 00 and t > 0,
A~ "--->
In turn, this leads to: 3.3. Let -1 < 0:,0:' < 00 and 0 < p, pi, q, q' < 00 be given. (a) Ifp ~ q, pi ~ q' and q. (0: + 2)/p = q'. (0: ' + 2)/p', then A~ "---> Lq(J-t) iff A~, "---> Lq' (J-t).
(b) Ifp> q and p/q = pi /q', then A~
Lq(J-t) iff A~
Lq' (J-t).
For a large range of parameters, this allows a reduction to Hilbert spaces as follows: 3.4. Suppose that -1 < 0:,0:' < 00 and 0 < p, q < 00. (a) Ifp ~ q and 0:' + 2 = q. (0: + 2)/p, then A~ "---> Lq
(J-t) iff A~,
(b) If p > q and pi /2 = 2/q' = p/q, then A~ A~ "---> L 2 (J-t). A special known case occurs when we take J-t Horowitz [7]).
Lq(J-t) iff A~
3.5. Suppose that -1 < 0:, (3 < 00 andO < p, q < (a) If p ~ q, then A~ "---> A~ iff (0: + 2)/p ~ ((3 + 2)/q. "--->
A~ iff (0: + l)/p
L 2 (J-t).
Lq' (J-t) iff
= C7fJ for some (3 > -1 (see C.
(b) If p > q, then A~
< ((3 + l)/q.
There are several ways to modify the domain space of a composition operator. In a systematic fashion, we may proceed as follows; cf. [4]. Each of the kernel functions K a ( Z, .) is bounded (z E lV),
and ._ ( 1 - IzI2 ) (a+2)/p
(1 _
ka,p,z(w),has (p-) norm one in representation
(0 < p <
The functions
E A~ which admit a
f(w) =
Vw E lV ,
where the scalars linear space, say
En lanl <
and the
are taken from lV, form a
This is a Banach space with norm
:= inf
{L lanl:
(*) holds} .
In fact, the map e1(lV)
A~) : (az)zEllJ
azka,p,z is a metric surjection.
It is immediate that
• if 0
are such that (a+2)/p
= (/3+2)/q,
Atomic decomposition is available for weighted Bergman spaces (e.g. [10], [1], [22]), hence
• A~
A~l) (with equivalent norms).
In particular, if a> p-2, then A~), alias Ab with /3+2 = (a+2)/p, is isomorphic to fl. Moreover: • If p 2: 1 then A~) '---+ A~ (boundedly and densely).
A~) = Ab (boundedly and densely). In fact, it was shown by J.H. Shapiro [19] that in the latter case Ab is the Banach • If 0
< p < 1, then A~
envelope of A~, that is, the Banach space which is obtained by completing A~ with respect to the biggest norm which is smaller than the given p-norm. PROPOSITION 3.6. Let -1 < a < 00 and 0 < p ::; q
< 00, q 2: 1. The following are equivalent: (i) J.L is an (a,p, q) - Carleson measure. (ii) A~)
Lq(J.L) (boundedly).
(iii) SUPzEV IIko,p,zlbu.t) < 00. PROOF. (i) =? (ii) is obvious if p 2: 1 and follows from the above result of J.H. Shapiro [19] if p < 1 ::; q: A~) is the Banach envelope of A~, and so the convex
hull of BAP is dense in B A(p), (ii) =? (iii) is trivial, and (iii) =? (i) is immediate from the followi~g estimate in ;hich C is a constant depending only on 0 < r < 1: J.L(B r (W))l/ q 1 )l/q Br{w)
dJ.L(z) (1 - IwI 2){o+2)/p = (1 -lwI 2){o+2)/p . - (1 -
C· (
(1 (1
1 d ( )) l/q (1 -lwI2)2q{o+2)/p J.L Z
r1(1(1 -- wz)2{o+2)/p I
IwI 2){o+2)/p qd
Z) l/q
= C ·lIko,p,wIILq{~).
Now apply Theorem 3.1(a).
Actually, in the last step, no restriction on p and q is needed: Ho,p,q(w) is bounded whenever A~) '---+ Lq(J.L). There is another interesting consequence of Theorem 3.1, Corollary 3.2, and
Proposition 3.6: COROLLARY 3.7. If 1 ::; q < p < 00 and (p/q) - 2 < a < L~(A) implies Ho,p,q(w) E LOO(A).
then Ho,p,q(w) E
PROOF. Define a' > -1 by a' + 2 = q(a + 2)/p. Then, with £T = £T(A),
Ho,p,q E Lpq/{p-q) <=> A~
=? A~/q) = A~, '---+
<=> H O ',l,l E L oo <=> Ho,p,q E L oo
L 1 (J.L)
EXAMPLE 3.8. (a) Let a, p, q be as in Corollary 3.7, let (an) be a sequence in foo \ fP/(p-q) , and let (1]n) be a sequence in llJ such that (!(1]n,1]k) :::: r· 8nk for all n, k, and such that llJ =
Un Br(1]n). Consider the measure JL = En bn 811n on llJ where bn = lanl' (1-I1]nI 2 )q("'+2)/p and note that (b n ) E fl. A calculation reveals that H""p,q is in LOO(A) but not in uq/(p-q) (A). We
conclude that the converse in Corollary 3.7 doesn't hold. (b) In Proposition 3.6, (ii) {::} (iii) is true for arbitrary 0 < p, q < 00, and (i) ~ (ii) holds trivially whenever q :::: 1. But (iii) ~
(i) fails for 1 ~ q < p and a + 2 > p. In fact, if JL is as in (a), then A~ 'f+ Lq(JL) since H""p,q tfLpq/(p-q)(A), but H-y,l,q = H""p,q E LOO(A) if we put "I = (a + 2)/p - 2. From A~) = A~l) = A~ we
conclude that A~ '----> Lq(JL). 4. Compactness We shall frequently make use of the following classical result: THEOREM 4.1 (Pitt's Theorem). If 0 < p < q < 00 then every operator f q --+ fP is
compact. This was obtained in 1936 by H.R. Pitt [16] for p :::: 1. For an extension see H.P. Rosenthal [17]. The result as stated was proved recently by E. Oja [14]. By atomic decomposition, A~ and
fP are isomorphic (see [10], [1]). Combining this with Pitt's theorem we see that in particular the embedding in Corollary 3.5.(b) is compact. It will follow from the next theorem that the embedding
in 3.5.(a) is compact iff (a + 2)/p < (f3 + 2)/q. Our characterization of compactness of Carleson measures splits into two parts. We consider the case p ~ q first. Here the characterization is as
expected: THEOREM 4.2. Let -1 < a < 00 and 0 < p ~ q < statements are equivalent: (i) JL is a compact (a,p, q) - Carles on measure. (ii) A~)
with 1 ~ q. The following
Lq(JL) compactly.
(iii) z-+ limlllk""p,zIILq(ll) (iv)
= O.
lim Ho.,p,q(w) = O.
PROOF. (i) ~ (ii): If p :::: 1 then nothing is to prove since A~) '----> A~. If p < 1 ~ q then, by [19], A~) is the Banach envelope of A~, and the convex hull of B A~ is dense in B A!:)' Hence
relative compactness of B A~ in the Banach space Lq(JL) entails relative compactness of B A<.!) in Lq(JL). - Here we have used Bx to denote the unit ball of a (quasi-) Banach space X. (ii) ~ (iii):
Suppose that (iii) doesn't hold. Then there exist an c > 0 and a sequence (zn) in llJ such that limn--+ oo IZnl = 1 and Ilko.,p,zn IIL-(Il) > c for all n. By (ii), we may assume that (k""p,zn)n
converges to some f E Lq(JL). But lim n -+ oo IZn I = 1 implies f = 0 since clearly (k""p,zn) tends to zero pointwise: contradiction. (iii) ~ (iv): The estimate proved in (iii) ~ (i) of Proposition
3.6 provides us with a constant C = C(r) > 0 such that JL(Br(W))l/q ~ C, (1 - IwI 2 )(",+2)/p . IIk""p,w IIL-(Il) for all W E llJ.
(iv) =} (i): We apply Lemma 4.3.6 of K. Zhu [22]: there exists an integer N such that for sufficiently small r there is a sequence (TJn) in llJ having the following properties:
(1) llJ = U~=l Br(TJn), (2) B r / 4 (TJm) n B r / 4 (TJn) = 0 whenever m -# n, (3) Every Z E llJ is contained in at most N of the sets B 2r (TJn)' Note that limn-+oo ITJnl = 1 follows from (2). Let
now (In) be a sequence in BA~' By Montel's theorem, some subsequence (Ink) converges uniformly on compact sets to some I E 1i(llJ). By Fatou's lemma, even I E BA~' Put gk := I - Ink' Vk E N. By our
hypothesis there exists, for any given c > 0, an integer ne such that J.L(Br(TJn)) :::; c' (1 - ITJnI 2)q(a+2)/p if n ~ ne' Therefore, with constants depending only on the indicated parameters,
n~, Lr(T/n) Igk(ZW dJ.L(z)
n~, L,.(T/,.) (lgk(ZW)q/PdJ.L(z)
< C(r)· n~E Lr(T/n) COa(B:r(TJn))
r < C(r)· C(r, 0)' "~ (1 _IJ.L(B 2(17n)) )q(a+2)/p 1
< C(r)·C(r,o)·c·
L ({ n~nE J
IgdwWdaa(w)) q/p dJ.L(z)
Igk(W)1 daa(w)
B 2r (T/n)
Igk (wWda a (w)r/ p
< C(r)· C(r, 0) . c· (
Igk(wWdaa(w) riP
Igk (wWda a (w) ) q/p :::;
< C(1')' C(r, 0) . Nq/p . c·
c· c ;
here C = C(r,o,N,p,q). If we choose now ke EN such that Ln
A slight modification of the argument used to prove (i) =} (ii) shows that compactness of A~ <--+ Lq(J.L) implies compactness of A~, <--+ Lq(J.L) if -1 < 0,0' < 00, 0< p < p' :::; q and (0 + 2)/p =
(0' + 2)/p'. It can be shown that (i) {::} (iii) {::} (iv) is true even for arbitrary 0 < q < 00. In the case p > q, we can prove: THEOREM 4.3. Suppose that -1 < 0 < 00 and 0 < q < p < 00. Regardless
01 the (0, p, q) - Carleson measure J.L, the embedding A~ <--+ Lq (J.L) is always compact. For composition operators C
Ah this is due to W. Smith and L. Yang
[21]; see also [4].
PROOF. Put s = p/(p - q) and recall that J.L(Br(-))/aa(Br (·)) E £B(aa) whenever J.L is (0, p, q) - Carleson.
Let Un) be a bounded sequence in A~. For some subsequence, f = limk-+oo fnk exists in 'H(lU). As before, f E A~, hence we may assume that Un) is a null sequence in 'H(lU) and that IIfnllo,p :::; 1
for all n. There are constants Cr , Cr > 0 such that, for each n,
Inf1lJ Ifnlq dp.
< Cr' = Cr'
l1lJ a o
(;r (W)) lBr(w) f Ifn(zWdao(z) dp.(w)
flao(B~(w)) I Br (w)(z) Ifn(zW dp.(w) dao(z)
(4.1) = Cr'
f Ifn(zW lBr(z)a f (B1 ( )) dp.(w) dao(z) rW - f q p.(Br(z))
< Cr' Cr" l1lJ1fn(z) I ao(Br(z)) dao(z) .
. f
By hypothesIs,
l1lJ ao(Br(Z))8dao(z) <
Therefore, given c > 0, there is an r E E (0,1) such that (4.2)
11lJ\r 1lJ
p.(Br(zW ( ) ( / )8 (B ( ))8 da o z < c 2 .
rZ uniformly on rEV, so that E
Un) tends to zero limn-+oo Ir,v Ifn (z)IPda o (z) = cordingly, we may choose n E EN such that, for n 2: n E ,
Combine (4.1), (4.4) and (4.5) to find a constant for n 2: ne:'
Cr > 0 such that I1lJ Ifnlq dp. :::; Cr·c 0
Essential parts of Theorem 4.3, if not the entire theorem, can be proved by using other methods. We sketch three possibilities: • Suppose that -1 < Q < 00, 1 :::; q :::; 2 < P < 00 and v is any
measure. Every operator u : A~ -+ Lq(v) is compact. In fact, by Kwapien's theorem (see [8], or [2], 12.19), the operator u admits a factorization u : A~ ~ 2 .2:. Lq(v). By atomic decomposition and
Pitt's theorem, w is compact, and so is u.
• Rosenthal's extension [17J of Pitt's theorem admits the same conclusion for q ~ 1 and p > max{2, q}. • Suppose that -1 < a < 00, 1 < p < 00, and pis (a,p, 1) - Carleson. Then I : A~ '--+ £1 (p) is
Since A~ is reflexive it suffices to verify that I is completely continuous. Accordingly, let (fn)n be a weakly null sequence in A~. Then fn(z) ----t 0 for each z E llJ. Being weakly null in L1(p),
(fn) is uniformly integrable. Now limn ..... oo IIfnl11 = 0, by a theorem of Vitali (see W. Rudin [18], p.133). Standard results from interpolation theory on the preservation of compactness by
interpolated operators lead from either of these special results to (at least parts of) Theorem 4.3. 5. Order bounded and absolutely summing operators
Our Banach lattices will be complex Banach lattices; see e.g. P. Meyer-Nieberg
[13J for the construction of such an object from a real Banach lattice. Let X be a Banach space and Y a closed subspace of a Banach lattice L. An operator u : X ----t Y is called order bounded if
there is a non - negative h E L such that lufl :::; h for f in Bx, the unit ball of X. Thus we require u to map Bx into the order interval {g E L : Igl :::; h} of L. Note that L is part of the
definition! Every 1.L E C(X, Y) is order bounded when Y is considered as a subspace of C(K) for some compact Hausdorff space K. Let I be an order interval in the Banach lattice L. Its span, Z, is a
Banach lattice with respect to L's order and (a multiple of) I's gauge functional as its norm. Z is an abstract M - space with unit and so, by a well-known theorem of S. Kakutani, isometrically
isomorphic (as a Banach lattice) to C(K) for some compact Hausdorff space K; see again [13J. It follows that every order bounded operator u : X ----t Y c L factorizes X ~
~ C(K)
L L where K is as before and j is the canonical embedding. In this paper, L will be a space LP(p) which results in close ties with absolutely summing operators. Recall that a Banach space operator u
: X ----t Y is (q,p)summing (p:::; q), written u E IIq.p(X, Y) , if there is a constant C such that, for every choice of n E N and x!, . . . ,X n EX, Z
In other words, u is (q,p)-summing iff every weak eP-sequence, i.e. every sequence (xn) in X which satisfies 2:::=11(x*,x n )iP < 00 for all x* E X*, is taken to a strong eq - sequence, i.e. 2:::=1
Iluxnllq < 00 holds. (p, p) - summing operators are called p - summing; the corresponding notation is IIp(X, Y) = IIp,p(X, Y) . We refer to [2J for details on these concepts and in particular for the
following facts: • If Hand K are Hilbert spaces and q ~ 2, then IIq,2(H, K) is the corresponding Schatten q - class. Moreover, for any 1 :::; p < 00, IIp(H, K) is the class of Hilbert - Schmidt
operators. • If 1 :::; P :::; 2, then every operator from C(K) to LP(lI) is 2 - summing.
• If p > 2, then every operator C(K) r - summing for every r > p.
U(v) is (p, 2) -summing, and
Moreover: • If u : X
Lp(v) is order bounded then u is p - summing. Here v is any measure. In the last statement, the converse fails. But: ----7
• If u* p - summing then u is order bounded.
More precisely, we have the following result due to D.J.H. Garling [5]: • Let 1 ::; p < 00. A Banach space operator u : X ----7 Y has a p - summing adjoint if and only if, for every measure v and
operator v : Y ----7 LP(v), the composition v 0 u : X ----7 LP(v) is order bounded.
In particular: • An operator u : L 2 (VI) Schmidt.
L 2 (V2) is order bounded iff it is Hilbert-
We are going to characterize order boundedness of Carleson embeddings
A~ ~
Lq(J.L). To this end we introduce, for s > 0, the Banach space
:= {f: lU
C: f measurable, sup(1-lzI 2 )Slf(z)1 < oo} . zE1U
and its closed subspace
Xs := Xs n H(lU) . It is easy to see that
A~ ~
X(o.+2)/p and that the index (o.+2)/p is best possible.
THEOREM 5.1. Let -1 < 0. < 00, 0 < p < 00 and 1 ::; q < s := (0. + 2)/p, the following statements are equivalent: (i)
A~ ~
Then, with
U(J.L) order boundedly.
(ii) A~) ~ U(J.L) order boundedly. (iii) (1-lzI 2 )-S E U(J.L). (iv)
XS ~ Lq(J.L)
(v) XS ~ Lq(J.L) order boundedly. (vi) XS ~ Lq(J.L) boundedly.
(vii) XS
Lq(J.L) order boundedly.
PROOF. (i)::::} (ii) is obtained as before, by considering separately the cases p ;:::: 1 and p < 1. In order to prove (ii)::::} (iii) it suffices to look at the functions ka,p,q' (iii)::::} (iv) and
(iv)::::} (vi) as well as (iv) <=?{v) and (vi) ¢:} (vii) are easily verified. Finally, for (vii)::::} (i), just observe that A~ ~ Xs. 0 Various statements related to boundedness of Carleson
embeddings do have 'order bounded counterparts'. The first example is:
COROLLARY 5.2. Suppose that -1 < 0.,0.' < 00,0 < p,p' < are such that q. (0. + 2)/p = q'. (0.' + 2)/p'. Then
A~ ~ U (J.L) order boundedly
and 1::; q,q' <
A~, ~ U' (J.L) order boundedly.
Again, in many cases, reduction to Hilbert spaces is possible.
COROLLARY 5.3. If -1 < a, ci < 00, 0 < p < 00 and 1 :::; q < 00 are such that a' +2 = q. (o+2)/p, then A~ '---+ Lq(f.-L) order boundedly if and only if A~, '---+ L2(f.-L)
exists as a Hilbert - Schmidt operator. A special case occurs when f.-L is a measure conclude:
From (iii) of Theorem 5.1 we
5.4. If -1 < a,(3 < 00, 0 < p < 00 and 1 < q < order boundedly if and only if (a + 2)/p < ((3 + l)/q .
In such case, we can even factorize: A~ '---+ Xs '---+ A~ where s = (a + 2)/p. - There is of course no problem in verifying this corollary directly. If (a + 2)/p < ((3 + l)/q, then A~ '---+ A~ is
order bounded and so q-summing; here -1 < a, (3 < 00 and 1 :::; p, q < 00. In many cases, the converse (which doesn't hold for general Banach space operators) is true for Carleson embeddings. To see
why, we need to look at special sets. With each z E U, we associate the 'interval' in au:
I(z) :=
{I:I ei9 :
-Izl) :::; () :::; 11"(1 - IZI)} ,
and the 'squares' in U:
R(z) := {w E U:
Izl < Iwl < 1, (w/lwl)
E I(z)}
Q(z) := {w E R(z) :
Iwl < (1 + Izl)/2} .
It can be shown that the sets Q(Tln) form a partition of U whenever (TIn) is a [sufficiently fine] sequence of points in U having the properties (1) - (3) listed in part (iii) =} (iv) of the proof of
Theorem 4.2. PROPOSITION 5.5. Let f.-L be a (a, p, q) - Carles on measure with -1 < a < 00 and 1 < p, q < 00. If p* :::; q < 00, then A~ '---+ L q(f.-L) is order bounded iff it is q - summing iff it
is (q, p*) - summing.
This is due to T. Domenig [3] for composition operators acting between weighted Bergman spaces. For the sake of completeness, we sketch a proof of the proposition which follows closely Domenig's
arguments. PROOF. If I : A~ '---+ Lq(f.-L) is order bounded, then it is q-summing and so (q,p*)-summing. Suppose now that I is (q, p*) - summing. By standard - but lengthy - calculations it can be
shown that the functions
form a weak ff -sequence in A~. Hence we get from our hypothesis that, with a suitable constant C, 00
L llvn(zW d/-L(z) 2 L 1 n
Ivn(zW d/-L(z)
L [ n
> C·
1Q (T/n)
-lzI 2 )-(<>+2)/p E Lq(/-L),
2 )l/p· )q(a+2)
11 -1Jn z
L1 n
Thus (1
((1 -11JnI
(1 -lzI 2)-q(<>+2)/Pd/-L(z) .
and so ] is order bounded by Theorem 5.1.
More is available. Consider the Rademacher functions
Tn : [0, 1]
1R : t
sign sin (2nrrt) , n E N
(or any sequence of independent symmetric Bernoulli variables). Given 0 < p < 00, Khinchin's inequality assures the existence of positive constants Ap and Bp such that, for any finite collection of
scalars al,"" an: Ap'
L n
lakl 2
1/2 ::; ([1 10 IL akTk(t) I dt n
) l/p
::; Bp'
L n
lakl 2
1/2 .
Pursuing the fate of this inequality within the framework of Banach spaces leads to the theory of type and cotype of Banach spaces, and to the following related class of operators (compare [2], Chs.
11- 12). A Banach space operator u : X -> Y is almost summing,
u E IIas(X, Y) , if there is a constant C such that, for any choice of finitely many vectors from X,
(10[III {; rk(t)UXk 112 dt )1/2 ::; C x.~~x. ( n
Xl, ... ,
I(x*, xk)1 2
It is known that each of the operator ideals IIp is properly contained in IIas. Moreover, if 1 ::; p < 00 and r = max{p,2} then IIash X) c II r ,2(', X) whenever X is an £P space, or the Schatten
p-class Sp(H) for some Hilbert space H. In addition, it was shown by S. Kwapien [9] that • if H is a Hilbert space and u is in IIas(H, Y) then the adjoint 11,* : Y* -> H
is 1 - summing. See [2], p.255 for details. We have the following application to Carleson embeddings. The argument is the same as for composition operators between weighted Bergman spaces [3].
PROPOSITION 5.6. Let /-L be an (a,p, q) - Carles on measure where 1 ::; q < 00., and 2 ::; p < 00. The embedding] : A~ '--+ Lq(/-L) is almost summing if and only if it is order bounded. PROOF. Define
"I > -1 by ("I + 2)/2 = (a + 2)/p. Since p 2 2, A~ :::::} ]: A~ '--+ Lq(/-L) is almost summing :::::} ]* is I-summing (Kwapien) :::::} ] is order bounded (Garling)
Combining the preceding two propositions yields: COROLLARY 5.7. Let -1 < ct < 00 and 1 ::; p, q < 00 be such that p 2: min {q* ,2} and let 11, be an (ct, p, q) - Carleson measure. The embedding I :
A~ "---> Lq(/.l) is q - summing iff it is order bounded. PROOF. Only sufficiency requires proof. If p 2: q*, then Proposition 5.5 settles the case. And if p 2: 2, then I, being q - summing, is almost
summing, and so order bounded by Proposition 5.6. 0
References [IJ R.R. Coifman, R. Rochberg, G. Weiss, Facto'rization theorems for Hardy spaces in seveml variables. Ann. of Math. (2) 103. (1976),611-635. [2J J. Diestel, H. Jarchow, A. Tonge:
Absolutely Summing Opemtors. Cambridge University Press 1995. [3J T. Domenig, CompoS'ition opemtors on weighted Beryman sp(,ces and Hardy spaces. Dissertation University of Zurich 1997. [4J T.
Domenig, H. Jarchow, R. Riedl, The domain space of an analytic composition opemtor. Journ. Austral. Math. Soc. 66 (1999), 56-65. [5J D.J.H. Garling, Lattice bounding, Radonifying and summing
mappings. Math. Proc. Camb. Phil. Soc. 77 (1975), 327-333. [6J W.W. Hastings, A Carleson measure theQrem for Beryman spaces. Proc. Amer. Math. Soc. 52 (1975), 237-241. [7J C. Horowitz Zeros of
functions in the Bergman spaces. Duke Math. Journ. 41 (1974), 693-710. [8J S. Kwapien, On a theorem of L. Schwartz and its applications to absolutely summing opemtors. Studia Math. 38 (1970),
193-201. [9J S. Kwapien, A remark on p - summing opemtors in fr - spaces. Studia Math. 34 (1970), 277278. [lOJ J. Lindenstrauss, A. Pelczynski, Contributions to the theory of classical Banach spaces.
Journ. Funct. Anal. 8 (1971), 225-249. [l1J D.H. Luecking, Multipliers of Bergman spaces into Lebesgue spaces. Proc. Edinb. Math. Soc. 29 (1986), 125-131. [12J D.H. Luecking, Embedding theorems for
spaces of analytic functions via Khinchine's inequality. Mich. Math. Journ. 40 (1993), 333-358. [13J P. Meyer-Nieberg, Banach Lattices. Springer-Verlag 1991. [14J E. Oja, Pitt Theorem for non-locally
convex spaces f p • Preprint. [15J V.L. Oleinik, B.S. Pavlov, Embedding theorems for weighted classes of harmonic functions. Zap. Nauchn. Sem. Leningrad Otdel. Mat. Inst. Steklov 22 (1971),94-102.
Transl. in Journ. Soviet Math. 2 (1974), 135-142. [16J H.R.Pitt, A note on bilinear forms. Journ. London Math. Soc. 11, 171-174 (1936). [17J H.P. Rosenthal, On quasi-complemented subspaces of Banach
spaces with an appendix on compactness of opemtors from LP(p,) to Lr(/I). Journ. Funct. Anal. 4 (1969), 176-214. [18J W. Rudin, Real and Complex Analysis. 3 rd ed., McGraw-Hill 1987. [19J J.H.
Shapiro, Mackey topologies, reproducing kernels, and diagonal maps on the Hardy and Bergman spaces. Duke Math. Journ. 43 (1976), 187-202. [20J W. Smith, Composition opemtors between Bergman and Hardy
spaces. Trans. Amer. Math. Soc. 348 (1996) 2331-2348. [21J W. Smith, L. Yang, Composition opemtors that impro1Je integmbility on weighted Beryman spaces. Proc. Amer. Math. Soc. 126 (1998) 411-420.
[22J K. Zhu, Opemtor Theory in Function Space.~. Marcel Dekker, New York 1990.
190, CH 8057
E-mail address:jarchowlDmath.unizh.ch INSTITUT FUR MATHEMATIK, UNIVERSITAT ZURICH, WINTERTHURERSTRASSE ZURICH, SWITZERLAND
E-mail address:kollbrunlDmath.unizh.ch
190, CH 8057
Contemporary Mathematics Volume 328, 2003
Weak* -extreme points of injective tensor product spaces Krzysztof Jarosz and T. S. S. R. K. Rao ABSTRACT. We investigate weak* -extreme points of the injective tensor product spaces of the form A ®.
E, where A is a closed subspace of C (X) and E is a Banach space. We show that if x E X is a weak peak point of A then f (x) is a weak*-extreme point for any weak*-extreme point f in the unit ball of
A ®. E C C (X, E). Consequently, when A is a function algebra, f (x) is a weak*-extreme point for all x in the Choquet boundary of A; the conclusion does not hold on the Silov boundary.
1. Introduction
For a Banach space E we denote by E1 the closed unit ball in E and by BeE1 the set of extreme points of E 1 . In 1961 Phelps [16] observed that for the space C(X) of all continuous functions on a
compact Hausdorff space X every point f in Be (C (X))1 remains extreme when C (X) is canonically embedded into its second dual C (X)**. The question whether the same is true for any Banach space was
answered in the negative by Y. Katznelson who showed that the disc algebra fails that property. A point x E OeE1 is called weak* -extreme if it remains extreme in BeEi*; we denote by B;E1 the set of
all such points in E 1. The importance of this class for geometry of Banach spaces was enunciated by Rosenthal when he proved that E has the Radon-Nikodym property if and only if under any renorming
the unit ball of E has a weak* -extreme point [19]. While not all extreme points are weak* -extreme the later category is among the largest considered in the literature. For example we have: strongly
exposed S;; denting S;; strongly extreme S;; weak* -extreme. We recall that x E E1 is not a strongly extreme point if there is a sequence Xn in E such that Ilx ± xnll ---t 1 while IIxnll ~ 0 (see [3]
for all the definitions). We denote by O;E1 the set of strongly extreme points of E 1. It was proved in [14] that e E O;E1 if and only if e E o;Ei* (see [9], [13], or [17] for related results).
Examples of weak* -extreme points that are no longer weak* -extreme in the unit ball of the bidual were given only recently in [6]. In this paper we study the weak* -extreme points of the unit ball
of the injective tensor product space A®,E, where A is a closed subspace of C(X). Since C(X)®,E Both authors were supported in part by a grant #0096616 from DST/INT/US(NSFRP041)/2000.
© 231
2003 American Mathematical Society
KRZYSZTOF JAROSZ AND T. S. S. R. K. RAO
can be identified with the space C(X, E) of E-valued continuous functions on X, equipped with the supremum norm, elements of A ®e E can be seen as functions on X. We are interested in the relations
between 1 E a; (A ®£ E) and 1 (x) E a;E1 , for all (some?) x E X. Since any Banach space can be embedded as a subspace A of a C (X) space no complete characterization should be expected in such very
general setting. For example if A is a finite dimensional Hilbert space naturally embedded in C (X), with X = Ai, and dim E = 1, then any norm one element 1 of A ®£ E is obviously weak* -extreme
however the set of points x where 1 (x) is extreme is very small consisting of scalar multiples of a single vector in Ai. Hence in this note we will be primarily interested in the case when A is a
sufficiently regular subspace of C (X) and/or x is a sufficiently regular point of X. It wa..<; proved in [5] that
1 E a:C(X, Eh
[I (x)
E a:EI , for all x EX].
It follows from the arguments given during the proof of Proposition 2 in [6] that for a function 1 E (A ®€ E)l we have
[I (x)
E a;El for all x E X with c5 x E aeAi]
=> 1 E a; (A ®e E)I'
where we denote by c5x the functional on A of evaluation at the point x. In this paper we obtain a partial converse of the above result (Theorem 1). Our proof also shows that if 1 E a; (A ®€ E)l then
1 (x) E a;El for any weak peak point x (see Def. 1), extending one of the implications of (1.1). It follows that when A is a function algebra then any weak* -extreme point of Al is of absolute value
one on the Choquet boundary ChA (and hence on its closure, the Shilov boundary) and consequently is a strongly extreme point [17]. Since we have concrete descriptions of the set of extreme points of
several standard function algebras (see e.g. [12], page 139 for the Disc algebra) one can give easy examples of extreme points that are not weak* -extreme. Recently several authors have studied the
extremal structure of the unit ball of function algebras ([1], [15], [18]). It follows from their results that the unit ball has no strongly exposed or denting points. Our description that strongly
extreme and weak* -extreme points coincide for function algebras and are precisely the functions that are of absolute value one on the Shilov boundary completes that circle of ideas. We also give an
example to show that the weak* -extreme points of (A ®, E)l in general need not map the Shilov boundary into aeE1 • Considering the more general case of the space of compact operators K (E, F) (we
recall that under assumptions of approximation property on E or F*, K (E, F) can be identified with E* ®£ F) we exhibit weak*-extreme points T E K (fP)1 for 1 < p =I=- 2 < 00 for which T* does not
map unit vectors to unit vectors. Our notation and terminology is standard and can be found in [3], [4], or [11]. We always consider a Banach space as canonically embedded in its bidual. By E(n) we
denote the n-th dual of E. By a function algebra we mean a closed subalgebra of a C (X) space separating the points of X and containing the constant functions; we denote the Choquet boundary of A by
2. The result
As noticed earlier, for A c C (X) and a point 1 E (A ®£ Eh we may not have 1 (x) E a;EI for all x E X even in a finite dimensional case. Hence we need to define a sufficiently regular subset of X in
relation to A.
DEFINITION 1. A point Xo E X is called a weak peak point of A C C (X) if for each neighborhood U of Xo and £ > 0 there is a E A with 1 = a (xo) = Iiall and la (xo)1 < £ for x E X\U; we denote by opA
the set of all such points in X.
There are a number of alternative ways to describe the set opA. If Xo E X is a weak peak point of A C C (X), /-l is a regular Borel measure on X annihilating A, and al' is a net in A convergent
almost uniformly to 0 on X\ {xo} and such that a.., (xo) = 1 then /-l({xo}) = liml'Jxal' = O. Hence if a* E A* and VI,V2 are measures on X representing a* we have VI ({Xo}) = V2 ({xo}), consequently
~ v ( {xo}) is a well defined functional on A * .
On the other hand if X {xo} E A ** then /-l ( {xo}) = 0, for any annihilating measure /-l, and Xo is a weak peak point. To justify the last claim notice that Al is weak*dense in Ai* so X{xo} is in
the weak*-closure of the set K = {f E Al : f (xo) = I}. Let U be an open neighborhood of Xo and Ax\U be the space of all restrictions of the functions from A to X\U. We define the norm on Ax\U as sup
on X\U. Lct K X\U be the set of restrictions ofthe functions from K and cl (Kx\U) be the norm closure of Kx\U C Ax\U. If 0 ~ clKx\U then there is G E (Ax\U)* , represented by a mcasure 1] on X\U and
separating Kx\U from 0: Re G (h) > a > 0
for all hE clKx\U·
The measure 1] extends G to a functional on A so K is functionally separated from 0 in A contrary to our previous observation. Hence 0 E clKx\u so there is a function in K that is smaller then £
outside U which means that Xo is a weak peak point. The concept of weak peak points is well known in the context of function algebras where opA coincides with the Choquet boundary ([8]' p. 58). For
more general spaces of the form Ao 'f1 {foa E C (X) : a E A} I where A C C (X) is a function algebra and fo a nonvanishing continuous function on X we have ChA ~ opAo. Spaces of these type appear
naturally in the study of singly generated modules and Morita equivalence bimodules in the operator theory [2J. THEOREM 1. Let E be a Banach space, X a compact Hausdorff space, and A a closed
subspace of C (X). If f E A ®, E is a weak* (strongly) extreme point of the unit ball then f (x) is a weak* (strongly) extreme point of the unit ball of E for any x E opA. In particular if f E (C
(X,E))I is a weak*(strongly) extreme point then f(x) is a weak* (strongly) extreme point of EI for all x EX.
We first need to show that for a weak peak point Xo E X there exists a function in A not only peaking at Xo but that is also almost real and almost positive. LEMMA
1. Assume X is
compact Hausdorff space, A is a closed subspace of
C (X), and Xo is a weak peak point of A. Then for each neighborhood U of Xo and £
> 0 there is g
E A such that
Ilgll = 1 = g (xo) , (2.1)
Ig (x)1
IIRe+ g where Re+ z
= max{O,Rez}.
< £, for all x
gil < £,
E X\U, and
KRZYSZTOF JAROSZ AND T. S. S.
R. K.
PROOF. Put U1 = U and let gl E A be such that IIg111 = 1 = gl (xo) and Ig1 (x)1 < e for x
Put U2 = {x E U1 : Ig1 (x) - 11 < e} and let g2 E A be such that IIg211
= 1 = g2 (xo)
and Ig2 (x)1 < e for x
Put U3 = {x E U2 : Ig2 (x) - 11 < e}. Proceeding this way we choose a sequence {gn}n>l _ in A. Fix a natural number k such that k> 1e: and put 1 k
k Lgj· j=l
We clearly have Ilgll = 1 = 9 (xo) and Ig (x)1 < e for x i U. Let x E U, then either x belongs to all of the sets Uj , j k, in which case Ig (x) - 11 < e, or there is a natural number p < k such that
x E Up \ Up+!. In the later case we have
tgj-(P-l) ;=1
.!. (
(lg1 (x) - 11
< -k- e + k Hence IIRe+ 9 -
+ ... + Igp-1 (x) -
+ Igp (x)1 + (lgp+dx)1 + ... + Igk (x)!)
- k
+ -k- e < e. o
< e.
We are now ready to finish the proof of the Theorem. PROOF. Suppose! (xo) is not a weak*-extreme point. Then by [9] there is a 1 + ~ and e* (en) ~ O. sequence en in E1 and e* E Ei such that II! (xo)
± enll By the Lemma there is a sequence gn in A such that
Ilgnll = 1 = gn (xo) , (2.2)
ign (x)1
<.!., n
IIRe+ gn - gnll
< .!., n
if II! (x) - ! (xo) II
~ .!., n
Hence II! (x) ± 9 (x) e II < max { sUPllf(x)-f(xo)II~,* {II! (x)11 + Ign (x)llIenll}, } n n sUPllf(x)- f(xo)lI< {II! (x) ± gn (x) en II}
:s max {I +.!.,.!. + II! (xo) ± Re+ gn (x) en II + .!.} n n n 3
< - 1+-. n Therefore II! ± gnenll ---> 1 but (8 (xo) ® e*)(gnen) = gn (xo) e* (en) ~ O. This contradiction shows that! (xo) is a weak*-extreme point. The same line of arguments shows that! (x) is
strongly extreme for any strongly extreme! E (A ®, E)l' 0 Since for a function algebra A the Choquet boundary C hA coincides with 8p A ([8], p. 58) and the Shilov boundary 8A is equal to the closure
of ChA we have:
1. Let A be a function algebra, E a Banach space and f a weak* -extreme point. Then
(A ®e
f(x) E 8;E1 , for x E ChA, and Ilf (x)11 = 1, for x E 8A. REMARK 1. Theorem 1 is not valid for the spaces WC(X, E) of E-valued continuous functions with E quipped with the weak topology. Even a
strongly extreme point of WC(X, Eh need not assume extremal values at all points of X [13].
We next give an example of a function algebra A and a 3-dimensional space showing that a weak* -extreme point f E (A ®e Eh need not take extremal values on the entire Shilov boundary. Since E is
finite dimensional this function f maps the Choquet boundary into the set of strongly extreme points but f is not a strongly extreme point. E
1. Put
Q = {(z,w,O) E (:3: Izl2 + Iwl 2 ::; I} U {(O,w,u) E (:3: max{lwl, lui}::; I}, and B = convQ. Let 11·11 be the norm on (:3 such that B is its unit ball. Note that (z,w,O) is an extreme point ofB
ifflwl =I 1 and Iz12+lw12 = 1. PutE = ((:3,11·11), X'!!:. {O} x {I} x lD>u {(sint,cost,O): 0::; t::; 11"},
fo : X
El, fo (x) = x, and A = {h E C (X) : h (0, 1,·) E A (lD>)},
where A (lD» is the disc algebra. We have ChA = {O} x {I} x 8lD> U {(sin t, cos t, 0) : 0 < t < 11"} . The function fo is in A ®e E and takes extremal values on the Choquet boundary of A so it is a
weak* -extreme points of (A ®e E)l. However fo (0, ±1, 0) = (0, ±1, 0) are not extreme points of Ei while (0, ±1, 0) are in the Shilov boundary of A. Since E is finite dimensional clearly the
function fo maps the Choquet boundary of A into the set of strongly extreme points of E 1 • We next show that f is not a strongly extreme point. Let gn E A be such that Ilgnll
gn (sint, cost, 0)
= 1 = gn
~,cos ~,o) ,
= 0, for ~ < t::; 1, and n
gn (0, 1, z) = 0, for z E lD>. Put fn = (O,O,gn) E A ®e E. We have (fo ± fn)(a, b, c) Hence IIf ± fnll
(0,1, c) for = { (a, b, ±gn) for
(a,b,c) E {O} x {I} x 8lD> (a, b, c) E {(sin t, cos t, 0) : 0 ::;
t ::; 11"} .
1 but Ilfnll ~ 0 so f is not a strongly extreme point.
In the next Proposition we consider a more general setting of compact operators. For a Banach space E we denote by C(E) the space of all linear bounded maps on E, by K(E) the set of all compact
linear maps, and by S(E) the set of
KRZYSZTOF JAROSZ AND T. S. S. R. K. RAO
unit vectors in E. Since K (E, C (X)) can be identified with C (X, E*) our result on weak* -extreme points taking weak* -extremal values can be interpreted as follows T E a;K(E,C(X))l
T* (a;C(X);) c a;Er.
Thus more generally one can ask whether T* (a; Ft) c a; Ei for any TEa; K(E, Fh. We give a class of counter examples with the help of the following proposition. PROPOSITION 1. Let E be an infinite
dimensional Banach space such that K(E) is an M-ideal in C(E). 1fT E K(Eh then T*(aeEi) ct. S(E*).
We recall that a closed subspace M of a Banach space E is an AI-ideal if there is a projection P E C (E*) such that ker P = Ml. and liP (e*)II+lle* - P (e*)11 = Ile*ll, for all e* E E* (see [11] for
an excellent introduction to 1\/-ideals). PROOF. Since qE) is an AI-ideal it follows from Corollary V1.4.5 in [11] that E* has the Radon-Nikodym property and hence the IP (see [10]). Also since qE)
is a proper M-ideal it fails the IP. It therefore follows from Theorem 4.1 in [10] that there exists a net {x~} c e Ei such that x~ ---> Xu in the weak* -topology with Ilxoll < 1. Suppose T*(aeEi) c
S(E*). Since T* is a compact operator by going through a subnet if necessary we may assume that T*(x~) ---> T*(xo) in the norm. Thus 1 = IIT*(xo)11 < 1 and the contradiction gives the desired
conclusion. 0
EXAMPLE 2. Banach spaces E for which K(E) is an AI ideal in C(E) have been well extensively studied. Chapter VI of [11] provides seveml examples including E = p , 1 < p < 00, as well as properties of
these spaces. It was observed in [6] that for p # 2 there are weak*-extreme points in the space K(ePh. It follows from the last proposition that the adjoint of these weak" -extreme points do not even
map extreme points to unit vectors.
A strongly extreme point remains extreme in all the dual spaces of arbitrary even order. A weak* -extreme point remains extreme in the second dual but may not be extreme in the fourth dual. Hence the
property of remaining extreme in all the duals of even order is placed between the strong and the weak* type of extreme points. It would be interesting to describe that property in terms of the
original Banach space alone. A procedure for generating extreme points which have this property but are not strongly extreme was described in [6]. PROPOSITION 2. Let X be a compact Hausdorff space, A
a closed subspace of C(X), and E a Banach space. Suppose Xo E X is a weak peak point and f E A®. E is an extreme point in the unit ball of all the duals of even order. Then f (xo) is an extreme point
of the unit ball of all the duals of E of even order. PROOF. Since the space A ®. E** can be canonically embedded in (A ®. E)** [7] we have, for any natural number n
A ®. E(2n) C (A ®( E(2n-2))** C (A ®. E)(2n).
If f E A ®. E is an extreme point of (A ®. E)(2n+2) then it is a weak*-extreme point of (A ®. E)(2n), as it also belongs to A ®. E(2n) it is a weak*-extreme point of A ®. E(2n). Hence by our theorem
f (xo) is an extreme point of E~2n). 0 The next proposition characterizes strongly extreme points in terms of ultrapowers.
WEAK' -EXTR.EME POINTS
of the unit ball
3. An element e of a Banach space E is a strongly extreme point if and only if (e).:F is an extreme point of (E.:Fh-
PROOF. If e ¢. O;El then there is a sequence {en}n~l eEl with lie ± enll --+ 1 and infnEN Ilenll > o. Thus II (e).:F ± (en).:F11 = 1 and II (en).:F11 =I- 0 so (e).:F is not an extreme point. If
(e).:F ¢. oe(E.:Fh then there is 0 =I- (en).:F E (E.:Fh with 1 = II (e).:F ± (en).:F11 = lim.:F lie ± enll· Thus for every € > 0 the set {n E N: lie ± enll ~ 1 + €} is none empty as an element of F.
Hence there exists a sequence {k n } such that lie ± ek n II --+ 1 but Ilek" II ---A- 0 so e is not a strongly extreme point. 0
References [1] P. Beneker and J. Wiegerinck, Strongly exposed points in 'Uniform algebras, Proc. Amer. Math. Soc. 127 (1999) 1567-1570. [2] D. Blecher and K. Jarosz, Isomorphisms of function modules,
and generalized approximation in modulus, Trans. Amer. Math. Soc. 354 (2002), 3663-3701 [3] R. D. Bourgin, Geometric aspects of convex sets with the Radon-Nikodym property, LNM 993, Springer, Berlin
1983. [4] A. Browder, [ntroduction to Function Algebras, W. A. Benjamin, New York 1969. [5] P. N. Dowling, Z. Hu and M. A. Smith, Extremal structure of the unit ball of C(K, X), Contemp. Math., 144
(1993) 81-85. [6] S. Dutta and T. S. S. R. K. Rao, On weak*-extreme points in Banach spaces, preprint 2001. [7] G. Emmanuele, Remarks on weak compactness of operators defined on injective tensor
products, Proc. Amer. Math. Soc., 116 (1992) 473-476. [8] T. Gamelin, Un'iform Algebras, Chelsea Pub. Comp., 1984. [9] B. V. Godun, Bor-Luh Lin and S. L. Troyanski, On the strongly extreme points of
convex bodies in separable Banach spaces, Proc. Amer. Math. Soc., 114 (1992) 673-675. [10] P. Harmand and T. S. S. R. K. Rao, An intersection property of balls and relations with M-ideals, Math. Z.
197 (1988) 277-290. [11] P. Harmand, D. Werner and W. Werner, M-ideals in Banach spaces and Banach algebras, Springer LNM No 1547, Berlin 1993. [12] K. Hoffman, Banach spaces of analytic functions,
Dover 1988. [13] Z. Hu and M. A. Smith, On the extremal structure of the unit balls of Banach spaces of weakly continuous functions and their duals, Trans. Amer. Math. Soc. 349 (1997) 1901-1918. [14]
K. Kunen and H. P. Rosenthal, Martingale proofs of some geometric results in Banach space theory, Pacific J. Math. 100 (1982) 153-175. [15] O. Nygaard and D. Werner, Slices in the unit ball of a
uniform algebra, Arch. Math. (Basel) 76 (2001) 441-444. [16] R. R. Phelps, Extreme points of polar convex sets, Proc. Amer. Math. Soc. 12 (1961) 291-296. [17] T. S. S. R. K. Rao, Denting and strongly
extreme points in the unit ball of spaces of operators, Proc. Indian Acad. Sci. (Math. Sci.) 109 (1999) 75-85. [18] T. S. S. R. K. Rao, Points of weak-norm continuity in the unit ball of Banach
spaces, J. Math. Anal. Appl., 265 (2002) 128-134. [19] H. Rosenthal, On the non-norm attaining functionals and the equivalence of the weak' -KMP with the RNP, Longhorn Notes, 1985-86. DEPARTMENT OF
IL 62026-1653, USA E-mail address: kjaroszlDsiue. edu URL: http://www.siue.edu/-kjarosz/
R. V. COLLEGE POST, BANGALORE 560059. INDIA E-mail address:tsslDisibang.ac . in
Contemporary Mathematics Volume 328. 2003
Determining Sets and Fixed Points for Holomorphic Endomorphisms Kang-Tae Kim and Steven G. Krantz The authors study the fixed point sets of a holomorphic endomorphism of a domain in complex space.
Sufficient (and necessary) conditions are given-on the number and configuration of the fixed points-for the endomorphism to be forced to be the identity. The proofs depend on certain key ideas from
differential geometry, particularly the notions of cut locus and Hadamard ABSTRACT.
length space.
1. Introduction
This article concerns the study of the concept of determining set for a collection of holomorphic mappings. We first give the definition. DEFINITION 1.1. Let M be a complex manifold, and let Aut (M)
be the collection of biholomorphic mappings of M into itself. We call a subset Z c M a determining set for Aut (M) (or, equivalently, an Aut (M)-determining set), if any map f E Aut (M) satisfying f
(p) = p for every p E Z is in fact the identity map of
M. We observe first that this article is related to the authors' collaboration with Burna Fridman and Daowei Ma (see [FKKM]), which was originally inspired by the following remarkable theorem in
complex dimension one. THEOREM 1.2. Let n be a domain in the complex plane C and let f : n --+ n be a biholomorphic (conformal) mapping. If there are three distinct points Pl,P2,P3 in n such that f
(pj) = Pj, for j = 1,2,3, then f is the identity map. The higher-dimensional analog of this theorem given in [FKKM] is as follows: THEOREM 1.3. (Fridman-Kim-Krantz-Ma [FKKM]) Let M be a connected,
complex manifold of dimension n admitting a complete invariant Hermitian metric. 2000 Mathematics Subject Classification. 32H02, 32H50, 32H99. Key words and phrases. fixed point set, holomorphic
mapping, cut locus, Hadamard length
space. K.- T. Kim supported in part by grant ROl-1999-00005 from The Korean Science and Engineering Foundation. Steven G. Krantz supported in part by grant DMS-9988854 from the National Science
© 239
2003 American Mathematical Society
KANG-TAE KIl\1 AND STEVEN G. KRANTZ
Then a determining set consisting of n + 1 points exists for the automorphisms of .M. Furthermore, the choice of such a determining set is generic. Throughout this paper we shall discuss both
endomorphisms and automorphisms. If M is a complex manifold then an endomorphism of .I'IJ is any holomorphic mapping
2. The Case of Convex Domains Let us consider a bounded strongly convex domain n in en with a smooth (Ck, k 2 6) boundary. By the well-known work of Lempert ([LEM]), for each pair of distinct points
p, q E n, there exists a unique holomorphic map 'P : ~ ---- n such that (1) 'P(O) = p and 'P(~) = q for some ~ E ~, and (2) 'P*dn = dLl, where d denotes the Kobayashi distance. We call such a map 'P
a complex geodesic joining p and q. We now consider the holomorphic endomorphisms of n fixing two given points. LEMMA 2.1. Let n be a bounded, strongly convex domain in en with Ck smooth boundary for
some k 2 6. Let p, q E n be two distinct points and let 'P denote a complex geodesic joining p and q. If a holomorphic mapping f : n ---- n satisfies the condition that f(p) = p and f(q) = q, then it
holds that f('P(()) = 'P(() for every (E ~. PROOF. Let f and p, q be as in the hypothesis. Let 'P : ~ ---- n be a complex geodesic joining p and q, with 'P(O) = P and 'P(~) = q. Then let "( : [0, f]
---- ~ be the unit speed geodesic in ~ with "((0) = 0 and "((e) =~, where £ = dLl(O,~). Let o :S t :S f and let r = "((t). Then we see that
dn(p, q)
< < <
dn(f(p), f(q)) dn (f(p), f('P(r))) + dn(f('P(r)), f(q)) dn('P("((O)), 'P("((t))) + dLl('P("((t)), cp("((£))) dLl(,,((O), "((t)) + dLl("((t), "((f)) dLl(O,~)
dn(p, q). Because of the distance-decreasing property of the Kobayashi metric and the fixed point conditions, we see from the above that
where dn(p, q) = £. Notice that every Kobayashi distance ball is strictly convex, as our domain n is a strongly convex domain with smooth boundary (see [LEM]). Hence the above observations together
with the fact that
dn(p,cp(r)) = t, dn('P(r)) = f - t imply that f('P(r)) = 'P(r). Consequently, the map f fixes every point in the set 'P 0 "(( [0, f]). Hence the two maps f 0 'P and 'P of ~ into n coincide along a
curve in the unit disc ~. Therefore f 0 'P(() = 'P(() for every ( E ~, as claimed. 0
In other words, we have shown that any holomorphic endomorphism of a bounded strongly convex domain in en fixing two distinct points must fix every point that belongs to the complex geodesic passing
through the two fixed points. We immediately ohtain the following general result on the determining sets for holomorphic endomorphisms of a bounded convex domain.
KANG-TAE KIM AND STEVEN G. KRANTZ
LEMMA 2.2. Let 0 be a bounded, strongly convex domain in en with C k (k ~ 6) smooth boundary. Let PO,Pl,'" ,Pn be points in 0 chosen in such a way that the complex geodesics passing through Po and Pj
(j = 1, ... n) have tangent vectors at Po that are linearly independent over IC. Then any holomorphic mapping f: 0 ---> 0 fixing po, ... ,Pn must fix every point ofO. PROOF. Notice that the current
hypothesis together with the preceding lemma implies that dfpo is the identity map. Therefore a theorem of H. Cartan implies that f is in fact the identity mapping. 0 We remark that the choice for
Po, ... ,Pn is generic. To formulate this notion more precisely, we consider the cartesian product rrnHo of (n + 1) copies of O. In fact it is shown in [FKKM] that there exists an open dense subset U
of rrn+10 such that any element of U gives (n + 1) points that satisfy the sufficiency condition of the preceding lemma. We summarize the result more elegantly in the following statement. THEOREM
2.3. For a bounded, strongly convex domain in en, there exists a collection of n + 1 points such that any holomorphic endomorphism of the domain fixing them must fix every point in the domain.
Moreover, the choice of such n + 1 points is generic. REMARK 2.4. We point out that the result of this section concerns the class of bounded, strongly convex domains, which is a rather special
collection of objects. However, in compensation, we emphasize that we have treated general holomorphic endomorphisms, rather than just biholomorphic self-maps.
3. Hadamard Spaces In light of the preceding section, we would like to present in this section a description of the underlying metric space principles that we use in the study of determining sets.
Let (X, d) be a metric space, equipped with the distance function d : X x X ---> lR. By an isometry we mean a self-mapping f : X ---> X satisfying the condition:
d(J(p), f(q)) = d(p, q), Vp, q E X. We denote by Isom(X) the collection of isometries of (X, d). Imitating the concept of "length spaces" that is commonly encountered in geometry (cf. [BUS]), we give
the following definition. DEFINITION 3.1. Let"(: (a,b)
X be a continuous curve in X. We call it
minimal if d("((x), ,,((y)) = t - x + d("((t), ,,((y)), for every t,x,y with a < x:::; t:::; y < b. DEFINITION 3.2. A metric space (X, d) is called a length space if, for every pair of points p, q
EX, there exists a minimal curve "( : [a, b] ---> X such that "((a) = P and "((b) = q. Furthermore, we call (X, d) a Hadamard space if the minimal curve joining each pair of points is unique up to a
reversal of parametrization. Notice that any convex subset of Euclidean space is a Hadamard space with respect to the standard Euclidean distance. A strongly convex domain in en, equipped with the
Kobayashi distance, is also a Hadamard space. Every complete,
simply connected Riemannian manifold with non-positive curvature is also an example of a Hadamard space. These are often called Hadamard manifolds; from this derives our terminology of Hadamard
(length) space. Now. for the study of determining sets, we present this lemma. LEMMA 3.3. Let (X, d) be a Hadamard space and let p, q E X be two distinct points. If an isometry f : X --> X fixes P
and q, tllen f fixes every point on the minimal curve passing through P and q. PROOF. Let'Y : [O,f] --> X be a minimal curve with 'Y(O) = p,'Y(f) = q. It is immediate to see that the isometry f of
(X, d) has the property that f 0 'Y is also a minimal curve. Since P and q are fixed by f, and since (X, d) is Hadamard, it follows that f 0 'Y(t) = 'Y(t) for every t E [0, fl. Now consider the
minimal curve passing through P and q. So far, we have shown that the portion of this minimal curve between P and q is pointwise fixed by f. We still must show that the other portion of the minimal
curve is fixed pointwise by f. But this is a simple matter using the uniqueness of the minimal curve passing through any two given points and the minimal-curve-preserving property of isometries. This
completes the proof. D We remark that it is possible to derive the same conclusions for locally Hadamard spaces but, in order to keep our exposition concise, we do not introduce the concept here.
LEMMA 3.4. Let (X, d) be a Hadamard space and let U be an open subset of X. Then any isometry f fixing every point in U must be the identity map. PROOF. Let P E U. Then, for every minimal curve 'Y
emanating from p, f fixes a point in 'Y n (U \ p). The preceding lemma implies now that f fixes every point of 'Y. Since every point in X can be joined to P by a minimal curve, this completes the
proof. D Now we consider the concept of convex hull in a Hadamard space. We say that a set Q in a Hadamard space X is convex if every minimal curve joining P and q in X is contained in Q. For a
subset A of a Hadamard space X, its convex hull W (A) is the smallest convex subset of X containing A. DEFINITION 3.5. Let Po, . .. ,Pm be points in a Hadamard space (X, d) with minimal geodesics 'Yl
.... ,'Ym such that 'Yj passes through Po and Pi for every j = 1, .... m. We call the points Po, ... ,Pm spanning if the convex hull Whl U ... U'Ym) has non-empty interior. Now we have the following
general result. PROPOSITION 3.6. If a Hadamard space (X, d) admits m + 1 points Po, PI, ... , Pm ill X that are spanning, then these m + 1 points constitute a determining set
for the isometries of( X, d). PROOF. Notice that the convex hull we obtain from the minimal curves through Po and Pi is fixed pointwise by any isometry that fixes the points Po,··· ,Pm. Then
the preceding lemma finishes the proof.
Observe that the full isometry condition is not really needed to prove the conclusion of the proposition. In fact, any distance-decreasing map will satisfy the
KANG-TAE KIM AND STEVEN G. KRANTZ
same conclusion (just use the unique continuation principle). Notice that this offers an underlying principle for the determining set theorem for the holomorphic endomorphisms of strongly convex
domains in the preceding section.
4. CH-Subsets and Automorphisms In this section we demonstrate how the principles of the preceding section reflect upon the main theorem of [FKKM] and its proof. Let M be a connected, complex
manifold that admits a smooth invariant Hermitian metric. Here the invariance refers to the property that every holomorphic automorphism of M is an isometry with respect to the Hermitian metric. For
a moment, we take the real part of the Hermitian metric, and consider everything in terms of lliemannian geometry. Let P EM. Then we call q E M a cut point of P if there are at least two distinct
geodesics joining P and q with the same minimal length. The collection of cut points for P is called the cut locus of p, which we denote by Cpo In [FKKMJ, a subset X of M was called Carlan-Hadamard
('CH' for short) if there exists Xo E X so that X does not intersect the cut locus C(xo) of Xo in M. Furthermore, we call such a CH-set X generating if the set
Ip(X) := {'Y~(O) I 'Y is the unique normal geodesic from Xo to p, Vp
spans TxoM over C. Suppose now that X is a set of finitely many points, and that a certain holomorphic automorphism f fixes every point of X. Then, by complex differentiability, one picks up more
geodesics than just the geodesics joining Xo and the other points of X. [That is to say, each geodesic tangent may be multiplied by i.] If x E X \ Xo and if 'Yx is the unit speed geodesic from Xo to
x, then 9x == expxo(i"(~(O)) is also fixed, point by point, by f. Now it is not hard to see, using the exponential map and the tangent space, that the convex hull W = Wbl U 91 U ... U 'Ym U 9m) has
non-empty interior. Notice that every point of the hull W is fixed by f pointwise. We obtain the following result as a consequence of Proposition 3.6. PROPOSITION 4.1. (Fridman/Kim/Krantz/Ma [FKKM])
Let M be a connected, complex manifold with an invariant Hermitian metric. Let X be a generating CHsubset of M. Then, whenever an automorphism fixes every point of X, it is in fact the identity map.
In other words, every generating CH-subset is a determining set for automorphisms. The method of choosing a smallest (in the sense of inclusion of sets) generating CH-subset in a complex manifold
with a complete invariant Hermitian metric has been explained in detail in [FKKM]. We briefly describe the paradigm. Choose an arbitrary p E M. Then the cut locus C(p) is a nowhere dense subset of M.
Thus choose PI E M \ (C (p) U {p}). Then choose P2 away from C (P) and the complete geodesic through p and Pl. Then P3 will be chosen away from C(p) and the geodesic cone generated by P,Pl and P2. An
inductive construction lets us choose p, PI, ... ,Pn which compose a spanning CH-subset of M. Thus we arrive at THEOREM 4.2. (Fridman-Kim-Krantz-Ma [FKKM]) Let M be a connected ndimensional complex
manifold admitting a complete invariant Hermitian metric. Then a determining set, consisting of n + 1 points, exists for the automorphisms of M. Furthermore, the choice of such a determining set is
Here, by "generic", we mean that the the collection of (n + I)-tuples of points in Al x ... x M that satisfy our conclusion form a dense, open set. From the discussion above, if the metric happens to
be distance-decreasing, in the sense that all holomorphic endomorphisms are distance-decreasing with respect to the given metric, then this theorem will hold for holomorphic endomorphisms. This
result of course uses the idea developed in the preceding section about the distance-decreasing property together with the unique continuation property. We remark at this point that the collection of
complex manifolds admitting a complete invariant Hermitian metric is rather large. For instance, every bounded pseudoconvex domain in is equipped with the complete Kiihler-Einstein metric. See [MOY]
(also [CHY], [OHS], [YAU]) for instance.
5. Examples, Counterexamples, and the Cut Locus One might have the impression that some transversality condition for m + 1 points might be sufficient for the determining set problem for holomorphic
automorphisms. However, it is shown in [FKKM] that a simplistic topological transversality assumption will not be sufficient; consider the following statement. THEOREM 5.1. ([FKKM]) Fix a finite set
K = {PI, ... , Pk} in n > 1. There exists a bounded domain containing K, and a subgroup H C Aut(D) isomorphic to the unitary group U(n - 1) of en-I, such that each element of H fixes every point of
K. Moreover, unlike the one-dimensional planar domain case, the consideration of the cut locus seems essential even for one-dimensional Riemann surfaces. If one considers the torus coming from the
lattice generated by {I, i}, then the map z --+ - z of e generates an automorphism on the torus. It is easy to see that it has 4 fixed points, and yet is different from the identity map. If one
considers a two-holed torus with a well-balanced fundamental domain centered at 0 in the Poincare disc, then the same map z --+ -z of the disc will generate a non-trivial automorphism with 6 fixed
points. In this way, one can generate arbitrarily many fixed points for a non-trivial automorphisms of compact Riemann surfaces of high enough genus. Since our discussion has not depended upon the
completeness of manifolds, simple puncturing will create an arbitrary number of fixed points. Notice that all these examples have fixed points in the cut loci.
KANG-TAE KIM AND STEVEN G. KRANTZ
References [ALK] G. Aladro and S. G. Krantz, A criterion for normality in Cn, Jour. Math. Anal. and Appl. 161(1991), 1-8. [BED] E. Bedford and J. Dadok, Bounded domains with prescribed group of
automorphisms., Comment. !'v'lath. Helv. 62 (1987), 561-572. [BUS] H. Busemann, The geometry of geodesics, Academic Press, New York, NY, 1955. [CHY] S.Y. Cheng and S.T. Yau, On the existence of a
compact Kahler metric, Comm. Pure App!. Math., 33 (1980), 507-544. [FIF] S. D. Fisher and John Franks, The fixed points of an analytic self-mapping, Proc. AMS, 99(1987), 76-78. [FKKM] B. Fridman,
K-T. Kim, S. G. Krantz, and D. Ma, On Fixed Points and Determining Sets for Holomorphic Automorphisms, Michigan Math. Jour., to appear. [FP] B. L. Fridman and E. A. Poletsky, Upper semicontinuity of
automorphism groups, Math. Ann., 299(1994), 615-628. [GRK] R. E. Greene and S. G. Krantz, Stability properties of the Bergman kernel and curvature properties of bounded domains, Recent Developments
in Several Complex Variables (J. E. Fornress, ed.), Princeton University Press (1979),179-198. [GRW] R. E. Greene and H. Wu, Function theory on manifolds which possess a pole, Lecture Notes in
Mathematics, 699, Springer, Berlin, 1979. [GKM] D. Gromoll, W. Klingenberg, and W. Meyer, Riemannsche Geometrie im Grossen, 2nd ed., Lecture Notes in Mathematics, v. 55, Springer-Verlag, New York,
1975. [ISK] A. Isaev and S. G. Krantz, Domains with non-compact automorphism group: A survey, Advances in Math. 146(1999), 1-38. [KLI] W. Klingenberg, Riemannian Geometry, 2nd ed., de Gruyter Studies
in Mathematics, Berlin, 1995. [KOB] S. Kobayashi, Hyperbolic complex spaces, Springer, 1999. [LEM] L. Lempert, La metrique Kobayashi et las representation des domains sur la boule, Bull. Soc. Math.
Prance 109(1981), 427-474. [LES] K Leschinger, Uber fixpunkte holomorpher Automorphismen, Manuscripta Math., 25 (1978), 391-396. [MA] D. Ma, Upper semicontinuity of isotropy and automorphism groups,
Math. Ann., 292(1992), 533-545. [MAS] B. Maskit, The conformal group of a plane domain, Amer. J. Math., 90 (1968), 718-722. [MOY] N. Mok and S. T. Yau, Completeness of the Kahler-Einstein metric on
bounded domains and the characterization of domains of holomorphy by curvature conditions, Symposia in Pure Math. The mathematical heritage ofH. Poincare, Amer. Math. Soc., 39, Part I. (1983),41-60.
[OHS] T. Ohsawa, On complete Kahler domains with C 1 boundary, Pub!. Res. Inst. Math. Sci., RIMS (Kyoto), 16 (1980), 929-940. [PEL] E. Peschl and M Lehtinen, A conformal self-map which fixes 3 points
is the identity, Ann. Acad. Sci. Fenn., Ser. A I Math., 4 (1979), no. 1, 85-86. [SUI] N. Suita, On fixed points of conformal self-mappings, Hokkaido Math. J., 10(1981), 667-671. [Vll] J.-P. Vigue,
Fixed points of holomorphic mappings in a bounded convex domain in Cn, Proceedings of Symposia in Pure Mathematics, 52(1991), Amer. Math. Soc., 579-582. [VI2] J.-P. Vigue, Fixed points of holomorphic
mappings, Complex Geometry and Analysis (Pisa, 1988), Lecture Notes in Mathematics, v. 1422, Springer, Berlin, 1990, pp. 101-106. [YAU] S. T. Yau, A survey on Kahler-Einstein metrics. Complex
analysis of several variables (Madison, Wis., 1982), 285-289, Proc. Sympos. Pure Math., 41, Amer. Math. Soc., Providence, RI, 1984. KANG-TAE KIM, DEPARTMENT OF MATHEMATICS, POHANG UNIVERSITY OF
SCIENCE AND TECHNOLOGY, POHANG 790-784, KOREA STEVEN G. KRANTZ, DEPARTMENT OF MATHEMATICS, CAMPUS Box 1146, WASHINGTON UNIVERSITY, ST. LOUIS, MISSOURI 63130 U.S.A. E-mail address:kimkt«lpostech.ac.kr
E-mail address: sklDmath. wustl. edu
Contemporary Mathenlatics Volume 328, 2003
Localization in the Spectral Theory of Operators on Banach Spaces T. L. Miller, V. G. Miller, and M. M. Neumann ABSTRACT. In the first two sections of this article, we survey some of the recent
progress in the local spectral theory of operators on Banach spaces with emphasis on the local spectrum and on restrictions and quotients of decomposable operators. In particular, the problem of
characterizing restrictions and quotients of generalized scalar operators with spectrum in the unit circle in terms of suitable growth conditions is addressed in detail, with emphasis on [11], [22],
and [23]. The last two sections center around certain localized versions of the single-valued extension property, Bishop's property (13), and the decomposition property (8), mainly in the spirit of
[2], [5], [6], and [13]. For each of these properties, we find a smallest closed set modulo which it holds. For these residual sets, we establish a spectral mapping theorem with respect to the Riesz
functional calculus. We also obtain precise information about the extent to which Bishop's property «(3) holds on the essential or the Kato resolvent set. Our results are exemplified in the case of
weighted shifts. Moreover, several of the outstanding open questions of the field are mentioned in their natural context.
1. Decomposable operators and the local spectrum
Among the various aspects and levels of localization in spectral theory, we choose decomposability as our starting point. Let X be a complex Banach space, and let L(X) denote the Banach algebra of
all bounded linear operators on X. For T E L(X), let, as usual, a(T), ap(T), aap(T), r(T), and p(T) denote the spectrum, point spectrum, approximate point spectrum, spectral radius, and resolvent set
of T, and let Lat(T) stand for the collection of all T-invariant closed linear subspaces of X. From [18J and [29J we recall that an operator T E L(X) is said to be decomposable provided that, for
each open cover {U, V} of C, there exist Y, Z E Lat(T) for which X = Y + Z, a(T Iy) ~ U, and a(T I Z) ~ V. By [18, 1.2.23J or [29, 4.4.28J, this simple definition is equivalent to the original notion
of decomposability, as introduced by Foi~ in 1963 and discussed in the classical book by Colojoara and Foi~ [IOJ. 2000 Mathematics Subject Classification. Primary 47All, 47B40; Secondary 47B37. © 247
2003 American Mathematical Society
T. L. MILLER, V. G. MILLER, AND M. M. NEUMANN
As witnessed, for instance, by the monographs [10], [16], [18], and [29], the theory of decomposable operators is now richly developed with many interesting applications and connections. Evidently,
the class of decomposable operators contains all normal operators on Hilbert spaces and, more generally, all spectral operators in the sense of Dunford on Banach spaces. Moreover, a simple
application of the Riesz functional calculus shows that all operators with totally disconnected spectrum are decomposable. In particular, all compact and all algebraic operators are decomposable. An
important subclass of the decomposable operators is formed by the generalized scalar operators, defined as those operators T E L(X) for which there exists a continuous unital algebra homomorphism
(Tu f)(J-t) := (T - J-t)f(J-t)
for all f E H(U, X) and J-t E U.
It turns out that this operator dominates large parts of spectral theory. The local resolvent set PT(X) of T at a vector x E X is defined to consist of all >. E C for which there exists some f E H(U,
X) on an open neighborhood U of >. for which Tuf = x. Clearly, f(J-t) = (T - J-t)-lX for all J-t E Un p(T), so that PT(X) is open and contains p(T). Hence the local spectrum aT(x) := C \ pT(X) is a
closed subset of a(T). In general, the various analytic functions that occur in the definition of pT(X) need not be consistent. This issue is addressed by the following definition. The operator T E L
(X) is said to possess the single-valued extension property (SVEP), if Tu is injective for all open sets U £; IC. By [18, 3.3.2], T has SVEP precisely when, for each x E X, there exists a unique
function f E H(PT(X), X) for which
(T - J-t)f(J-t) = x
for all J-t E PT(X).
This function is then called the local resolvent function for T at x. In remarkable contrast to the usual resolvent function, such functions may well be bounded; this recent discovery of Bermudez and
Gonzalez will be exemplified below. One might expect a(T) to be the union of the local spectra aT(x) over all x E X, but this is not true in general. In fact, this union coincides with the
surjectivity spectrum asu(T) of T, the set of all >. E C for which T - >. fails to be surjective. However, if T has SVEP, then asu(T) = a(T), and aT(x) is non-empty for all non-zero x E X, [18,
1.2.16 and 1.3.2]. As a powerful application, we obtain
that every surjective operator with SVEP is actually bijective, [18, 1.2.10]. A more precise version of this result will be discussed in Section 3. For arbitrary T E L(X) and F ~ C, let XT(F) := {x
EX: aT(x) ~ F} denote the corresponding local spectml subspace. Evidently, XT(F) is a linear subspace of X, but need not be closed. The following classical result illustrates that these subspaces
playa basic role for spectral decompositions, see [18, 1.2] and [29,4.4]. THEOREM 1. Suppose that T E L(X) is decomposable. Then T has SVEP, and, for each closed set F ~ C, the space XT(F) is closed
and satisfies a(T I XT(F)) ~ F. In fact, XT(F) is the largest among all spaces Y E Lat(T) for which aT(T IY) ~ F. Moreover, XT(F) ~ XT(Ut}+·· +XT(U n ) for every finite open cover {U1 , ... , Un} ~F
The following examples may illuminate how spectral decompositions work in some important cases; for details see [18] and, for the last assertion of Example 4, also [26, Th.16]. The extent to which
the compactness of the group is essential here remains a challenging open problem. EXAMPLE 2. Let T E L(X) be a normal opemtor with spectml measure ~ on a complex Hilbert space X. Then T is
decomposable, and XT(F) = ran ~(F) for every closed set F ~ Co Moreover, there exists a non-zero bounded local resolvent function for T precisely when int a(T) =f:. 0. 0 EXAMPLE 3. Let X := C(O) be
the space of continuous functions on a compact Hausdorff space 0, and let T E L(X) denote the opemtor of multiplication by a given function g E X. Then T is decomposable, and XT(F) = {f E C(O) : g
(supp f) ~ F} for every closed set F ~ Co Also in this case, there exists a non-zero bounded local resolvent function for T precisely when int a(T) =f:. 0. 0 EXAMPLE 4. Let X := Ll(G) be the group
algebm of a locally compact abelian group G, and let T E L(X) denote the opemtor of convolution by a given function g E X. Then T is decomposable, and XT(F) = {f E Ll(G) : g(suppj) ~ F} for every
closed set F ~ C, where j denotes the Fourier tmnsform. Moreover, at least when G is compact, there exists a non-zero bounded local resolvent function for T precisely when int a(T) =f:. 0. 0 On the
other hand, there are important classes of operators which are not covered by decomposability. For instance, by [18, 1.6.14] and [22], a unilateral weighted right shift on the sequence space fP(N o)
for arbitrary 1 ::::; p < 00 is decomposable, or, equivalently, the quotient of a decomposable operator, only in the trivial case when it is quasi-nilpotent, while unilateral weighted right shifts
are never generalized scalar. Moreover, as we shall see, there are many examples of unilateral and bilateral weighted left shifts without SVEP. Another illuminating case is that of isometries. By
[18, 1.6.7], an arbitrary Banach space isometry is decomposable, or, equivalently, the quotient of a decomposable operator or generalized scalar, precisely when it is invertible. On the other hand,
every isometry may be extended, by a classical result due to Douglas, recorded in [18, 1.6.6], to an invertible isometry, and hence has a decomposable extension. In the next section, we shall discuss
a more general version of this result.
T. L. MILLER, V. G. MILLER, AND M. M. NEUMANN
2. Moving beyond decomposability Several years before decomposability was formally introduced by Foi~, Bishop [9] investigated a number of spectral decomposition properties in an attempt to extend
some of the features of the theory of normal operators to the general setting of Banach spaces. Among these properties, one turned out to be particularly important. We now say that an operator T E L
(X) on a complex Banach space X has Bishop's property ((J) provided that, for each open set U ~ C, the operator Tu on H(U, X) is injective with closed range, equivalently, if, for each sequence (fn)
nEN in H(U, X) with (T - )..)fn(>\) -> 0 asn->oo, uniformly on each compact subset of U, it follows that fn()..) -> 0 as n -> 00, again uniformly on the compact subsets of U, [18, 1.2.6]. Actually,
by [18, 3.3.5], the injectivity condition in this definition is redundant. Obviously, property ({J) implies SVEP. It was shown a long time ago by Foi~ that all decomposable operators share property
((J), but the precise relationship was discovered only recently by Albrecht and Eschmeier [6]. THEOREM 5. An operator T E L(X) has Bishop's property ((J) precisely when T is similar to the
restriction of a decomposable operator to a closed invariant subspace. Moreover, in this case, there exists a decomposable extension 8 for which aCT) ~ a(8). 0
The result was, in part, inspired by the work of Putinar [27] who proved that every hyponormal operator is subscalar, in the sense that it has a generalized scalar extension. Thus all hyponormal
operators have property ((J). In particular, all unilateral weighted right shifts on f2(N o) with an increasing weight sequence w have property ({J), but a characterization of ({J) in terms of w
seems to be an intriguing open problem. For partial results, see [11], [18], [22], and [23]. To discuss the dual notion of Bishop's property ((J), we need a slight variant of the local spectral
subspaces. For arbitrary T E L(X) and a closed subset F of
x E ran TC\F }.
In this definition, the point is that the local resolvent function is defined globally on the entire complement of F. Clearly, XT(F) is a linear subspace contained in XT(F). Moreover, by [18, 3.3.2],
the identity XT(F) = XT(F) holds for all closed sets F ~
THEOREM 6. Property (8) chamcterizes, up to similarity, the quotients of decomposable opemtors by closed invariant subspaces. Moreover. the properties (13) and (8) are dual to each other. in the
sense that an opemtor T E L(X) has one of these properties precisely when the adjoint T* E L(X*) on the dual space X*has the other one. D
Here the hardest assertion to prove is that property (13) for T* implies property (8) for T. The construction of decomposable extensions and liftings uses two powerful functional models for operators
on Banach spaces of independent interest. These models are in the spirit of function-theoretic operator theory, and involve the operator of multiplication by the independent variable on certain
Sobolev-type spaces together with the theory of topological tensor products. The complete duality between the properties (/3) and (8) employs the Grothendieck-K6the duality for spaces of
vector-valued analytic functions. All of this is described, in considerable detail, in [18, Ch.2]. There are interesting applications to the invariant subspace problem. Indeed, if the operator T E L
(X) has either property (/3) or property (8), then Eschmeier and Prunaru [14] established that Lat(T) is non-trivial provided that a(T) is thick, and that Lat(T) is rich in the sense that it contains
the lattice of all closed subspaces of some infinite-dimensional Banach space provided that the essential spectrum ae(T) is thick. Here we skip the formal definition of thick subsets of the complex
plane, but note that all compact sets with non-empty interior are thick. A streamlined approach to this result and further references may be found in [18, 2.6]. Since all hyponormal operators have,
by Putinar's result [27], property (/3), the preceding result subsumes, in particular, Brown's celebrated invariant subspace theorem for hyponormal operators with thick spectrum. In light of Read's
recent construction of a quasi-nilpotent, and hence decomposable, operator on a Banach space without non-trivial invariant subspaces, it is clear that the condition of thick spectrum cannot. be
dropped in general. However, the invariant subspace problem for operators on Hilbert spaces remains open, even for the class of hyponormal operators. As discussed in the monograph by Eschmeier and
Putinar [16], there are also interesting connections between property (/3) and the theory of analytic sheaves. These connections are not only important for the spectral theory of several commuting
operators, but they are also at the heart of some of the recent developments in the case of single operators. Although, as witnessed by the exposition of local spectral theory in [18], the explicit
use of sheaf theory can be avoided in the case of single operator theory, the reader should be aware of these connections. The basic idea is sketched in [18, 2.2]. A classical issue of local spectral
theory is to derive spectral decomposition properties from growth conditions on the powers or the resolvent function of a given operator. For instance, Levinson's log-log theorem from complex
analysis may be used to show that, for operators with spectrum in the real line or the unit circle 'll', a very weak logarithmic growth condition on the resolvent function suffices to ensure
decomposability. The short approach from [18, 1.7] to this classical result due to Radjabalipour is based on the fact that, by Theorem 6, an operator T E L(X) is decomposable precisely when both T
and T* have property (13). A very attractive account of the local spectral t.heory for operators with thin spectrum may be found in a recent survey article by Albrecht and Ricker [7].
T. L. MILLER, V. G. MILLER, AND M. M. NEUMANN
Here we focus only on one aspect that leads to interesting open problems. For arbitrary T E L(X), let K(T) := inf{IITxll : Ilxll = I} denote the lower bound of T. Evidently, K(T)-l = II T-lil when T
is invertible. Similar to the case of the spectral radius, it is known that the sequence of numbers K(Tn) lin converges to its supremum, denoted by i(T), and that aap(T) ~ {>' E C : i(T) ~ 1>'1 ~ r
(T)}, [18, 1.6.1 and 1.6.2]. By a classical result due to Colojoara and Foi~, [10, 5.1] or [18, 1.5.12], a generalized scalar operator T satisfies a(T) ~ T precisely when T is £(T) -scalar, in the
sense that T admits a continuous functional calculus on the Frechet algebra £(T) of all COO-functions on T. Moreover, T is £(T)-scalar if and only if T is invertible and satisfies the condition of
polynomial growth (P), in the sense that there exist constants c, s > 0 such that 1
-s ~
K(Tn) ~ IITnl1 ~ cn s for all n E Nj cn indeed, in this case, a functional calculus cP for T is given by the formula 00
!(n) Tn
for all
E £ (T) ,
where !(n) denotes the nth Fourier coefficient of f. Evidently, all invertible isometries have property (P), and hence are £ (T)scalar. Also, it follows from the preceding characterization that an
operator T E L(X) is £(T)-scalar precisely when its adjoint T* is £(T)-scalar. Moreover, since property (P) implies that i(T) = r(T) = 1 and consequently aap(T) ~ T, and since aap(T) = a(T) when T
has property (6), we are led to the following result. PROPOSITION
7. For every T E L(X) with property (P), the following equiva-
lences hold: T is invertible ¢:} a(T)
T has (6)
T is decomposable
T is £(T)-scalar.
Moreover, ifT is not invertible, then aap(T) = T and a(T) is the closed unit disc.o
Evidently, every restriction of an £(T)-scalar operator has property (P), but the converse is open in general. The preceding proposition shows that this problem is equivalent to the problem of
extending an arbitrary operator with property (P) to an invertible operator with property (P) for possibly larger constants c, s > O. Since the extension provided by the Albrecht-Eschmeier functional
model in Theorem 5 increases the spectrum, a different approach is needed here. As noted above, for isometries, the desired extension is possible by a result of Douglas. Also, for a certain class of
operators that includes all unilateral weighted right shifts, a positive solution was recently provided by Didas [11] and the authors [23]. While Didas exploits the theory of topological tensor
products in the spirit of Eschmeier and Putinar [16], the more elementary approach from [23] uses a modification of a construction provided by Bercovici and Petrovic [8] to characterize compressions
of £(T)-scalar operators. For unilateral weighted right shifts on fP(N o), the method developed in [23] leads to extensions as bilateral weighted shifts on fP(Z) with sharp growth estimates. To
reduce the case of quotients of £(T)-scalar operators to that of restrictions, we recall that the minimum modulus 'Y(T) of a non-zero operator T E L(X) is defined as 'Y(T):= inf{IITxll/dist(x,kerT):
x (j kerT}. Clearly, 'Y(T) = K(T)
when T is injective. It is also well known that 'Y(T) = 'Y(T*), and that 'Y(T) > 0 precisely when T has closed range. Standard duality theory now leads to the following result. PROPOSITION 8. An
operator T E L(X) is the restriction of an £(1l')-scalar operator if and only if its adjoint T* is the quotient of an £(1l')-scalar operator. Moreover, ifT is the quotient of an £(1l')-scalar
operator, then T* is the restriction of an £(1l')-scalar operator, and hence there exist constants c, s > 0 for which
~::;'Y(Tn)::;IITnll::;cns cn
In general, it is not known if the last growth condition characterizes the quotients of £(1l')-scalar operators, but, by Proposition 8 and [23, Prop.5], this is the case for the class of all
unilateral weighted left shifts on fP(N o ) for arbitrary 1 < p < 00. More precisely, a unilateral weighted left shift on fP(N o) satisfies the growth condition of Proposition 8 if and only if it
admits a bilateral weighted shift lifting on fP(Z) that is £(1l')-scalar. Similar results hold for more general growth conditions; see [22] and [23]. For instance, by another classical result due to
Colojoara and Foi~, an invertible operator T E L(X) is decomposable provided that T satisfies Beurling's condition (B), in the sense that
L 00
1 n 2 (llogK(Tn)1
+ IlogliTnll1) < 00,
[10, 5.3.2] and [18,4.4.7]. Clearly, property (B) is inherited by restrictions, but it remains open, if every operator with property (B) has an invertible extension with property (B). In fact, it is
not known, if property (B) implies property ((3). For certain unilateral weighted right shifts, a positive answer was recently given in [22] and [23].
3. Localization of the single-valued extension property For an arbitrary operator T E L(X) on a complex Banach space X, here the spaces K(T) := XT(C\ {O}) and Ho(T) := X T ( {O}) will be of
particular importance. Both spaces were, in some disguise, studied by Mbekhta and also by Vrbova; see [19], [20], and [30]. By [18, 3.3.7], K(T) coincides with the analytic core of T, defined to
consist of all x E X for which there exist a constant c > 0 and elements Xn E X such that for all n E N. By this characterization and the open mapping theorem, K(T)
if and only if
T is surjective. In terms of local spectral theory, this follows also from the fact that asu(T) is the union of all local spectra of T. On the other hand, by [18, 3.3.13], Ho(T) is the
quasi-nilpotent part of T, defined as the set of all x E X for which
IITnx11 1/ n _ 0
as n -
In general, neither K(T) nor Ho(T) need to be closed, but, if 0 is isolated in a(T), then, by [19, 1.6], both spaces are closed and X = K(T) EB Ho(T). For more on operators with closed K(T) and Ho
(T), see [1], [2], and [24]. For instance, by [24, Cor.6], for any non-invertible decomposable operator T, the point 0 is isolated in
T. L. MILLER, V. G. MILLER, AND M. M. NEUMANN
a(T) precisely when K(T) is closed. In particular, the analytic core of a compact or, more generally, a Riesz operator T is closed exactly when T has finite spectrum, [24, Cor.9]. The spaces Ho(T)
and K(T) are related to the kernel N(T) and the range R(T) of T as follows. With the notation oc
N(T) :=
U N(Tn)
n 00
for the generalized kernel and range of T, there is an increasing chain of kernel-type spaces
N(T) ~ N(Tn) ~ N(T) ~ Ho(T) and a decreasing chain of range-type spaces
~ X T ({O})
R(T) 2 R(Tn) 2 R(T) 2 K(T) 2 X T (0) for arbitrary n E N. [18, 1.2.16 and 3.3.1]. The geometric position of the kerneltype spaces versus the range-type spaces turns out to be intimately related to a
certain localized version of SVEP for the operator T and its adjoint T*. An operator T E L(X) is said to have SVEP at a point A E C, if, for every open disc U centered at A, the operator Tu is
injective on H(U, X). This notion dates back to Finch [17], and was pursued further, for instance, in [1], [2], [3], [4], [5], and [20]. Evidently, T has SVEP at A precisely when T - A has SVEP at 0,
while SVEP for T is equivalent to SVEP for T at A for each A E C. Local spectral theory leads to a variety of characterizations of this localized version of SVEP that involve the kernel-type and
range-type spaces introduced above. Our starting point is the following characterization from [3, 1.9]. The result shows, in particular, that every injective operator T E L(X) has SVEP at 0, and may
be viewed as a local version of the classical fact that T has SVEP if and only if X T (0) = {O}, [18, 1.2.16]. For completeness, we include a short new proof that uses nothing but local spectral
theory. THEOREM
9. For every operator T
T has SVEP at 0 <=} N(T)
L(X), the following equivalences hold:
n X T (0) = {O} <=} aT(x) = {O}
for all 0 =f. x E N(T).
Proof. First suppose that T has SVEP at 0, and consider an arbitrary x E N(T) for which aT(x) is empty. Then 0 E pT(X) so that there exists an f E H(U, X) on some open disc U with center 0 for which
(T-A)f(A) = x for all A E U. It follows that (T - A)Tf(A) = Tx = 0 for all A E U, and therefore Tf(A) = 0 for all A E U, since T has SVEP at O. Thus x = Tf(O) = 0, and hence N(T) n X T (0) = {O}.
Next observe that, for each x E N(T), the definition f(A) := -xl A yields an analytic function for which (T - A)f(A) = x for all non-zero A E C. Thus aT(x) ~ {O} for all x E N(T). Consequently, the
second and third assertions are equivalent. Finally suppose that N(T) n XT(0) = {O}, let U be an open disc with center 0, and consider a function f E H(U, X) for which Tu f = O. By [18, 1.2.14], aT(f
(A)) = aT(O) = 0 for all A E U. Now, for the power series representation f(A) = L:~=o an An for all A E U, our task is to show that each of the coefficients an E X is zero. For the case n = 0, this
is immediate, since ao = f(O) E N(T) nXT (0) = {O}. But then it follows that for all A E U,
and therefore (T - >.) (at + a2 >. + a3 >.2 + ... ) = 0 first for all non-zero>. E U, and then, by continuity, also for>. = O. Exactly as before, we conclude that at = 0 and hence, by induction, an =
0 for all n ~ O. Thus f == 0 on U, as desired. 0 Since N(T) n K(T) ~ X T ( {O}) n XT(C \ {O}) = X T (0), it clearly follows that N(T) n K(T) = N(T) n X T (0) for every T E L(X). Thus, by Theorem 9, T
has SVEP at 0 if and only if N(T) n K(T) = {O}. In particular, if T is surjective, then, as noted above, K(T) = X, so that T has SVEP at 0 precisely when T is injective. This characterization from
[3, 1.11] extends a classical result due to Finch [17]. As another immediate consequence of Theorem 9, we obtain the following result. COROLLARY 10. An operator T E L(X) has SVEP at 0 provided that
either Ho(T) n K(T) = {O} or N(T) n R(T) = (0). 0 Recent counter-examples in [2] show that, in general, none of the latter conditions is equivalent to SVEP of T at 0, thus disproving a claim made in
[20, 1.4]. However, by [1, 2.7], [5, 1.3], and Theorem 12 below, equivalences do hold for certain classes of operators. We now describe how the localized SVEP behaves under duality. For a linear
subspace M of X, let Ml. := {cp E X* : cp(x) = 0 for all x E M}, and for a linear subspace N of X*, let l.N := {x EX: cp(x) = 0 for all cp E N}. By the bipolar theorem, l.(Ml.) is the norm-closure of
M, and (l.N)l. is the weak-*-closure of N. Moreover, for every T E L(X), it is well known that N(T*) = R(T)l. and N(T) = l.R(T*), while R(T) is a norm-dense subspace of l.N(T*), and R(T*) is a
weak-*-dense subspace of N(T)l.. An elementary short proof of the following result may be found in [2,4.1]. PROPOSITION 11. For every operator T E L(X), the following assertions hold: (a) K(T) ~ l.Ho
(T*) and K(T*) ~ Ho(T)l.; (b) if Ho(T) + R(T) is norm-dense in X, then T* has SVEP at 0; (c) if Ho(T*) + R(T*) is weak-*-dense in X*, then T has SVEP at o. 0 Even in the Hilbert space setting, the
inclusions in part (a) of Proposition 11 need not be identities, and the implications of parts (b) and (c) cannot be reversed in general; see [2] for counter-examples in the class of weighted shifts.
However, for suitable classes of operators, the results can be improved. As usual, an operator T E L(X) is said to be a semi-Fredholm operator, if either N(T) is finite-dimensional and R(T) is
closed, or R(T) is of finite codimension in X. Also, an operator T E L(X) is said to be semi-regular, if R(T) is closed and N(T) ~ R(T); see [18], [19], and [21] for a discussion of these operators.
THEOREM 12. Suppose that the operator T E L(X) is either semi-Fredholm or semi-regular. Then the following assertions hold: (a) R(T) = K(T) = l.Ho(T*) = l.N(T*); (b) R(T*) = K(T*) = Ho(T)l. = N(T)l.;
(c) N(T) n R(T) = {O} <=> T has SVEP at 0; (d) N(T*) n R(T*) = {O} <=> T* has SVEP at 0; (e) N(T)
+ R(T) = X<=>
(f) N-;-;:-;:(T=*""7"")-+-:R~(T=*"-;-) w'
+ R(T) = X<=> T* has SVEP at 0; <=> Ho(T*) + R(T*) w' = X* <=> T has SVEP
= X*
where w* indicates the closure with respect to the weak-*-topology.
at 0,
T. L. MILLER,
V. G.
MILLER, AND M. M. NEUMANN
Theorem 12 was recently obtained in [2], see also [5]. An important ingredient of the proof is the fact that T has SVEP at 0 if and only Tn has SVEP at 0 for arbitrary n E N. This equivalence is a
special case of a spectral mapping formula for the set 6(T) of all A E C at which T fails to have SVEP, namely 6(f(T)) = f(6(T)) for every analytic function f on some open neighborhood of a(T); see
[2, 3.1] and also Theorem 18 below. Further developments may be found in [1], [3], [4], [5], [17], and [20]. Here we mention only one simple consequence of Theorem 12 for semi-regular operators from
[3, 2.13]. For T E L(X), let PK(T) consist of all A E C for which T - A is semi-regular. The Kato spectrum aK(T) := C \ PK(T) is a closed subset of a(T) and contains oa(T); see [18, 3.1] and [21] for
details. We include a short proof of the following result, since the dichotomy for the connected components of the Kato resolvent set PK (T) with respect to the localized SVEP will play an essential
role in Section 4. THEOREM 13. Let T E L(X) be semi-regular. Then T has SVEP at 0 precisely when T is injective, or, equivalently, when T is bounded below, while T* has SVEP at 0 precisely when T is
surjective. Moreover, for arbitrary T E L(X), each connected component n of PK(T) satisfies either n ~ 6(T) or n n 6(T) = 0. The inclusion n ~ 6(T) OCC1J.rs precisely when n ~ ap(T), or,
equivalently, when n n aap(T) i=- 0, while the identity n n 6(T) = 0 occurs precisely when n n ap(T) = 0, or, equivalently, when n \ aap(T) i=- 0.
Proof. If T is semi-regular, then N(T) n n(T) = N(T) and N(T) + R(T) = R(T) = R(T). Hence the first assertions follow from parts (c) and (e) of Theorem 12. For the last claim, it suffices to see that
injectivity of T - A for some A E n entails that T - f.J. is injective for every f.J. E n. But this is clear, since, by part (b) of Theorem 12, N(T - f.J.) = J..n(T* -f.J.) and, by [18, 3.1.6 and
3.1.11], n(T* - f.J.) = n(T* -A) for all f.J. E n. 0 It is well known that the approximate point spectrum and the surjectivity spectrum of an arbitrary operator T E L(X) are related by the duality
formulas aap(T) = asu(T*) and asu(T) = aap(T*), [18, 1.3.1]. Moreover, by [18, 1.3.2 and 3.1.7], asu(T) = a(T) and aap(T) = aK(T) if T has SVEP, and aap(T) = a(T) and asu(T) = aK(T) if T* has SVEP.
The following local version of these results is immediate from Theorem 13. PROPOSITION 14. For every operator T E L(X), the following assertions hold: (a) If A E a(T) \ aap(T), then T has SVEP at A,
but T* fails to have SVEP at A; (b) if A E a(T) \ asu(T), then T* has SVEP at A, but T fails to have SVEP at A.
o The next result from [2, 5.2] is a straightforward consequence of Proposition 14. For instance, it follows that 6(T*) is the open unit disc for every non-invertible operator T with property (P) or
(B). Further examples including analytic Toeplitz operators, composition operators on Hardy spaces, and weighted shifts may be found in [2]. COROLLARY 15. If aap(T) ~ oa(T), then T has SVEP and 6(T*)
= int a(T). Similarly, if asu(T) ~ oa(T), then T* has SVEP and 6(T) = int a(T). 0
4. Localization of the properties ((:J) and (8) There is a natural extension of the class of decomposable operators for which spectral decompositions are only required with respect to a given open
subset U of the complex plane. These operators were introduced by Vasilescu as residually decomposable operators in 1969, shortly after the publication of the seminal monograph [10]. They became also
known as S-decomposable operators with S = c \ U, and were studied by Bacalu, Nagy, Vasilescu, and others; see [29, eh.4]. As in [6] and [13], we now say that an operator T E L(X) on a complex Banach
space X is decomposable on an open subset U of C provided that, for every finite open cover {Vl , ... , Vn } of C with C \ U ~ Vl , there exist Xl,"" Xn E Lat(T) with the property that X
= Xl + ... + Xn and
a(T I X k ) ~
for k
= 1, ... , n.
It is known, although certainly not obvious, that, in this definition, it suffices to consider the case n = 2; see [6] and [29]. Evidently, classical decomposability occurs when U = C. On the other
hand, every operator T E L(X) is at least decomposable on its resolvent set p(T). Among the remarkable early accomplishments of the theory is the following result due to Nagy [25] from 1979: For
every T E L(X), there exists a largest open set U ~ C on which T is decomposable. The complement of this set is Nagy's spectral residuum Sr(T), a closed, possibly empty, subset of a(T). In the
present section, we shall employ the recent results of Albrecht and Eschmeier [6] to obtain a short proof for the existence and a useful description of Nagy's spectral residuum. In particular, we
shall see how Sr(T) is related to the Kato spectrum aK(T) and the essential spectrum ae(T). For this, we shall work with certain localized versions of property ((:J) and property (8) from [6]. An
operator T E L(X) is said to possess Bishop's property ((:J) on an open set U ~ C, if, for every open subset V of U, the operator Tv is injective with closed range, equivalently, if, for every
sequence of analytic functions In: V ---+ X for which (T->')In(>\) ---+ 0 as n ---+ 00 locally uniformly on V, it follows that In(>') ---+ 0 as n ---+ 00, again locally uniformly on V. It is
straightforward to check that this condition is preserved under arbitrary unions of open sets. This shows that there exists a largest open set on which T has property ((:J), denoted by U{3(T). Its
complement S{3(T) := C \ U{3(T) is a closed, possibly empty, subset of a(T). In fact, T satisfies Bishop's classical property ((:J) precisely when S{3(T) = 0. Moreover, the operator T is said to have
property (8) on U, if X
= XT(C \ V) + XT(W)
for all open sets V, W ~ C for which C \ U ~ V ~ V ~ W; see [6] and [13]. Quite remarkably, as shown in [6, Th.3], this condition holds precisely when, for each closed set F ~ C and every finite open
cover {VI"'" Vn } of F with F \ U ~ VI, it follows that XT(F) ~ XT(V d + ... + XT(V n); see also [18, 2.2.2] for the case U=C, These localized versions of ((:J) and (8) already proved to be useful in
the theory of invariant subspaces for operators on Banach spaces, [14]. The following result summarizes the main accomplishments from [6].
T. L. MILLER, V. G. MILLER, AND M. M. NEUMANN
THEOREM 16. For every operator T following equivalences hold:
L(X) and every open set U
C, the
(a) T has (13) on U {::} T* has (8) on U; (b) T has (8) on U {::} T* has (/3) on U;
(c) T is decomposable on U {::} T has both (13) and (8) on U; (d) T has (13) on U {::} T is the restriction of a decomposable operator on U; (e) T has (15) on U {::} T i,~ the quotient of a
decomposable operator on U.
It is not at all obvious from the definition of (8) that there exists a largest open set, say Uc5(T), on which the operator T has property (8), but this now follows from the corresponding result for
(13) by duality. In fact, Uc5(T) = U{3(T*) by part (b) of the preceding result. More precisely, Theorem 16 leads to the following result. COROLLARY 17. For every operator T E L( X), there exists a
smallest closed set Sc5(T) so that T has property (8) on its complement. Moreover, Sc5(T) = S{3(T*), S{3(T) = Sc5(T*), and Sr(T) = S{3(T) U Sc5(T) = S{3(T) U S{3(T*) = Sr(T*). 0
Perhaps somewhat surprisingly, it will be possible to obtain general information about the location of S{3(T), and hence of Sc5(T) and Sr(T). For this, another localized version of SVEP will play a
crucial role. For consistency, we say that the operator T E L(X) has SVEP on an open set U ~ C, if, for every open subset V of U, the operator Tv is injective, [13]. It is straightforward to check
that T has SVEP on U precisely when T has SVEP at each point >. E U, as defined in the previous section. Obviously, there exists a largest open set on which T has SVEP, and the analytic spectral
residuum S(T) is defined to be the complement of this set; see [29, 4.3.2] and [30]. Clearly, 6(T) ~ S(T), but, since 6(T) is open and S(T) is closed, equality occurs only in the trivial case when T
has SVEP. Nevertheless, as noted in [2], a simple verification shows that 6(T) = S(T) ~ S{3(T). It is interesting to observe that all these residual sets behave canonically with respect to the Riesz
functional calculus. As usual, for T E L(X) and any analytic complex-valued function f on an open neighborhood 0 of a(T), the operator f(T) E L(X) is defined by
f(>.)(>. - T)-l d>', 2m where r denotes an arbitrary contour in 0 that surrounds a(T), [12] or [18, A.2]. The standard spectral mapping theorem asserts that a(f(T)) = f(a(T)). The next result has a
similar flavor, and may be viewed as an extension of the fact that the classical versions of SVEP, property (/3), property (8), and decomposability are all preserved under the Riesz functional
calculus, [18, 3.3.6 and 3.3.9]. The fact that the Riesz functional calculus respects Bishop's classical property (13) was established by Eschmeier and Putinar [15]. The following proof involves a
different approach to this result. f(T) :=
THEOREM 18. Let T E L(X) be an arbitrary operator, let f : 0 -+ C be an analytic function on an open neighborhood 0 of a(T), and suppose that f is nonconstant on each connected component ofO. If E
denotes any of the symbols 6, S, S{3, Sc5, or S., then E(f(T)) = f(E(T)).
Proof. In the case of 6(T), the spectral mapping formula was recently obtained in [2, 3.1]. The result for S(T) is a standard fact that may be found in [29, 4.3.14] and [30, 1.6]. Note, however, that
the formula for S(T) is also an immediate consequence of that for 6(T), because 6(T) = S(T). While the existence of the residual set So(T) was most conveniently established by using S{3(T) and the
Albrecht-Eschmeier duality between the localized versions of (f3) and (8), for the issue at hand it seems appropriate to switch the order. Indeed, since f(T*) = f(T) *, Corollary 17 ensures that it
suffices to prove the claim for So(T). For this, fortunately, we may proceed as in the proof of [18,3.3.6 and 3.3.9], where property (8) is shown to be stable under the Riesz functional calculus.
First, consider arbitrary open sets V, W ~ C for which f(So(T)) ~ V ~ V ~ W. Then {f-l(C \ V), f- 1(W)} is an open cover of a(T) for which So(T) ~ f-l(W). Thus, by the characterization of the
localized property (8) mentioned above, we obtain that
X = XT(a(T))
= XT
(1-1(C \ V)
n a(T)) + XT (1-1(W) n a(T)) .
Clearly, f- 1(C \ V) n a(T) ~ f- 1(C \ V) n a(T) and, similarly, f-l(W) n a(T) ~ f-l(W) n a(T). Since, by [18,3.3.6], the formula XT (f-l(F) n a(T)) = Xf(T) (F) holds for every closed set F ~ C, we
conclude that X = Xf(T) (C \ V)
+ Xf(T) (W).
This shows that f(T) has (8) on C \ f(So(T)), thus C \ f(So(T)) ~ Uo(f(T)), and hence So(f(T)) ~ f(So(T)). Note that this inclusion even holds without the requirement that f be non-constant on each
of the connected components of its domain. The reverse inclusion is less obvious, but may be obtained by a suitable modification of the proof of [18, 3.3.9]. Let S := f- 1 (So(f(T))) n a(T). Then the
desired inclusion f(So(T)) ~ So(f(T)) means precisely that the decomposition X = XT(C \ V) + XT(W) holds for all open sets V, W ~ C for which S ~ V ~ V ~ W. Evidently, it suffices to show that X = XT
(G) + XT(H) for every open cover {G,H} of a(T) for which S ~ G, SnH = 0, and both G and H are compact subsets of n. Ignoring momentarily the exceptional set S, we may proceed word by word along the
lines of the proof of [18, 3.3.9] to obtain a finite open cover {WI, ... , W n } of a(T) in n for which fork=I, ... ,n. To handle the residual set, we note that the identity SnH = 0 may be
reformulated in the form So(f(T)) n f(a(T) n H) = 0. Hence, by continuity and compactness, there exists an open neighborhood V of So (f(T)) for which V n f(a(T) n H) = 0. This implies that f- 1 (V) n
a(T) n H = 0, hence f- 1 (V) n a(T) ~ G, and therefore, by [18, 3.3.6], Xf(T)(V) = XT (J-l(V)
n a(T))
~ XT(G).
Now, since f(T) has (8) on C \ So(f(T)), and since {V, f(W1), ... , f(Wn )} is an open cover of a(f(T)) = f(a(T)) for which So(f(T)) ~ V, we conclude that X
= Xf(T)(a(T)) = Xf(T)(V) + Xf(T)
+ ... + Xf(T)
(f(Wn)) '
L. MILLER, V. G. MILLER, AND M. M. NEUMANN
again by the basic characterization of the localized version of (8) provided in [6, Th.3]. Thus X = XT(G) + XT(H), as desired. 0 In the following results, we show how to verify property ({3) on the
Kato resolvent set and its Fredholm counterpart. THEOREM 19. For an arbitrary operator T E L(X) and every connected component n of PK(T), the following equivalences hold:
T has ({3) on
n {:}
T has SVEP on
n {:} n n O'p(T) = 0 {:} n \ O'ap(T) =F 0j
in particular, T has property ({3) on PK(T) precisely when T has SVEP on PK(T). Proof. Clearly, the first of the displayed conditions implies the second one, and the equivalence of the last three
conditions follows from Theorem 13. Conversely, if these three conditions hold, then T - A is injective with closed range for all A E n. Thus f'i,(T - A) = "((T - A) > 0 for all A E n. Moreover, by
[18, 3.1.10], for every compact subset K of n, there exists a constant c > 0 such that f'i,(T - A) > 0 for all A E Kj in fact, as shown, for instance, in [21,4.1], the function A t-+ "((T - A) is
continuous and strictly positive on PK(T). From this it is immediate that T has ({3) on n. 0 The next result is clear from Corollary 17, Theorem 19, and the well-known identity PK(T) = PK(T*), [18,
3.1.6]. Part (c) of Corollary 20 may be viewed as an extension of the fact that, by [18, 3.1.7], p(T) = PK(T) whenever both T and T* have SVEP. COROLLARY 20. For every operatorT E L(X), the following
assertions hold:(a) T has SVEP on PK(T) {:} S{3(T) ~ O'K(T)j (b) T* has SVEP on PK(T) {:} SIi(T) ~ O'K(T)j (c) T and T* have SVEP on PK(T) {:} Sr(T) ~ O'K(T). 0 To derive the companion result for the
essential spectrum, we employ the fact that, for every operator T E L(X) and every open subset V of the essential resolvent set Pe(T) := C\O'e(T), the operator Tv has closed range (but need not be
injective). This interesting result was recently obtained by Eschmeier [13, 3.1], based on sheaftheoretic methods developed by Putinar [28] to show that quasi-similar operators with property ({3)
have the same essential spectrum. In tandem with Corollary 17, we obtain the following extension of [13, 3.9]. COROLLARY 21. For every operator T E L(X), the following assertions hold: (a) T has SVEP
on Pe(T) {:} S{3(T) ~ O'e(T)j (b) T* has SVEP on Pe(T) {:} SIi(T) ~ O'e(T)j (c) T and T* have SVEP on Pe(T) {:} Sr(T) ~ O'e(T).
We close with an application to the spectral theory of weighted shifts. EXAMPLE 22. Let W := (Wn)nEN"o be a bounded sequence of strictly positive real numbers, and let T E L(X) denote the
corresponding unilateral weighted right shift on the sequence space X := fP(N o) for some 1 ::; p < 00. Clearly,
i(T) = lim inf
n-+oo k~O
(Wk· .. Wk+n_d 1 / n
r(T) = lim sup (Wk· n-+oo k~O
.. Wk+n_d 1 / n .
Since T has no eigenvalues, T has SVEP and asu(T) = a(T) = {A E C : IAI by [18, 1.3.2 and 1.6.15J. Moreover, as noted in [18,3.7.7],
s r(T)},
siAl s r(T)} , and therefore, by Corollaries 20 or 21, the annulus {A E C : i(T) siAl s r(T)} contains S(3(T). On the other hand, by [2, 6.1J, 6(T*) = {A E C : IAI < c(T)} , where ae(T) = aK(T) = aap
(T) = {A E C : i(T)
c(T):= liminf(wl" n->oo
Thus, by Corollary 17, it follows that S5(T) = S(3(T*) 2 {A E C : IAI S c(T)}. We finally note that, by [22, 2.7J, condition (f3) on T implies that i(T) = r(T) and aT(x) = a(T) for all non-zero x E
X, while, by [22, 3.3J or [23, Prop.5], a certain growth condition of exponential type for the weight sequence w suffices to ensure that T has (f3). 0
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Neumann, On a localized single-valued extension property, to appear in Proc. Royal Irish Acad. [3J P. Aiena and O. Monsalve, Opemtors which do not have the single valued extension property, J. Math.
Anal. Appl. 250 (2000),435-448. [4J P. Aiena and O. Monsalve, The single valued extension property and the genemlized Kato decomposition property, Acta Sci. Math. (Szeged) 67 (2001), 791-807. [5J P.
Aiena and F. Villafane, Components of resolvent sets and local spectml theory, submitted to this volume. [6J E. Albrecht and J. Eschmeier, Analytic functional models and local spectml theory, Proc.
London Math. Soc. (3) 75 (1997), 323-348. [7J E. Albrecht and W. J. Ricker, Local spectml theory f01· opemtors with thin spectrum, preprint, University of Saarbriicken, 2002. [8J H. Bercovici and S.
Petrovic, Genemlized scalar opemtors as dilations, Proc. Amer. Math. Soc. 123 (1995), 2173-2180. [9J E. Bishop, A duality theory for an arbitmry opemtor, Pacific J. Math. 9 (1959), 379-397. [lOJ I.
Colojoara and C. Foi~, Theory of Genemlized Spectml Opemtors, Gordon and Breach, New York, 1968. [11J M. Didas, E(]"n )-subscalar n-tuples and the Cesaro opemtor on HP, Annales Universitatis
Saraviensis, Series Mathematicae 10 (2000), 285-335. [12J N. Dunford and J. T. Schwartz, Linear Opemtors III, Wiley-Interscience, New York, 1971. [13J J. Eschmeier, On the essential spectrum of
Banach-space opemtors, Proc. Edinburgh Math. Soc. (2) 43 (2000), 511-528. [14J J. Eschmeier and B. Prunaru, Invariant subspaces for opemtors with Bishop's property ({3) and thick spectrum, J. Funct.
Anal. 94 (1990), 196-222. [15J J. Eschmeier and M. Putinar, Bishop's condition ({3) and rich extensions of linear opemtors, Indiana Univ. Math. J. 37 (1988), 325-348. [16J J. Eschmeier and M.
Putinar, Spectml Decompositions and Analytic Sheaves, Clarendon Press, Oxford, 1996. [17J J. K. Finch, The single valued exten.9ion property on a Banach space, Pacific J. Math. 58 (1975),61-69. [18J
K. B. Laursen and M. M. Neumann, An Introduction to Local Spectml Theory, Clarendon Press, Oxford, 2000. [19J M. Mbekbta, Genemlisation de la decomposition de Kato aux opemteurs pamnormaux et
spectmux, Glasgow Math. J. 29 (1987), 159-175. [20J M. Mbekhta, Sur la theorie spectmle locale et limite des nilpotents, Proc. Amer. Math. Soc. 110 (1990), 621-631.
T. L. MILLER, V. G. MILLER, AND M. M. NEUMANN
[21] M. Mbekhta and A. Ouahab, Operateurs s-regulier dans un espace de Banach et theorie spectrale, Acta Sci. Math. (Szeged) 59 (1994), 525-543. [22] T. L. Miller, V. G. Miller, and M. M. Neumann,
Local spectral properties of weighted shifts,
to appear in J. Operator Theory. [23] T. L. Miller, V. G. Miller, and M. M. Neumann, Growth conditions and decomposable exten-
sions, to appear in Contemp. Math. [24] T. L. Miller, V. G. Miller, and M. M. Neumann, On operators with closed analytic core, to appear in Rend. Cire. Mat. Palermo (2) 51 (2002). [25] B. Nagy, On
S-decomposable operators, J. Operator Theory 2 (1979),277-286. [26] M. M. Neumann, Recent developments in local spectral theory, Rend. Circ. Mat. Palermo (2) Suppl. 68 (2002), 111-131. [27] M.
Putinar, Hyponormal operators are subsealar, J. Operator Theory 12 (1984), 385-395. [28] M. Putinar, Quasi-similarity of tuples with Bishop's property (,8), Integral Equations Operator Theory 15
(1992), 1047-1052. [29] F.-H. Vasilescu, Analytic FUnctional Calculus and Spectral Decompositions, Editura Aeademiei and D. Reidel Publishing Company, Bucharest and Dordreeht, 1982. [30] P. Vrbova.,
On local spectral properties of operators in Banach spaces, Czechoslovak Math. J. 23 (98) (1973),483-492. DEPARTMENT OF MATHEMATICS AND STATISTICS, MISSISSIPPI STATE UNIVERSITY, MISSISSIPPI STATE, MS
39762, USA E-mail address: neumannOmath.msstate.edu
Contemporary Mathematics Volume 328, 2003
Abstract harmonic analysis, homological algebra, and operator spaces Volker Runde ABSTRACT. In 1972, B. E. Johnson proved that a locally compact group G is amenable if and only if certain Hochschild
cohomology groups of its convolution algebra Ll(G) vanish. Similarly, G is compact if and only if Ll(G) is biprojective: In each case, a classical property of G corresponds to a cohomological propety
of Ll(G). Starting with the work of Z.-J. Ruan in 1995, it has become apparent that in the non-commutative setting, i.e. when dealing with the Fourier algebra A(G) or the Fourier-Stieltjes algebra B
(G), the canonical operator space structure of the algebras under consideration has to be taken into account: In analogy with Johnson's result, Ruan characterized the amenable locally compact groups
G through the vanishing of certain cohomology groups of A(G). In this paper, we give a survey of historical developments, known results, and current open problems.
1. Abstract harmonic analysis, ...
The central objects of interest in abstract harmonic analysis are locally compact groups, i.e. groups equipped with a locally compact Hausdorff topology such that multiplication and inversion are
continuous. This includes all discrete groups, but also all Lie groups. There are various function spaces associated with a locally compact group G, e.g. the space Co(G) of all continuous functions
on G that vanish at infinity. The dual space of Co(G) can be identified with M(G), the space of all regular (complex) Borel measures on G. The convolution product * oftwo measures is defined via
(1,11-* v):= LLf(XY)dJ.L(X)V(Y)
(J.L,V E M(G), f E Co(G))
and turns M(G) into a Banach algebra. Moreover, M(G) has an isometric involution given by
(I,J.L*):= Lf(x-1)dJ.L(X)
(J.L E M(G), f E Co(G)).
1991 Mathematics Subject Classification. 22D15, 22D25, 43A20, 43A30, 46H20 (primary), 46H25, 46L07, 46M18, 46M20, 47B47, 47L25, 47L50. Key words and phrases. locally compact groups, group algebra,
Fourier algebra, FourierStieltjes algebra, Hochschild cohomology, homological algebra, operator spaces. Financial support by NSERC under grant no. 227043-00 is gratefully acknowledged. © 263
2003 American Mathematical Society
The most surprising feature of an object as general as a locally compact group is the existence of (left) Haar measure: a regular Borel measure which is invariant under left translation and unique up
to a multiplicative constant. For example, the Haar measure of a discrete group is simply counting measure, and the Haar measure of ]RN, is N-dimensional Lebesgue measure. The space Ll(G) of all
integrable functions with respect to Haar measure can be identified with a closed *-ideal of M(G) via the Radon-Nikodym theorem. Both M(G) and Ll(G) are complete invariants for G: Whenver Ll(G 1 )
and £1(G 2 ) (or M(Gt} and M(G 2 )) are isometrically isomorphic, then G 1 and G 2 are topologically isomorphic. This means that all information on a locally compact group is already encoded in Ll
(G) and M(G). For example, Ll(G) and M(G) are abelian if and only if G is abelian, and Ll (G) has an identity if and only if G is discrete. References for abstract harmonic analysis are [Fol], [H-R],
and [R-St]. The property of locally compact groups we will mostly be concerned in this survey is amenability. A a mean on a locally compact group G is a bounded linear functional m: LOC(G) ---+ C
such that (1, m) = Ilmil = 1. For any function I on G and for any x E G, we write LxI for the left translate of I by x, i.e. (Lxf)(y) := I(xy) for y E G. DEFINITION 1.1. A locally compact group G is
called amenable if there is a (left) translation invariant mean on G, i.e. a mean m such that
(¢, m) = (L x ¢, m)
(¢ E LOC(G), x E G).
EXAMPLE 1.2. (1) Since the Haar measure of a compact group G is finite, LOC(G) C £1(G) holds. Consequently, Haar measure is an invariant mean on G. (2) For abelian G, the Markov-Kakutani fixed point
theorem yields an invariant mean on G. (3) The free group in two generators is not amenable ([Pat, (0.6) Exanlple]). Moreover, amenability is stable under standard constructions on locally compact
groups such as taking subgroups, quotients, extensions, and inductive limits. Amenable, locally compact groups were first considered by J. v. Neumann ([Neu]) in the discrete case; he used the term
"Gruppen von endlichem MaB". The adjective amenable for groups satisfying Definition 1.1 is due to M. M. Day ([Day]), apparently with a pun in mind: They are amenable because they have an invariant
mean, but also since they are particularly pleasant to deal with and thus are truly amenable - just in the sense of that adjective in everyday speech. For more on the theory of amenable, locally
compact groups, we refer to the monographs [Gre], [Pat], and [Pie].
2. homological algebra, ... We will not attempt here to give a survey on a area as vast as homological algebra, but outline only a few, basic cohomological concepts that are relevant in connection
with abstract harmonic analysis. For the general theory of homological algebra, we refer to [C-E], [MacL], and [Wei]. The first to adapt notions from homological algebra to the functional analytic
context was H. Kamowitz in [Kam]. Let 2l be a Banach algebra. A Banach 2l-bimodule is a Banach space E which is also an 2l-bimodule such that the module actions of 2l on E are jointly continuous.
ABSTRACT HARMONIC ANALYSIS A derivation from 2l to E is a (bounded) linear map D: 2l D(ab) = a . Db + (Da) . b
E satisfying
(a, bE !2l);
the space of all derivation from 2l to E is commonly denoted by ZI(!2l, E). A derivation D is called inner if there is x E E such that Da = a·x-x·a
(a E !2l).
The symbol for the subspace of ZI (!2l, E) consisting of the inner derivations is B 1 (2l,E); note that B 1 (!2l,E) need not be closed in ZI(!2l,E). DEFINITION 2.1. Let !2l be a Banach algebra, and
let E be a Banach 2l-bimodule. Then then the first Hochschild cohomology group 'HI (2l, E) of 2l with coefficients in E is defined as 'H 1 (!2l, E) := ZI(!2l, E)/B 1 (2l, E). The name Hochschild
cohomology group is in the honor of G. Hochschild who first considered these groups in a purely algebraic context ([Hoch 1] and [Hoch 2]). Given a Banach !2l-bimodule E, its dual space E* carries a
natural Banach 2l-bimodule structure via (x,a· ¢) := (x· a,¢)
(x,¢· a) := (a· x,¢)
(a E !2l, ¢ E E*, x E E).
We call such Banach !2l-bimodules dual. In his seminal memoir [Joh 1], B. E. Johnson characterized the amenable locally compact groups G through Hochschild cohomology groups of Ll(G) with
coefficients in dual Banach £l(G)-bimodules ([Joh 1, Theorem 2.5]): THEOREM 2.2 (B. E. Johnson). Let G be a locally compact group. Then G is amenable if and only if'Hl(Ll(G),E*) = {O} for each Banach
Ll(G)-bimodule E. The relevance of Theorem 2.2 is twofold: First of all, homological algebra is a large and powerful toolkit - the fact that a certain property is cohomological in nature allows to
apply its tools, which then yield further insights. Secondly, the cohomological triviality condition in Theorem 2.2 makes sense for every Banach algebra. This motivates the following definition from
[Joh 1]: DEFINITION 2.3. A Banach algebra 2l is called amenable if 'Hl(!2l, E*) = {O} for each Banach 2l-bimodule E. Given a new definition, the question of how significant it is arises naturally.
Without going into the details and even without defining what a nuclear C* -algebra is, we would like to only mention the following very deep result which is very much a collective accomplishment of
many mathematicians, among them A. Connes, M. D. Choi, E. G. Effros, U. Haagerup, E. C. Lance, and S. Wassermann: THEOREM 2.4. A C* -algebra is amenable if and only if it is nuclear. For a relatively
self-contained exposition of the proof, see [Run, Chapter 6]. Of course, Definition 2.3 allows for modifications by replacing the class of all dual Banach 2l-bimodules by any other class. In [B-C-D],
W. G. Bade, P. C. Curtis, Jr., and H. G. Dales called a commutative Banach algebra !2l weakly amenable if and only if 'HI (2l, E) = {O} for every symmetric Banach !2l-bimodule E, i.e. satisfying (a E
!2l, x E E). a·x=x·a
This definition is of little use for non-commutative 21. For commutative 21, weak amenability, however, is equivalent to 'Jtl(21, 21*) = {O} ([B-C-D, Theorem 1.5]), and in [Joh 2], Johnson suggested
that this should be used to define weak amenability for arbitrary 21: DEFINITION 2.5. A Banach algebra 21 is called weakly amenable if 'Jtl(21, 21*) =
{O}. REMARK 2.6. There is also the notion of a weakly amenable, locally compact group ([C-H]). This coincidence of terminology, however, is purely accidental. In contrast to Theorem 2.2, we have:
THEOREM 2.7 ([Joh 3]). Let G be a locally compact group. Then £l(G) is weakly amenable. For a particularly simple proof of this result, see [D-Gh]. For M(G), things are strikingly different: THEOREM
2.8 ([D-Gh-H]). Let G be a locally compact group. Then M(G) is weakly amenable if and only if G is discrete. In particular, M (G) is amenable if and only if G is discrete and amenable. Sometime after
Kamowitz's pioneering paper, several mathematicians started to systematically develop a homological algebra with functional analytic overtones. Besides Johnson, who followed Hochschild's original
approach, there were A. Guichardet ([Gui]), whose point of view was homological rather than cohomological, and J. A. Taylor ([Tay]) and - most persistently - A. Ya. Helemskil and his Moscow school,
whose approaches used projective or injective resolutions; Helemskil's development of homological algebra for Banach and more general topological algebras is expounded in the monograph [He} 2]. In
homological algebra, the notions of projective, injective, and flat modules play a pivotal role. Each of these concepts tranlates into the functional analytic context. Helemskil calls a Banach
algebra 21 biprojective (respectively biflat) if it is a projective (respetively flat) Banach 21-bimodule over itself. We do not attempt to give the fairly technical definitions of a projective or a
flat Banach 21-bimodule. Fortunately, there are equivalent, but more elementary characterizations of biprojectivity and biflatness, respectively. We use ®-y to denote the completed projective tensor
product of Banach spaces. If 21 is a Banach algebra, then 21 ®-y 21 has a natural Banach 21-bimodule structure via a·(x®y):=ax®y and (x®y)·a=:x®ya (a, x, y E 21). This turns the multiplication
A: 21 ®-y 21
a ® b f-+ ab
into a homomorphism of Banach 21-bimodules. DEFINITION 2.9. Let 21 be a Banach algebra. Then: (a) 21 is called biprojective if and only if A has bounded right inverse which is an 21-bimodule
homomorphism. (b) 21 is called biftat if and only if A * has bounded left inverse which is an 21-bimodule homomorphism.
Clearly, biflatness is a property weaker than biprojectivity. The following theorem holds ([Hell, Theorem 51]): THEOREM 2.10 (A. Ya. Helemskil). Let G be a locally compact group. Then Ll (G) is
biprojective if and only if G is compact. Again, a classical property of G is equivalent to a cohomological property of Ll(G). The question for which locally compact groups G the Banach algebra Ll(G)
is biflat seems natural at the first glance. However, any Banach algebra is amenable if and only if it is biflat and has a bounded approximate identity ([Hel 2, Theorem Vii.2.20]). Since Ll(G) has a
bounded approximate identity for any G, this means that Ll (G) is biflat precisely when G is amenable. Let G be a locally compact group. A unitary representation of G on a Hilbert space jj is a group
homomorphism 7r from G into the unitary operators on jj which is continuous with respect to the given topology on G and the strong operator topology on B(jj). A function G-+C,
~, TJ
E jj is called a coefficient function of 7r.
EXAMPLE 2.11. The left regular representation A of G on L2(G) is given by A(X)~ := LX-l~
(x E G, ~ E L2(G)).
DEFINITION 2.12 ([Eym]). Let G be a locally compact group. (a) The Fourier algebra A(G) of G is defined as A(G) := {f: G
C : f is a coefficient function of A}.
(b) The Fourier-Stieltjes algebra B( G) of G is defined as B( G) := {f: G
C : f is a coefficient function of a unitary representation of G}.
It is immediate that A(G) c B(G), that B(G) consists of bounded continuous functions, and that A(G) C Co(G). However, it is not obvious that A(G) and B(G) are linear spaces, let alone algebras.
Nevertheless, the following are true ([Eym]): • Let C*(G) be the enveloping C*-algebra of the Banach *-algebra Ll(G). Then B(G) can be canonically identified with C*(G)*. This turns B(G) into a
commutative Banach algebra. • Let VN(G) := A(G)" denote the group von Neumann algebra of G. Then A(G) can be canonically identified with the unique predual of VN(G). This turns A( G) into a
commutative Banach algebra whose character space is G. • A(G) is a closed ideal in B(G). If G is an abelian group with dual group r, then the Fourier and FourierStieltjes transform, respectively,
yield isometric isomorphisms A( G) ~ Ll (r) and B(G) ~ M(r). Consequently, A(G) is amenable for any abelian locally compact group G. It doesn't require much extra effort to see that A(G) is also
amenable if G has an abelian subgroup of finite index ([L-L-W, Theorem 4.1] and [For 2, Theorem 2.2]). On the other hand, every amenable Banach algebra has a bounded approximate identity, and hence
Leptin's theorem ([Lep]) implies that the amenability of A(G) forces G to be amenable. Nevertheless, the tempting conjecture that A( G) is amenable if and only if G is amenable is false:
THEOREM 2.13 ([Joh 4]). The Fourier algebra of SO(3) is not amenable. This leaves the following intriguing open question: QUESTION 2.14. Which are the locally compact groups G for which A(G) is
amenable? The only groups G for which A( G) is known to be amenable are those with an abelian subgroup of finite index. It is a plausible conjecture that these are indeed the only ones. The
corresponding question for weak amenability is open as well. B. E. Forrest has shown that A( G) is weakly amenable whenever the principal component of G is abelian ([For 2, Theorem 2.4]). One can, of
course, ask the same question(s) for the Fourier-Stieltjes algebra: QUESTION 2.15. Which are the locally compact groups G for which B(G) is amenable? Here, the natural conjecture is that those groups
are precisely those with a compact, abelian subgroup of finite index. Since A( G) is a complemented ideal in B( G), the hereditary properties of amenability for Banach algebras ([Run, Theorem 2.3.7])
yield that A( G) has to be amenable whenever B( G) is. It is easy to see that, if the conjectured answer to Question 2.14 is true, then so is the one to Question 2.15. Partial answers to both
Question 2.14 and Question 2.15 can be found in [L-L-W] and [For 2]. 3. and operator spaces Given any linear space E and n E N, we denote the n x n-matrices with entries from E by Mn(E); if E = C, we
simply write Mn. Clearly, formal matrix multiplication turns Mn(E) into an Mn-bimodule. Identifying Mn with the bounded linear operators on n-dimensional Hilbert space, we equip Mn with a norm, which
we denote by I· I· DEFINITION 3.1. An operator space is a linear space E with a complete norm II· lin on Mn(E) for each n E N such that (R 1) II
~ I~
= max{llxll n, IIYllm}
(n, mEN, x E Mn(E), Y E Mm(E))
(R 2) EXAMPLE 3.2. Let fJ be a Hilbert space. The unique C*-norms on Mn(13(SJ)) 13(fJn) turn 13(SJ) and any of its subspaces into operator spaces.
Given two linear spaces E and F, a linear map T: E --+ F, and n E N, we define the the n-th amplification T(n) : Mn(E) --+ Mn(F) by applying T to each matrix entry. DEFINITION 3.3. Let E and F be
operator spaces, and let T E 13(E, F). Then: (a) T is completely bounded if IITllcb := sup nEN
IIT(n) IIB(Mn(E),Mn(F))
(b) T is a complete contraction if IITlicb ~ 1. (c ) T is a complete isometry if T( n) is an isometry for each n EN. The following theorem due to Z.-J. Ruan marks the beginning of abstract operator
space theory: THEOREM 3.4 ([Rna 1]). Let E be an operator space. Then there is a Hilbert space Sj and a complete isometry from E into B(Sj). To appreciate Theorem 3.4, one should think of it as the
operator space analogue of the elementary fact that every Banach space is isometrically isomorphic to a closed subspace of C(O) for some compact Hausdorff space O. One could thus define a Banach
space as a closed subspace of C(O) some compact Hausdorff space O. With this definition, however, even checking, e.g., that £1 is a Banach space or that quotients and dual spaces of Banach spaces are
again Banach spaces is difficult if not imposssible. Since any C* -algebra can be represented on a Hilbert space, each Banach space E can be isometrically embedded into B(Sj) for some Hilbert space
Sj. For an operator space, it is not important that, but how it sits inside B(Sj). There is one monograph devoted to the theory of operator spaces ([E-R]) as well as an online survey article ([Wit et
al.]). The notions of complete boundedness as well as of complete contractivity can be defined for multilinear maps as well ([E-R, p. 126]). Since this is somewhat more technical than Definition 3.3,
we won't give the details here. As in the category of Banach spaces, there is a universallinearizer for the right, i.e. completely bounded, bilinear maps: the projective operator space tensor product
([E-R, Section 7.1]), which we denote by ®. DEFINITION 3.5. An operator space 2l which is also an algebra is called a completely contractive Banach algebra if multiplication on 2l is a complete
(bilinear) contraction. The universal property of ® ([E-R, Proposition 7.1.2]) yields that, for a completely contractive Banach algebra 2l, the multiplication induces a complete (linear) contraction
~: 2l®2l --+ 2l. EXAMPLE 3.6. (1) For any Banach space E, there is an operator space maxE such that, for any other operator space F, every T E B(E,F) is completely bounded with IITlicb = IITII ([E-R,
pp. 47-54]). Given a Banach algebra 2l, the operator space max2l is a completely contractive Banach algebra ([E-R, p. 316]). (2) Any closed subalgebra of B(Sj) for some Hilbert space Sj is a
completely contractive Banach algebra. To obtain more, more interesting, and - in the context of abstract harmonic analysis - more relevant examples, we require some more operator space theory. Given
two operator spaces E and F, let
CB(E, F) := {T: E
F : T is completely bounded}.
It is easy to check that CB(E, F) equipped with 11·llcb is a Banach space. To define an operator space structure on CB(E, F), first note that Mn(F) is, for each n E N, an operator space in a
canonical manner. The (purely algebraic) identification
Mn(CB(E, F)) := CB(E, Mn(F))
(n E N)
then yields norms 1I·lln on the spaces Mn(CB(E, F)) that satisfy (R 1) and (R 2), which is not hard to verify. Since, for any operator space E, the Banach spaces E* and CB(E, C) are isometrically
isomorphic ([E-R, Corollary 2.2.3]), this yields a canonical operator space structure on the dual Banach space of an operator space. In partiuclar, the unique predual of a von Neumann algebra is an
operator space in a canonical way. We shall see how this yields further examples of completely contractive Banach algebras. We denote the W* -tensor product by ®. DEFINITION 3.7. A Hop/-von Neumann
algebra is a pair (rot, V), where rot is a von Neumann algebra, and V is a co-multiplication: a unital, injective, w*continuous *-homomorphism V: rot --+ rot®rot which is co-associative, i.e. the
diagram rot
vl rot®rot
commutes. EXAMPLE 3.8. Let G be a locally compact group. (1) Define V: £oo(G) --+ £oo(G x G) by letting
(V»(xy) := >(xy)
(> E £oo(G), x, y E G).
Since £oo(G)®£oo(G) ~ £oo(G x G), this turns £oo(G) into a Hopf-von Neumann algebra. (2) Let W*(G) be the enveloping von Neumann algebra of C*(G). There is a canonical w* -continuous homomorphism w
from G into the unitaries of W* (G) with the following universal property: For any unitary representation 7r of G on a Hilbert space, there is unique w* -continuous *homomorphism (J: W*(G) --+ 7r(G)"
such that 7r = (J 0 w. Applying this universal property to the representation G
yields a co-multiplication V: W*(G)
w(x) ® w(x) W*(G)®W*(G).
Given two von Neumann algebras rot and 1)1 with preduals rot* and 1)1*, their W*-tensor product rot®1)1 also has a unique predual (rot®I)1)*. Operator space theory allows to identify this predual in
terms of rot* and 1)1* ([E-R, Theorem 7.2.4]): (rot®I)1)* ~ rot*ci~m*. Since VN(G)® VN(H) ~ VN(G x H) for any locally compact groups G and H, this implies in particular that A(G x H) ~ A(G)®A(H).
Suppose now that rot is a Hopf-von Neumann algebra with predual rot*. The comultiplication V : rot --+ rot®rot is w· -continuous and thus the adjoint map of a complete contraction V. : rot*®rot. --+
rot.. This turns rot. into a completely contractive Banach algebra. In view of Example 3.8, we have:
EXAMPLE 3.9. Let G be a locally compact group. (1) The multiplication on L1(G) induced by \7 as in Example 3.8.1 is just the usual convolution product. Hence, L1 (G) is a completely contractive
Banach algebra. (2) The multiplication on B( G) induced by \7 as in Example 3.8.2 is pointwise multiplication, so that B( G) is a completely contractive Banach algebra. Since A (G) is an ideal in B
(G) and since the operator space strucures A ( G) has as the predual of VN(G) and as a subspce of B(G) coincide, A(G) with its canonical operator space structure is also a completely contractive
Banach algebra. REMARK 3.10. Since A(G) fails to be Arens regular for any non-discrete or infinite, amenable, locally compact group G ([For 1]), it cannot be a subalgebra of the Arens regular Banach
algebra B(f)). Hence, for those groups, A(G) is not of the form described in Example 3.6.2. We now return to homological algebra and its applications to abstract harmonic analysis. An operator
bimodule over a completely contractive Banach algebra Il is an operator space E which is also an !!-bimodule such that the module actions of Il on E are completely bounded bilinear maps. One can then
define operator Hochschild cohomology groups 01t 1 (!!, E) by considering only completely bounded derivations (all inner derivations are automatically completely bounded). It is routine to check that
the dual space of an operator !!-bimodule is again an operator Il-bimodule, so that the following definition makes sense: DEFINITION 3.11 ([Rua 2]). A completely contractive Banach algebra Il is
called operator amenable if 01t 1 (Il,E*) = {O} for each operator Il-bimodule E. The following result ([Rua 2, Theorem 3.6]) shows that Definition 3.11 is indeed a good one: THEOREM 3.12 (Z.-J.
Ruan). Let G be a locally compact group. Then G is amenable if and only if A( G) is operator amenable. REMARK 3.13. A Banach algebra!! is amenable if and only if max!! is operator amenable ([E-R,
Proposition 16.1.5]). Since L1(G) is the predual of the abelian von Neumann algebra LOO(G), the canonical operator space structure on L1(G) is maxL1(G). Hence, Definition 3.11 yields no information
on L1(G) beyond Theorem 2.2. The following is an open problem: QUESTION 3.14. Which are the locally compact groups G for which B(G) is operator amenable? With Theorem 2.8 and the abelian case in
mind, it is reasonable to conjecture that B(G) is operator amenable if and only if G is compact. One direction is obvious in the light of Theorem 3.12; a partial result towards the converse is given
in [R-Sp]. Adding operator space overtones to Definition 2.5, we define: DEFINITION 3.15. A completely contractive Banach algebra Il is called operator weakly amenable if 01t 1 (!!, Il *) = {O}.
In analogy with Theorem 2.7, we have: THEOREM 3.16 ([Spr]). Let G be a locally compact group. Then A(G) is operator weakly amenable.
One can translate Helemskir's homological algebra for Banach algebras relatively painlessly to the operator space setting: This is done to some extent in [Ari] and [Woo 1]. Of course, appropriate
notions of projectivity and flatness play an important role in this operator space homological algebra. Operator biprojectivity and biflatness can be defined as in the classical setting, and an
analogue - with ® instead of Q$)-y - of the characterization used for Definition 2.9 holds. The operator counterpart of Theorem 2.10 was discovered, independently, by O. Yu. Aristov and P. J. Wood:
THEOREM 3.17 ([Ari], [Woo 2]). Let G be a locally compact group. Then G is discrete if and only if A(G) is operator biprojective.
As in the classical setting, both operator amenability and operator biprojectivity imply operator biflatness. Hence, Theorem 3.17 immediately supplies examples of locally compact groups G for which A
( G) is operator biflat, but not operator amenable. A locally compact group is called a [SIN]-group if Ll(G) has a bounded approximate identity belonging to its center. By [R-X, Corollary 4.5], A(G)
is also operator biflat whenever G is a [SIN]-group. It may be that A( G) is operator biflat for every locally compact group G: this question is investigated in [A-R-Sp]. All these results suggest
that in order to get a proper understanding of the Fourier algebra and of how its cohomological properties relate to the underlying group, one has to take its canonical operator space structure into
References O. Yu. Aristov, Biprojective algebras and operator spaces. J. Math. Sci. (to appear). O. Yu. Aristov, V. Runde, and N. Spronk. Operator biflatness of the Fourier algebra. In preparation.
[B-C-D] W. G. Bade, P. C. Curtis, Jr., and H. G. Dales, Amenability and weak amenability for Beurling and Lipschitz algebras. Proc. London Math. Soc. (3) 55 (1987), 359-377. [C-E] H. Cartan and S.
Eilenberg, Homological algebra. Princeton University Press, Princeton, 1956. M. Cowling and U. Haagerup, Completely bounded multipliers of the Fourier algebra [C-H] of a simple Lie group of real rank
one. Invent. Math. 96 (1989), 507-549. [D-Gh-H] H. G. Dales, F. Ghahramani, and A. Va. HelemskiY, The amenability of measure algebras. J. London Math. Soc. 66 (2002), 213-226. M. M. Day, Means on
semigroups and groups. Bull. Amer. Math. Soc. 55 (1949), [Day] 1054-1055. [D-Gh] M. Despic and F. Ghahramani, Weak amenability of group algebras of locally compact groups. Ganad. Math. Bull. 37
(1994), 165-167. E. G. Effros and Z.-J. Ruan, Operator spaces. Clarendon Press, Oxford, 2000. [E-R] [Eym] P. Eymard, L'algebre de Fourier d'un groupe localement compact. Bull. Soc. Math. France 92
(1964), 181-236. G. B. Folland, A course in abstract harmonic analysis. CRC Press, Boca Raton, [Fol] Florida, 1995. B. E. Forrest, Arens regularity and discrete groups. Pacific J. Math. 151 (1991),
[For 1] 217-227. B. E. Forrest, Amenability and weak amenability of the Fourier algebra. Preprint [For 2] (2000). [Gre] F. P. Greenleaf, Invariant means on locally compact groups. Van Nostrand, New
York-Toronto-London, 1969. [Ari] [A-R-Sp]
A. Guichardet, Sur I'homologie et la cohomologie des algebres de Banach. C. R. Acad. Sci. Paris, Ser. A 262 (1966), 38-42. [Her] C. S. Herz, Harmonic synthesis for subgruops. Ann. Inst. Fourier
(Grenoble) 23 (1973),91-123. [H-R] E. Hewitt and K. A. Ross, Abstract harmonic analysis, I and II. Springer Verlag, Berlin-Heideberg-New York, 1963 and 1970. [Hell] A. Ya. Helemskil', Flat Banach
modules and amenable algebras. Trans. Moscow Math. Soc. 47 (1985), 199-224. [HeI2] A. Ya. Helemskil, The homology of banach and topological algebras (translated from the Russian). Kluwer Academic
Publishers, Dordrecht, 1989. G. Hochschild, On the cohomology groups of an associative algebra. Ann. of Math. [Hoch 1] (2) 46 (1945), 58-67. [Hoch 2] G. Hochschild, On the cohomology theory for
associative algebras. Ann. of Math. (2) 47 (1946), 568-579. B. E. Johnson, Cohomology in Banach algebras. Mem. Amer. Math. Soc. 127 (1972). [Joh 1] B. E. Johnson, Derivations from Ll(G) into Ll(G)
and LOO(G). In: J. P. Pier (ed.), [Joh 2] Harmonic analysis (Luxembourg, 1987), pp. 191-198. Lectures Notes in Mathematics 1359. Springer Verlag, Berlin-Heidelberg-New York, 1988. [Joh 3] B. E.
Johnson, Weak amenability of group algebras. Bull. London Math. Soc. 23 (1991),281-284. [Joh 4] B. E. JOHNSON, Non-amenability of the Fourier algebra of a compact group. J. London Math. Soc. (2) 50
(1994),361-374. [Kam] H. Kamowitz, Cohomology groups of commutative Banach algebras. Trans. Amer. Math. Soc. 102 (1962), 352-372. [L-L-W] A. T.-M. Lau, R. J. Loy, and G. A. Willis, Amenability of
Banach and C"-algebras on locally compact groups. Studia Math. 119 (1996), 161-178. H. Leptin, Sur l'algebre de Fourier d'un groupe localement compact. C. R. Acad. Sci. [Lep] Paris, Ser. A 266
(1968), 1180-1182. S. MacLane, Homology. Springer Verlag, Berlin-Heidelberg-New York, 1995. [MacL] J. von Neumann, Zur allgemeinen Theorie des MaBes. Fund. Math. 13 (1929), 73-116. [Neu] A. L. T.
Paterson, Amenability. American Mathematical Society, Providence, 1988. [Pat] J. P. Pier, Amenable locally compact groups. Wiley-Interscience, New York, 1984. [Pie] H. Reiter and J. D. Stegeman,
Classical harmonic analysis and locally compact [R-St] groups. Clarendon Press, Oxford, 2000. [Rua 1] Z.-J. Ruan, Subspaces of C"-algebras. J. Funct. Anal. 76 (1988), 217-230. [Rua2] Z.-J. Ruan, The
operator amenability of A(G). Amer. J. Math. 117 (1995), 14491474. Z.-J. Ruan and G. Xu, Splitting properties of operator bimodules and operator ame[R-X] nability of Kac algebras. In: A. Gheondea, R.
N. Gologan and D. Timotin, Operator theory, operator algebras, and related topics, pp. 193-216. The Theta Foundation, Bucharest, 1997. v. Runde, Lectures on amenability. Lecture Notes in Mathematics
1774. Springer [Run] Verlag, Berlin-Heidelberg-New York, 2002. V. Runde and N. Spronk, Operator amenability of Fourier-Stieltjes algebras. Preprint [R-Sp] (2001). N. Spronk, Operator weak amenability
of the Fourier algebra. Proc. Amer. Math. [Spr] Soc. 130 (2002), 3609-3617. [Tay] J. A. Taylor, Homology and cohomology for topological algebras. Adv. in Math. 9 (1970), 137-182. [Wei] C. A. Weibel,
An introduction to homological algebra. Cambridge University Press, Cambridge, 1994. [Wit et al.] G. Wittstock et al., What are operator spaces? - An online dictionary. URL: http://wwv.math.uni-sb.de
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operator biprojectivity of the Fourier algebra. Canadian .1. Math. [Woo 2] (to appear).
T6G 2Gl
E-mail address:vrundeClualberta.ca
Contemporary Mathematics Volume 328, 2003
Relative Tensor Products for Modules over von Neumann Algebras David Sherman ABSTRACT. We give an overview of relative tensor products (RTPs) for von
Neumann algebra modules. For background, we start with the categorical definition and go on to examine its algebraic formulation, which is applied to Morita equivalence and index. Then we consider
the analytic construction, with particular emphasis on explaining why the RTP is not generally defined for every pair of vectors. We also look at recent work justifying a representation of RTPs as
composition of unbounded operators, noting that these ideas work equally well for LP modules. Finally, we prove some new results characterizing preclosedness of the map (~, 7) f-> ~ 181",7).
1. Introduction
The purpose of this article is to summarize and explore some of the various constructions of the relative tensor product (RTP) of von Neumann algebra modules. Alternately known as composition or
fusion, RTPs are a key tool in subfactor theory and the study of Morita equivalence. The idea is this: given a von Neumann algebra M, we want a map which associates a vector space to certain pairs of
a right M-module and a left M-module. If we write module actions with subscripts, we have (XM,M!i)) f-> X 0M!i). This should be functorial, covariant in both variables, and appropriately normalized.
Other than this, we only need to specify which modules and spaces we are considering. In spirit, RTPs are algebraic; a ring-theoretic definition can be found in most algebra textbooks. But in the
context of operator algebras, the requirement that the output be a certain type of space - typically a Hilbert space - causes an analytic obstruction. As a consequence, there are domain issues in any
vector-based construction. Fortunately, von Neumann algebras have a sufficiently simple representation theory to allow a recasting of RTPs in algebraic terms. The analytic study of RTPs can be
related nicely to noncommutative £P spaces. Indeed, examination of the usual (£2) case reveals that the technical difficulties 2000 Mathematics Subject Classification. Primary: 46LIO; Secondary:
46M05. Key words and phrases. relative tensor product, von Neumann algebra, bimodule. © 275
2003 American Mathematical Society
come from a "change of density". (We say that the density of an LP-type space is lip.) Once this is understood, it is easy to handle LP modules [JS] as well. Modular algebras ([Y], [S]) provide an
elegant framework, so we briefly explain their meaning. The final section of the paper investigates the question, "When is the map (~, 1]) f--+ ~ ®
2. Notations and background The basic objects of this paper are von Neumann algebras, always denoted here by M, N, or P. These can be defined in many equivalent ways: • C*-algebras which are dual
spaces. • strongly-closed unital *-subalgebras of B(i)). B(i)) is the set of bounded linear operators on a Hilbert space i); the strong topology is generated by the seminorms x f--+ Ilx~ll, ~ E i);
the * operation is given by the operator adjoint. • *-closed subsets of B(i)) which equal their double (iterated) commutant. The commutant of a set S c B(i)) is {x E B(i)) I xy = yx, 'l:/y E S}. As
one might guess from the definitions, the study of von Neumann algebras turns on the interplay between algebraic and analytic techniques. Finite-dimensional von Neumann algebras are direct sums of
full matrix algebras. At the other extreme, commutative von Neumann algebras are all of the form Loo(X, J-l) for some measure space (X, J-l)' so the study of general von Neumann algebras is
considered "noncommutative measure theory." Based on this analogy, the (unique) predual M* of M is called Ll(M); it is the set of normal (= continuous in yet another topology, the a-weak) linear
functionals on M c B(i)), and can be thought of as "noncommutative countably additive measures". A functional r.p is positive when x> 0 =} r.p(x) 2: 0; the set of positive normal functionals is
denoted M;t. The support s( r.p) of a positive normal linear functional r.p is the smallest projection q E M with r.p(l - q) = O. So if M is abelian, r.p corresponds to a measure and q is the
(indicator function of the) usual support. For simplicity, all modules in this paper are separable Hilbert spaces (except in Section 6), all algebras have separable predual, all linear functionals
are normal, and all representations are normal and nondegenerate (MSJ or SJM is all of SJ). Two projections p, q in a von Neumann algebra are said to be (Murray-von Neumann) equivalent if there
exists v E M with v*v = p, vv* = q. Such an element v is called a partial isometry, and we think of p and q as being "the same size". Subscripts are used to represent actions, so XM indicates that X
is a right M-module, i.e. a representation of the opposite algebra MOP. It is implicit in the term "bimodule", or in the notation Mi)N, that the two actions commute. The phrase "left (resp. right)
action of' is frequently abbreviated to L (resp. R) for operators or entire algebras, so that we speak of L(x) or R(M). Finally, we often write Moo for the von Neumann algebra of all bounded
operators on a separable infinite-dimensional
Hilbert space, and Moc(M) for the von Neumann tensor product Moc®M. One can think of this as the set of infinite matrices with entries in M; we will denote by eij the matrix unit with 1 in the ij
position and 0 elsewhere. The (left) representation theory of von Neumann algebras on Hilbert spaces is simple, so we recall it briefly. (Most of this development can be found in Chapters 1 and 2 of
[JoS].) First, there is a standard construction, due to Gelfand-Neumark and Segal (abbreviated GNS), for building a representation from 'P E Mt. To each x E M we formally associate the vector x'P 1/
2 (various notations are in use, e.g. 7]",(x) or A",(.'1:), but this one is especially appropriate ([C2] V.App.B, [S])). We endow this set with the inner product
< x'P1/2, Y'P 1/ 2 >= 'P(Y* x), and set fJ", to be the closure in the inherited topology, modulo the null space. The left action of M on fJ", = M'P1/2 is bounded and densely defined by left
composition. When 'P is faithful (meaning x > 0 => 'P(x) > 0), the vector 'P 1/ 2 = I'P1/2 is cyclic (M'P1/2 = fJ",) and separating (x =f. 0 => X'P1/2 =f. 0). Now all representations with a cyclic
and separating vector are isomorphic - a sort of "left regular representation"; we will denote this by ML 2 (M). It is a fundamental fact that the commutant of this action is antiisomorphic to M, and
when we make this identification we call ML2(M)M the standard form of M. If 'P is not faithful, the GNS construction produces a vector 'Pl/2 which is cyclic but not separating, and a representation
which is isomorphic to ML2(M)s('P) ([T2], Ch. VIII, IX). Now let us examine an arbitrary (separable, so countably generated) module MfJ. Following standard arguments (e.g. [TI] I.9), fJ decomposes
into a direct sum of cyclic representations M(M~n), each of which is isomorphic to the GNS representation for the associated vector state w~n (=< ·~n, ~n ». With qn = s(w~J, we have MfJ ~ EBMM~n ~
EBMfJw~n ~ EB M L 2(M)qn. (Here and elsewhere, "~" means a unitary equivalence of (bi)modules.) Since this is a left module, it is natural to write vectors as rows with the nth entry in L 2(M)qn:
We will call such a decomposition a row representation of MfJ. Here e nn are diagonal matrix units in Moo, so (Eqn ®e nn ) is a diagonal projection in Moc(M). The left action of M is, of course,
matrix multiplication (by 1 x 1 matrices) on the left. The module (L2(M)L2(M) ... ) will be denoted R2(M) (for "row"). Since the standard form behaves naturally with respect to restriction - L2(q,Nq)
~ qL 2(,N)q as bimodules - it follows that L2(Moo(M)) is built as infinite matrices over L2(M) (see (3.3)). Thus R2(M) can be realized as ellL 2(Moo (M)). PROPOSITION 2.1. Any countably generated
left representation of M on a Hilbert space is isomorphic to R 2(M)q for some diagonal projection q E Moc(M). Any projection ,in Moc(M), diagonal or not, defines a module in this way, and two such
modules are isomorphic exactly when the projections are equivalent. In fact
So isomorphism classes correspond to equivalence classes of projections in lVIoo(M), which is the monoid V(Moo(M)) in K -theoretic language [W-O}. The direct sum of isomorphism ciasses of modules
corresponds to the sum of orthogonal representatives in V(Moo(M)), giving a monoidal equivalence.
We denote the category of separable left M-modules by Left L 2(M). For us, the most important consequence of (2.2) is that (2.3)
C.(MR2(M)q) = R(qMoo(M)q),
where "c." stands for the commutant of the M-action. (In particular, the case = ell is just the standard form.) The algebra qMoo(M)q is called an amplification of M, being a generalization of a
matrix algebra with entries in M. Of course everything above can be done for right modules - the relevant abbreviations are C 2(M), for "column," and Right L 2(M). Example. Suppose M = M3(C). In this
case the standard form may be taken as M3L2(M3hf3; L2(M3) ~ (M3, < .,. », where < x, y >= Tr(y*x). Note that this norm, called the Hilbert-Schmidt norm, is just the e2 norm of the matrix entries, and
that the left and right multiplicative actions are commutants. (If we had chosen a nontracial positive linear functional, we would have obtained an isomorphic bimodule with a "twisted" right action
... this is inchoate Tomita-Takesaki theory.) The module R2(M3) is M 3xoo , again with the Hilbert-Schmidt norm, and the commutant is Moo(M3) ~ Moo. According to Proposition 2.1, isomorphism classes
of left M 3-modules should be parameterized by equivalence classes of projections in Moo. These are indexed by their rank n E (1:+Uoo); the corresponding isomorphism class of modules has
representative M 3xn . In summary, we have learned that any left representation of M3 on a Hilbert space is isomorphic to some number of copies of C 3. The same argument shows that V(Moo(Mk)) ~ (1:+
U 00) for any k. Properties of the monoid V ( Moo (M)) determine the so-called type of the algebra. For a factor (a von Neumann algebra whose center is just the scalars), there are only three
possibilities: (1:+ U 00), (lR.+ U 00), and {O,+oo}. These are called types I, II, III, respectively; a fuller discussion is given in Section 7. q
3. Algebraic approaches to RTPs When R is a ring, the algebraic R-relative tensor product is the functor, covariant in both variables, which maps a right R-module A and left R-module B to the vector
space (A ®alg B)/N, where N is the subspace generated algebraically by tensors of the form ar ® b - a ® rb. In functional analysis, where spaces are usually normed and infinite-dimensional, one
obvious amendment is to replace vector spaces with their closures. But in the context of Hilbert modules over a von Neumann algebra M, this is still not enough. Surprisingly, a result of Falcone
([FJ, Theorem 3.8) shows that if the RTP L2(M) ®M L2(M) is the closure of a continuous (meaning III(~ ® 7])11 < ClI~IIII7]11) nondegenerate image ofthe algebraic M-relative tensor product, M must be
atomic, Le. M ~ EBnB(f.>n). We will discuss the analytic obstruction further in Section 5. For now, we take Falcone's theorem as a directive: do not look for a map which is defined for every pair of
vectors. If we give up completely on a vector-level construction, we can at least make the functorial
DEFINITION 3.1 (Sa). Given a von Neumann algebra M, a relative tensor product is a junctor, covariant in both variables,
RightL 2(M) x LeJtL2(M)
n ®M.ft,
which satisfies (3.2)
as bimodules. Although at first glance this definition seems broad, in fact we see in the next proposition that there is exactly one RTP functor (up to equivalence) for each algebra. The reader is
reminded that functoriality implies a mapping of intertwiner spaces as well, so it is enough to specify the map on representatives of each isomorphism class. In particular we have the bimodule
structure '£:(j) .••ll(n
®M .ft)C(M.ll)·
PROPOSITION 3.2. Let n ~ P C 2(M) E Right L2 M od(M) and.ft ~ R 2(M)q E Left L2Mod(M) Jor some projections p,q E Moo(M). Then
n ®M.ft ~ P L2(Moo(M))q with natural action oj the commutants. PROOF. By implementing an isomorphism, we may assume that the projections are diagonal: p = LPi ®eii, q = Lqj ®ejj. Using (3.2) and
functoriality, we have the bimodule isomorphisms
n ®M.ft ~ (EBPi L2(M)) ®M (EBL2(M)qj) ~
E9Pi L2(M) ®M L2(M)qj ~ E9Pi L2(M)qj ~ p L2(Moo(M))q. i,j
i,j D
Visually, (3.3) where of course the £2 sums of the norms of the entries in these matrices are finite. After making the categorical definition above, Sauvageot immediately noted that it gives us no
way to perform computations. We will turn to his analytic construction in Section 5; here we discuss an approach to bimodules and RTPs due to Connes. In his terminology a bimodule is called a
correspondence. (The best references known to the author are [C2l and [Pl, but there was an earlier unpublished manuscript which is truly the source of Connes fusion.) Consider a correspondence MnN.
Choosing a row representation R2(M)q for n, we know that the full commutant of L(M) is isomorphic to R(qMoo(M)q). This gives us a unital injective *-homomorphism p : N '---+ qMoo(M)q, and from the
map p one can reconstruct the original bimodule (up to isomorphism) as M(R 2(M)q)p(N)'
What if we had chosen a different row representation R 2 (M)q' and obtained P' : N ---+ q'Moo(M)q'? By Proposition 2.1, the module isomorphism MR 2(M)q ~ MR 2(M)q'
is necessarily given by the right action of a partial isometry v between q and q' in Moo(M). Then P and P' differ by an inner perturbation: p(x) = v*p'(x)v. We conclude that the class of M - N
correspondences, modulo isomorphism, is equivalent to the class of unital injective *-homomorphisms from N into an amplification of M, modulo inner perturbation. (These last are called sectors in
subfactor theory.) The distinctions between bimodules, morphisms, and their appropriate equivalence classes are frequently blurred in the literature; our convention here is to use the term
"correspondence" to mean a representative *-homomorphism for a bimodule. Notice that a unital inclusion N eM gives the bimodule ML 2(M)N. The RTP of correspondences is extremely simple. PROPOSITION
3.3. Consider bimodules MYJN and N.ftP coming from correspondences PI : N '----> qMoo(M)q and P2 : P '----> q'Moo(N)q'. The bimodule M(YJ®N.ft)P is the correspondence PI 0 P2, where we amplify PI
We pause to mention that it is also fruitful to realize correspondences in terms of completely positive maps. We shall have nothing to say about this approach; the reader is referred to [P] for
basics or [A2] for a recent investigation. 4. Applications to Morita equivalence and index An invertible *-functor from Left L2 M od(N) to Left L2 M od(M) is called a Morita equivalence [R]. Here a
*-functor is a functor which commutes with the adjoint operation at the level of morphisms. One way to create *-functors is the following: to the bimodule MYJN, we associate (4.1) FSj: Left L2 Mod(N)
Left L2 Mod(M);
N.ft 1---+ (MYJN) ®N (N.ft).
The next theorem is fundamental. THEOREM 4.1. When L(M) and R(N) are commutants on YJ, the RTP functor FSj is a Morita equivalence. Moreover, every Morita equivalence is equivalent to an RTP functor.
This type of result - the second statement is an operator algebraic analogue of the Eilenberg-Watts theorem - goes back to several sources, primarily the fundamental paper of Rieffel [R]. His
investigation was more general and algebraic, and his bimodules were not Hilbert spaces but rigged self-dual Hilbert C*-modules, following Paschke [Pal. From a correspondence point of view, rigged
self-dual Hilbert C*-modules and Hilbert space bimodules give the same theory; the equivalence is discussed nicely in [A1]. (And the former is nothing but an L oo version of the latter, as explained
in [JS].) Our Hilbert space approach here is parallel to that of Sauvageot [Sa], though modeled more on [R], and is streamlined by our standing assumption of separable preduals. We will need
DEFINITION 4.2. The contragredient of the bimodule MYJN is the bimodule NfJM' where fJ is conjugate linearly isomorphic to fj (the image of ~ is written (J, and the actions are defined by n~m =
LEMMA 4.3. Suppose L(M) and R(N) are commutants on fl. Then
NfJM ®M MflN ~ NL2(N)N. PROOF. If Mfl ~ MR 2(M)q, then N ~ qMoo(M)q by (2.3), and fJM qC 2 (M)M. By Proposition 3.2 and the comment preceding Proposition 2.1,
NfJM ®M MflN ~ N(qL2(Moo(M))q)N ~ NL2(qMoo(M)q)N ~ NL2(N)N. D Lemma 4.3 was first proven by Sauvageot (in another way). In our situation it means
FSj(Nfi) ~ L2(N) ®N Nfi ~ Nfi.
(Here we have used the associativity of the RTP, which is most easily seen from the explicit construction in Section 5.) We conclude that Fi) 0 FSj is equivalent to the identity functor on Left L2Mod
(N), and by a symmetric argument FSj 0 Fi) is equivalent to the identity functor on Left L2Mod(M). Thus FSj is a Morita equivalence, and the first implication of Theorem 4.1 is proved. Now let F be a
Morita equivalence from Left L2 M od(N) to Left L2 M od(M). Then F must take NR2(N) to a module isomorphic to MR2(M), because each is in the unique isomorphism class which absorbs all other modules.
(This is meant in the sense that NR2(N) ffiNfl ~ NR2(N); R2(N) is the "infinite element" in the monoid V (Moo (N)).) Being an invertible *-functor, F implements a *-isomorphism of commutants - call
it a, not F, to ease the notation: (4.2) Apparently we have (4.3) Before continuing the argument, we need an observation: isomorphic algebras have isomorphic standard forms. Specifically, we may
write L2(Moo(N)) as the GNS construction for tp E Moo(N)t and obtain the isomorphism (a-1)t : L2(Moo(N)) ..::. L2(Moo(M)), (a-1)t : xtp1/2 ........ a(x)(tp 0 a- 1)1/2. Note that (a- 1 )t(x~y) = a(x)
[(a- 1 )t(~)la(y). Now consider the RTP functor for the bimodule
MflN =
u- 1 (M)a- 1(et'{)C 2(N)N.
By Proposition 3.2 and the comment preceding Proposition 2.1, its action is
R2(N)q ........
17-1 (M)a- 1 (et'{)L2(Moo (N))q (17;:)'
Met'{ L 2(Moo (M))a(q)
~ MR 2(M)a(q) ~ F(R 2(N)q).
We conclude that F is equivalent to FSj, which finishes the proof of Theorem 4.1. Notice that (4.2) and (4.3) can also be used to define a functor; this gives us COROLLARY 4.4. For two von Neumann
algebras M and N, the following are equivalent: (1) M and N are Morita equivalent;
(2) Moo(N) ~ Moo(M); (3) there is a bimodule Mf)N where the actions are commutants of each other; (4) there is a projection q E Moo(M) with central support 1 such that
(The central support of x E M is the least projection z in the center of M satisfying x = zx.) Example continued. M3 and M5 are Morita equivalent. This can be deduced easily from condition (2) or (4)
of the corollary above. It also follows from the (Hilbert) equivalence bimodule M3HS(M3X5)Ms, where "HS" indicates the HilbertSchmidt norm; this bimodule gives us an RTP functor which is a Morita
equivalence. Regardless of the construction, the equivalence will send (functorially) n copies of C 5 to n copies of C 3 . Apparently Morita equivalence is a coarse relation on type I algebras - it
only classifies the center of the algebra (up to isomorphism). At the other extreme, Morita equivalence for type III algebras is the same as algebraic isomorphism; Morita equivalence for type II
algebras is somewhere in the middle ([RJ, Sec. 8). For a bimodule Mf)N where the algebras are not necessarily commutants, the functor (4.1) still makes sense. To get a more tractable object, we may
consider the domain and range to be isomorphism classes of modules:
71'1) :
V(Moo(N)) ---. V(Moo(M));
F1)(R 2(N)q) = Mf)N ®N R2(N)q ~ R2(M)7I'1) ([q]). We call this the bimodule morphism associated to f), a sort of "skeleton" for the correspondence. It follows from Proposition 3.3 that if the
bimodule is p : N '---+ qMoo(M)q, then 71'1) is nothing but poo, the amplification of p to Moo(N), restricted to equivalence classes of projections. This has an important application to inclusions of
algebras. We have seen in M is equivalent to a bimodule ML2(M)N. Section 3 that a unital inclusion N When the correspondence p is surjective, the module generates a Morita equivalence via its RTP
functor, and the induced bimodule morphism is an isomorphism of monoids. When N i= M, it is natural to expect that the bimodule morphism gives us a way to measure the relative size, or index, of N in
M. (For readers unfamiliar with this concept, the index of an inclusion N c M is denoted [M : N] and is analogous to the index of a subgroup. The kernel of this idea goes back to Murray and von
Neumann, but the startling results of Jones [J] in the early 1980's touched off a new wave of investigation. We recommend the exposition [K] as a nice starting point.) For algebras of type I or II,
the index can be calculated in terms of bimodule morphisms. (There are also broader definitions of index which require a conditional expectation (=norm-decreasing projection) from M ontoN.) This
amounts largely to rephrasing and extension of the paper [Jol] , and we do not give details here. Very briefly, let 71' : V(Moo(M)) ---. V(Moo(M)) be the bimodule morphism for
When M is a factor, 11" is a monoidal morphism on the extended nonnegative integers (type I) or extended nonnegative reals (type II). It must be multiplication by a scalar, and this scalar is the
index. If M is not a factor, the index is the spectral radius of 11", provided that V(Moo(M)) is endowed with some extra structure (at present it is not even a vector space). Example. Consider the
The image of M6L2(M6) under the RTP functor for (4.5) is
~ M6 L2 (M6)M3 ®M3 M3 L2 (M6) (now counting the dimensions of the Hilbert spaces) ~
M6HS(M12X3)M3 ®M3 M3 HS (M3xd ~ M6HS(M12X12) ~ M6HS(M6X24).
We have gone from 6 copies of (:6 to 24 copies; that is, 61-+24. Apparently the index is 4, which is also the ratio of the dimensions of the algebras. 5. Analytic approaches to RTPs
As mentioned in Section 3, we cannot expect the expression ~ ®M T/ to make sense for every pair of vectors ~,T/. In essence, the problem is that the product of two L2 vectors is Ll, and an Ll space
does not lie inside its corresponding L2 space unless the underlying measure is atomic. Densities add, even in the noncommutative setting, and so the product in (3.3) "should" be an Ll matrix. To
make this work at the vector level, we need to decrease the density by 1/2 without affecting the "outside" action of the commutants ... and the solution by Connes and Sauvageot ([Sa], [C2]) is almost
obvious: choose a faithful cp E Mt and put cp-l/2 in the middle of the product. That is, (5.1) This requires some explanation. Fix faithful cp E Mt and row and column representations of 5) and .It as
in (2.1). We define
D(S;, 1") ~ {
E S; ,
~>~Xn ex;"" in M}
V(5), cp) is dense in 5), and its elements are called cp-left bounded vectors [C1]. Now by (5.1) we mean the following: for ~ E V(5), cp), we simply erase the symbol cpl/2 from the right of each
entry, then carry out the multiplication. The natural domain is V(5), cp) x .It. Visually, .
( ( ::::;:) ,(" "' '" ))
~ (::::;:) (,-"') (" "' '" ) ~ (X;"»,
For cp =I- 1jJ E Mt, we cannot expect € ®cp TJ = € ®,p TJ even if both are defined, although the reader familiar with modular theory will see that (5.3)
€ ®cp TJ = (€cp-l/2)TJ = (€cp-l/21jJl/21jJ-l/2)TJ = €(Dcp : D1jJ)i/2 ®,p TJ.
(An interpretation of the symbols cpl/2, cp-l/2 as unbounded operators will be discussed in the next section.) Now we define S) ®cp .!'t to be the closed linear span of the vectors € ®cp TJ inside L2
(Moo(M)). Up to isomorphism, this is independent of cp. (We know this because of functoriality; the "change of weight" isomorphism is densely defined by (5.3).) The given definition for V(S), cp) c
S) makes it seem dependent on the choice of column representation. That this is not so can be seen by noting (as in (3.4)) that the intertwining isomorphism is given by L( v) for some partial
isometry v E Moo(M). But let us also mention a method of defining the same RTP construction without representing S) and.!'t. First notice that V(S), cp) can also be defined as the set of vectors €
for which 7rf(€) : L 2(M)M -+ S)M. cpl/2x 1-+ €x, is bounded. (A more suggestive (and rigorous) notation would be L(€cp-l/2).) Now we consider an inner product on the algebraic tensor product V(S),
cp) ®.!'t, defined on simple tensors by (5.4) The important point here is that 7rt(€2)*7rf(6) E .c(L2(M)M) = M. The closure of V(S), cp) ®.!'t in this inner product, modulo the null space, is once
again S) ®cp .!'t. (If we do choose a row representation as in (5.2), we have
The paper [F] contains more exposition of this approach, including some alternate constructions. 6. Realization of the relative tensor product as composition of unbounded operators
In this section we briefly indicate how (5.1) can be rigorously justified and extended. Readers are referred to the sources for all details. In his pioneering theory of noncommutative LP spaces,
Haagerup [H] estab-. lished a linear isomorphism between Mt and a class of positive unbounded operators affiliated with the core of M. (The core, well-defined up to isomorphism, is the crossed
product of M with one of its modular automorphism groups.) We will denote the operator corresponding to the positive functional cp by cp also. These operators are r-measurable (see the next section),
where r is the canonical trace
on the core, and so they generate a certain graded *-algebra: positive elements of LP(M) are defined to be operators of the form '{)1/ p . The basic development of this theory can be found in [Tej;
our choice of notation is influenced by [Yj, where it is called a modular algebra. The composition of two L2 operators is an Ll operator, and it turns out that (5.1) can be rigorously justified [Sj
as an operator equation. (This is not automatic, as operators like ,{)-1/2 are not 7-measurable and require more delicate arguments.) In fact, there is nothing sacred about half-densities. With the
recent development of noncommutative LP modules [JS], one can allow relative tensor products to be bifunctors on Right U(M) x Left Lq(M), with range in a certain L'" space. The mapping is
densely-defined by
~ ®cp 1/ ~ (~,{)~-*-~)1/. In the case of an RTP of L oo modules (or more generally, Hilbert C*-modules), the middle term is trivial and there is no change of density. This explains why there are no
domain issues in defining a vector-valued RTP of Hilbert C*-modules [Rj. Let us mention that the recent theory of operator bimodules, in which vectors can be realized as bounded operators, allows a
variety of relative tensor products over C*-algebras [APj. This can be naturally viewed as a generalization of the theory of Banach space tensor products, which corresponds to a C*-algebra of
scalars. 7. Preclosedness of the map
(~, 1/)
®cp 1/
Our purpose in this final section is to study when the relative tensor map is preclosed. This is a weaker condition than that of Falcone, who studied (effectively) when the map was bounded. We begin
with a base case: a fa.ctor, two standard modules, and a simple product. With the usual notation Bcp for V(L 2 (M), '{)), the relevant map is
Bcp x L2(M) :3 (~, 1/) 1-+ ~ ®cp 1/ E L 2(M). This is bilinear: we take "preclosed" to mean that if ~n -+ ~ E Bcp, 1/n -+ 1/, ~n ®cp 1/n -+ (, then necessarily ( = ~ ®cp 1/. We will also consider
several variations: changing the domain to an algebraic tensor product, allowing non-factors, and allowing arbitrary modules. Readers unfamiliar with von Neumann algebras will find this section more
technical, and any background we can offer here is sure to be insufficient. Still, we introduce the necessary concepts in hopes that the non-expert will at least find the statements of the theorems
accessible. A weight is an "unbounded positive linear functional": a linear map from M+ to [0, +ooj. We will always assume that weights are normal, so Xc< / x strongly => '{)(xc<) / '{)(x); and
semifinite, so {x E M+ I '{)(x) < oo} is a-weakly dense in M+. We can still define RTPs for faithful weights, but now Bcp = {x'{)I/2 I '{)(x*x) < oo} C L 2(M). For details of the representations
associated to weights, see [T2j. A weight 7 which satisfies 7(XY) = 7(YX) on its domain of definition will be called a trace (more properly called a "tracial weight"). An algebra which admits a
faithful trace 7 is semifinite; if in addition we can have 7(1) < 00, it is finite. This facilitates the following classification of factors: a factor with n orthogonal
minimal projections is type In (possibly n = 00); a semifinite factor without minimal projections is type III if finite and type 1100 if not; a factor which is not semifinite is type III. The reader
will note that this refines our previous definitions of type, as a trace is exactly the object which orders the equivalence classes of projections. Obviously, there is much more to be said, and most
of it can be found in [Tl]. For a faithful trace T on semifinite M, it is useful to consider the T-measure topology [N]. This is a uniform topology with neighborhoods of 0 given by
N(6, f) = {x E M 13p E P(M) with T(P.L) < 6, Ilxpll < fl. The closure of M in this topology can be identified as a space of closed, denselydefined operators affiliated with M. It is denoted VJ1(M)
and actually forms a *-algebra to which T extends naturally. (The T-measurability of an operator T is equivalent to the assertion that T(e(A).L) < 00 for some spectral projection e(A) of ITI, so we
get that VJ1(M) = M if M is atomic.) It follows from modular theory that every weight on (M, T) is of the form Th ="T(h·)" for some closed, densely-defined, and positive operator h. In case h is not
T-measurable, this is to be interpreted as 1 / 2) where h = h(1 lim T(h e1/ 2 . he ' e
e-+ O
Finally, the presence of a faithful trace
LP(M, T) = {T
+ ch)-l .
allows us to introduce the spaces
VJ1(M) I T(ITIP) =
IITIIP < oo},
which are antecedent to Haagerup's. Exposition can be found in [N]. Here we will only need L2(M, T), which is a standard form and in particular isomorphic as a left module to any faithful GNS
representation SJ",. It is easy to check that the norm topology in L2(M, T) is stronger than the T-measure topology. THEOREM 7.1. Let M be a factor. The map
(7.1) is preclosed iff M
~'" x L2(M) - L2(M) :
= (M, T)
(~, TJ)
~ 0'" TJ
is semifinite and h- 1 is T-measurable, where
= Th.
PROOF. The proof is by consideration of cases.
M is type III: Choose a projection eo so
M = (M,r) is semifinite and h- 1 is not r-measurable: First note that the measurability of h- 1 does not depend on the choice of r. Writing h = J >.de(>.) , the hypothesis is that r(e(>.)) = 00, V>..
Choose a projection eo with cp(eo) < 00 and r(eo) < 00. Then e(1/n 3) has a subprojection en which is equivalent to eo. The above construction again shows that the map is not preclosed, except that
M = (M, r) is semifinite and h- 1 is r-measurable: We assume (7.2) and want to show ( n = {x E M
= xTJ. Set
I xh 1/ 2 E L2(M,r)};
= {x
I cp(x·x) < oo},
both of which are strongly dense in M. (The bar stands for graph closure.) Then 7r : ncpl/2 ~ L2(M, r);
densely defines a left module Hilbert space isomorphism from fJ IP to L2(M, r); denote its extension by 7r as well. Recalling that h- 1 / 2 is r-measurable by assumption,
p: nIP ~ !m(M); x 1-+ 7r(Xcpl/2)h- 1/ 2 is well-defined and the identity map on n. It is also strong-measure continuous: Xa
~ X '* Xa cpl/2 ~ Xcpl/2
'* 7r(xacpl/2) £ 7r(xcpl/2)
'* 7r(Xacpl/2) ~ 7r(Xcpl/2) '* 7r(xacpl/2)h-l/2 ~ 7r(Xcpl/2)h-1/2, where we used that multiplication is jointly continuous in the measure topology. We may conclude that p is the identity on all of
nIP' Implementing the isomorphism 7r, (7.2) becomes
7r(Xncpl/2) ~ 7r(Xcpl/2),
7r(TJn) ~ 7r(TJ),
Xn7r(TJn) ~ 7r(().
The convergences in (7.3) are also in measure; by the foregoing discussion we have
Xn7r(TJn) = 7r(x ncpl/2)h- 1/ 27r(TJn) ~ 7r(xcpl/2)h- 1/ 27r(TJ) = X7r(TJ) in measure as well. The measure topology is also Hausdorff, so 7r(() = X7r(TJ) and therefore ( = XTJ. D The map p suggests a
schematic recovery of the "operators" in fJ IP : (7.4) Such operators are densely-defined but in general not closable (or may have multiple closed extensions [Sk]). Not surprisingly, then, the
right-hand side of (7.4) may be only formal. The condition on h in Theorem 7.1 makes the equality (7.4) rigorous, as the products on the right-hand side are well-defined r-measurable operators. Note
that hand h- 1 are automatically r-measurable when M is finite, and in
this case all multiplications and isomorphisms between GNS representations stay within W1(M), and all operators are closed - a version, somewhat oblique, of the T-theorem of Murray and von Neumann.
THEOREM 7.2. Let M be a factor, and consider 23"" 0alg L2(M) as a subspace of the Hilbert space tensor product L2(M) 0 L 2(M). The linear map (7.5)
23"" 0
al g
L 2(M):
~n 0 TIn
is preclosed iff M = (M, T) is atomic and T(h- 1 ) < it is actually a bounded map, with norm T(h- 1 )1/2.
~n 0"" TIn
where c.p
= Th.
In this case
PROOF. When M is type III, the map is not preclosed by the previous theorem. We will therefore fix a trace T, set c.p = Th, use the decomposition h = f Ade(A), and view all vectors as elements of L2
(M, T). (When M is type I, we assume that T is normalized so that T(p) = 1 for any minimal projection p.) The rest of the proof is again by cases. M is type II: Choose p < e(A) for some A with T(p) =
C < 00. For each k, break up p into equivalent orthogonal projections as L~=l p~. Consider the tensors
= LP~hl/2 0
= p.
Since the p~ are orthogonal,
JJTkJJ2 =
LT(p~h)T(p~) ~ L (~c) (~) = A~2
and the map is not preclosed. M is type leo and T(e(A)) = 00 for some A: Fix an orthogonal sequence of minimal projections {Pn}, Pn < e(A). The equivalence gives partial isometries with v~vn = PI,
VnV~ = Pn· Then
JJTkJJ = k 2 L...J T(Pnh)T(Pl) ~ k2 L...JA =
and the map is not preclosed. In the only remaining situation, M is type I and h is diagonalizable. Let {An} be the eigenvalues (with repetition), arranged in nondecreasing order. We will write all
matrices with respect to the basis of eigenvectors. If Sk
= L~=l
)..In /'00:
and the map is not preclosed.
If Sk = L~=l >L / C < 00; that is, T(h- 1 ) < 00: We show that the map is bounded on finite tensors of the form T = L i j eij ® yij. We have
S =
h- 1/ 2 Y ij =
\-1/2 Y ij
=~ ~
(~\-1/2 ~I'\j
ij ) Yjkeik
By Cauchy-Schwarz,
:s cl: IY~{12 :s Cl: ly;tl 2= CIITI12. ijk
Since such tensors are dense in the Hilbert space tensor product, we may conclude that the norm of the map is C 1 / 2 • But the tensors Tk from (7.6) show that the norm is at least Cl/2. 0
We now extend Theorem 7.1 to the non-factor case. A general von Neumann algebra is a direct integral of factors (see [Tl] for details), and weights on the algebra disintegrate as well. PROPOSITION
7.3. Let M be a von Neumann algebra with central decomposition
frtf! M(w)dIL(W). The map (7.7)
L 2(M):
(~, 11)
~ ®
is preclosed iff M = (M, T) is semijinite and (*) h(W)-1 is T(w)-measurable for JL-a.e. w, where 'P = Th· PROOF. If M contains a summand of type III, the construction from Theorem 7.1, with the added
restriction that fn and gn are chosen with equal central support, demonstrates that the map is not preclosed. If there is a trace T for which 'P = Th and h- 1 is T-measurable, then the argument in
Theorem 7.1 still shows that the map is preclosed. We will see that this possibility is equivalent to (*). First note that (*) is independent of the trace chosen, as the choice of a different trace
changes a.e. h(w) by a constant factor. If (*) does not hold, fix any trace T, write 'P = Th, and let {e(A)} be the spectral projections of h. By hypothesis, we can find a nonzero central projection
z with ze(A) a properly infinite projection for all A. The second construction of Theorem 7.1 shows that the map is not preclosed, where we choose all en, including eo, with central support z. Now
suppose that (*) holds. We may choose a trace T which factors as 70 q" where q, is an extended center-valued trace and 7 is a trace on the center with 7(1) < 00. Let hand {e(A)} be as before. Now by
assumption, the function
z(w) = max{l/n I T(w)(e(l/n)(w)) < I} is a.e. defined, non-zero, and finite. It is measurable by construction, so z and z-1 represent elements of the extended center. Now write 'P = (Tz ) z - 1 h.
Let f be the
spectral projection of z-lh for [0,1]. We have f(w) = e(z(w))(w), so T(W)(f(W)) < 1. Then
7.4. Let M be a factor with left module Mit and right module
SjM' The map
(7.8) is preclosed only under the same conditions as in Theorem 7.1; i.e. M = (M, T) is semijinite and h- 1 is T-measurable, where
~ (:~~~~:), (11~ 11~ ~ ( :~::~:) · . ... )
(111712 ... ),
k~ (ij), -+00
we also have L2 convergence in each coordinate. By Theorem 7.1, (ij = Xi"7j' When h- 1 is not T-measurable, M must be 100 or 1100 , In this case M Moo(M), and we do not need row and column matrices:
Sj ~ ql L2(M) and it ~ L2(M) q2 for appropriate projections ql, q2' Fix equivalent finite projections ft ~ ql, h ~ q2 with v*v = ft, vv* = h. By assumption e(1/n 3 ) is infinite for all n; let gn be
a subprojection equivalent to the fi with vin Vin = fi, vinvin = 9n. Then
11(1/n)v2nI1 2 = (1/n 2 )T(h) --> 0, and the map is not preclosed.
o References [AI] [A2] [AP] [C1] [C2] [F] [H] [Jol]
C. Anantharaman-Delaroche, Atomic correspondences, Indiana Univ. Math. J. 42 (1993), no. 2, 505-531. C. Anantharaman-Delaroche, Amenability of bimodules and opemtor algebms, in Opemtor algebms and
quantum field theory, Internat. Press, Cambridge, MA, 1997, 225-235. C. Anantharaman-Delaroche and C. Pop, Relative tensor products and infinite C*-algebras, J. Operator Theory 47 (2002), 389-412. A.
Connes, On the spatial theory of von Neumann algebms, J. Funct. Anal. 35 (1980), 153-164. A. Connes, Noncommutative geometry, Harcourt Brace & Co., San Diego, 1994. T. Falcone, L2-von Neumann
modules, their relative tensor products and the spatial derivative, Illinois J. Math. 44 (2000), no. 2,407-437. U. Haagerup, LP-spaces associated with an arbitmry von Neumann algebm, Algebres
d'operateurs et leurs applications en physique mathematique, CNRS 15 (1979), 175-184. P. Jolissaint, Index for pairs of finite von Neumann algebms, Pac. J. Math. 146 (1990), 43-70.
[J] [JoS] [JS] [KR] [K] [N] [Pal [P] [R] [Sa] [S] [Sk] [Tl] [T2] [Te] [W] [y]
V. Jones, Index for subfactors, Invent. Math. 72 (1983), 1-25. V. Jones and V. Sunder, Introduction to subfactors, London Mathematical Society Lecture Note Series 234, Cambridge University Press,
Cambridge, 1997. M. Junge and D. Sherman, Noncommutative LP modules, J. Operator Theory, to appear. R. Kadison and J. Ringrose, FUndamentals of the theory of operator algebras I,ll, Graduate Studies
in Mathematics 15, 16, AMS, Providence, 1997. H. Kosaki, Index theory for operator algebras, Sugaku Expositions 4 (1991), no. 2, 177-197. E. Nelson, Notes on non-commutative integration, J. Funct.
Anal. 15 (1974), 103-116. W. Paschke, Inner product modules over B*-algebras, Trans. AMS 182 (1973),443-468. S. Popa, Correspondences, notes, 1986. M. Rieffel, Morita equivalence for C*-algebras and
W*-algebras, J. Pure and Appl. Algebra 5 (1974), 51-96. J.-L. Sauvageot, Sur Ie produit tensoriel relatif d'espaces de Hilbert, J. Operator Theory 9 (1983), 237-252. D. Sherman, Applications of
modular algebras, in preparation. C. Skau, Positive self-adjoint extensions of operators affiliated with a von Neumann algebra, Math. Scand. 44 (1979), 171-195. M. Takesaki, Theory of Operator
Algebras I, Springer-Verlag, New York, 1979. M. Takesaki, Theory of Operator Algebras II, Springer-Verlag, to appear. M. Terp, LP-spaces associated with von Neumann algebras, notes, Copenhagen
University, 1981. N. E. Wegge-Olsen, K -theory and C*-algebras, Oxford University Press, Oxford, 1993. S. Yamagami, Algebraic aspects in modular theory, Publ. RIMS 28 (1992), 1075-1106. DEPARTMENT OF
MATHEMATICS, UNIVERSITY E-mail address: dashermalDmath. ui uc . edu
ILLINOIS, URBANA, IL 61801-2975
Contemporary ~1athematics
Volume 328. 2003
Uniform Algebras Generated by Unimodular Functions Stuart J. Sidney ABSTRACT. Our main result is the following reduction theorem: If A is a uniform algebra on X that is generated by unimodular
functions, in order to verify the strong corona property for A on its spectrum E(A), it suffices to verify it when the corona data are unimodular functions from A. This is a step in the direction of
finding a simpler proof of Carleson's Corona Theorem [C], and of extending it to higher dimensions. The main tool is a proof that the algebra of bounded sequences from A, regarded as a Banach algebra
of continuous functions on the Stone-Cech compactification of ]\I! x X. is itself a uniform algebra generated by unimodular functions.
1. Introduction
One of the jewels of twentieth-century analysis is Lennart Carleson's Corona Theorem, which asserts that the open unit disc lDl in the complex plane is dense in the maximal ideal space or spectrum of
H OO = HOO(lDl), the Banach algebra of bounded analytic functions on lDl in the supremum norm; here each point ( of lDl is identified with the complex homomorphism "evaluation at (." The publication
of this result in 1962 [C] generated a search for a more transparent proof, and for comparable theorems with other domains (both in complex dimension 1 and in higher dimensions) in place of the disc.
Progress has been made in both directions, but all proofs of Carleson's theorem are still fairly involved, and the other domains to which it has been extended are I-dimensional. One approach to the
problem was introduced in the 1960s by a group at the Institut Fourier in Grenoble, France under the leadership of Alain Bernard. It involves a sequence algebra naturally associated to a uniform
algebra. In this paper we shall revisit the Grenoble approach and add to it some new results that we hope will lead toward its eventual sucess. Our main result is the following: THEOREM 1 (Reduction
Theorem). Let A be a uniform algebra on a compact Hausdorff space X. Assume that A is generated as a Banach algebra by the unimodular (on X) functions in A. Then in order to verify the strong corona
property for A on its spectrum ~(A), it suffices to verify it for unimodular corona data. 2000 Mathematics Subject Classification. Primary 46JlO; Secondary 46E15, 46E25, 46J15. Key words and phrases.
Uniform algebra, corona property, corona data, unimodular.
© 293
2003 American Mathematical Society
s. J. SIDNEY
The terms strong corona property and corona data will be defined below. Note that functions in A that we are calling unimodular take values of modulus 1 on X, but (in general) only of modulus:::; 1
on E(A). Such functions are often called inner. Observe that for the disc algebra (see below), the reduction theorem says that the strong corona property need only be verified for corona data
consisting of
finite Blaschke products. In section 2 we recall the relationship between corona problems of density in spectra, and corona problems of solving systems of equations in a uniform algebra. Section 3
introduces sequence algebras and presents a proof that the property of being generated by unimodular functions passes from a uniform algebra to its associated sequence algebra. In section 4 we prove
the reduction theorem.
2. Background on corona problems The key abstract result that is just about always used in tackling corona problems is the following easy consequence of elementary Gelfand theory. PROPOSITION 2. Let
A be a uniform algebra and let E be a subset of its spectrum E(A). Then E is dense in E(A) if and only if whenever h, ... , fn are finitely many functions in A and there is a positive constant 8 such
that max{lh(x)I, ... , Ifn(x)l} ~ 8 for every x E E, it follows that there are functions gl, ... ,gn in A such that hgl + ... + fngn = l.
In this proposition, the fJ are known as corona data (for A on E), and the assertion equivalent to density of E is the corona property (again, of A on E). In particular, if A is a uniform algebra on
X, then X = E(A) if and only if A enjoys the following property: whenever h, ... , fn are finitely many functions in A that do not all vanish simultaneously at any point of X, there exist functions
gl, ... ,gn in A for which hgl + ... + fngn = 1. Carleson actually proved a stronger version of the corona property for HOO on Jl)), one in which there are bounds on the gj: DEFINITION 1. A uniform
algebra A has the strong corona property on a subset E of E(A) if for all positive integers n and positive numbers 8 < 1 there are finite constants C(n, 8) such that whenever h, ... ,fn are functions
in A satisfying IlfJ II :::; 1 and maxj IfJ(x)1 ~ 8 for all x E E, there exist functions gl, . .. , gn in A that satisfy E j fJgj = 1 and Ilgjll :::; C(n,8).
Clearly if A has the strong corona property on E then A has the corona property on E, so E is dense in E(A). In this definition, the fJ are strong corona data for A on E. Consider now the disc
algebra A(Jl))) consisting of all continuous complex-valued functions on the closed unit disc ii} that are analytic on Jl)). It is standard, and not hard to prove, that E(A(Jl)))) = ii}, and so A
(Jl))) has the corona property on ii} (equivalently, on Jl))). Suppose one can show that the disc algebra actually has the strong corona property on Jl)) with constants C(n,8). If h, ... , fn are
strong corona data for H OO on Jl)) for some 8, we can for each natural number k produce strong corona data hk, ... fnk for A(Jl))) and this same 8 such that for each j, fJk -+ fJ pointwise on Jl))
as k -+ 00 (for instance, fJk(() = fJ((l - k- 1 )()). By assumption there are functions glk, ... , gnk in A(Jl))) such that E j fJkgjk = 1 and Ilgjkll :::; C(n,8). By a normal families argument, we
may assume that for each j
there is gj in Hoc such that gjk ---- gj pointwise as k ---- 00, and clearly I:j hgj = 1 and Ilgjll :::; C(n,8). We see that HOC enjoys the strong corona property with the same constants as the disc
algebra. (Conversely, the reader may show that if Hoc has the strong corona property on J]), then A(J]}) also has it on J]}, with any choice of constants greater than those for HOC.) The above
argument works just as well on many complex domains S1 other than the unit disc J]}, in particular on unit balls and polydiscs in Cd. For these domains it is elementary that the spectrum of A(S1),
the algebra of continuous functions on Ii that are analytic on S1, is just fl, and the same argument as for S1 = J]} shows that if A(S1) enjoys the strong corona property on S1 then so does HOC(S1),
the algebra of bounded analytic functions on S1. Clearly the reduction theorem applies to the polydisc algebra A = A(S1) when S1 is the unit polydisc (and X is taken to be the torus consisting of
points in Cd all of whose coordinates have modulus 1). Thus the reduction theorem aims directly at the goal of proving the corona theorem for polydisc algebras, namely, that the polydisc S1 is dense
in the spectrum of HOO(S1). One last word before moving on to Bernard's technique. A natural question proposed by Walter Rudin ([Bil, page 347) asks whether, inasmuch as every uniform algebra has the
corona property on its spectrum, perhaps it also has the strong corona property on its spectrum. That is, in fact, the conclusion that the sequence spaces we are about to study was designed to prove.
Unfortunately, an ingenious example produced by another member of the Grenoble team, Jean-Pierre Rosay [RJ, shows that not every uniform algebra has the strong corona property. Furthermore, Brian
Cole ([Gal, chapter 4) has exhibited an open Riemann surface R that is not dense in the spectrum of HOO(R). 3. Bernard's sequence algebras and unimodular functions To any uniform algebra A we
associate the unital Banach algebra A consisting = (h) with h E A and IIJII == sUPk Ilhll < 00. If A is of those sequences a uniform algebra on X (where X is any compact subset of E(A) that contains
the Silov boundary of A), then every J E A may be naturally identified with the bounded continuous function on N x X that takes the value h(x) at the point (k, x) of N x X, and so with a continuous
function (also denoted J) on X = ,B(N x X), the Stone-tech compactification of N x X. In this way, A becomes a uniformly closed algebra of continuous functions on X that contains the constant
functions; in general, A need_not separate the points of so is a uniform algebra on some quotient space of X, but not necessarily on X itself. These sequence algebras and corona problems are related
by the following result, the original raison d 'etre for the study of A:
PROPOSITION 3. If A is a uniform algebra, A has the strong corona property on E(A) if and only ifN x E(A) is dense in E(A). PROOF. One direction is trivial: If A has the strong corona property on E
(A), then A has the strong corona property on N x E(A) with the same constants. To go in the opposite direction, suppose that A does not have the strong corona property on E(A), so for some nand 8 no
appropriate constant C(n,8) exists. Each natural number k cannot serve as C(n,8), so there are flk,"" fnk in A for which Ilhkll :::; 1 and maxj Ihkl ~ 8 throughout E(A), but if gl, ... ,gn are
S . .1. SIDNEY
functions in A for which Lj /jkgj = 1, then maxj !!gj!! > k. Let ij = (fjk)k, so i j E A, lIijll ::; 1, and maxj !ij! ~ 8 throughout N x E(A). There can be no ih (glk)k, ... , 9n (gnk)k in A for
which Lj i j 9j 1, for this equality would mean that for each k we would have Lj fjkgjk = 1, and for k > maxj IIgjll this would yield maxj IIgjk II > k > maxj 119j 1/, which is impossible. Thus N x E
( A) cannot be dense in E(A). D
Proposition 3 offers the potential to prove strong corona theorems by proving density of N x E(A) in E(A). It was hoped that a "soft" Banach algebra argument might accomplish this. However, such an
argument never materialized, as (in view of Rosay's example) it could not in complete generality. Instead, A found a central role in the theory of functions that operate on function spaces. The
seminal document here is Bernard's paper [Be], and a recent introductory account of both the relation to corona problems and the applications to functions that operate may be found in [HS]. That the
Grenoble program cannot work in general does not imply that it cannot work in special situations. Our goal in this paper is to begin movement toward a positive outcome in one important special
situation, namely, that in which the uniform algebra is generated by unimodular functions. Let us now establish some notation. If A is a uniform a~ebra on X, we let U denote the set of unimodular
functions in A and we let U denote the set of all functions i = (ik) such that ik E U for every k; viewed as functions on X, ii consists of precisely the unimodular (on X) functions in A. We shall
need the following "trick" developed by Bernard in a context related to ours but not involving sequence algebras. LEMMA 4 (Bernard trick). [BGM] Suppose v is in the subalgebra (equivalently, linear
subspace) of A generated algebraically by U, and that IIvll < 1. There are functions u E U s'u.ch that u'iJ E A. Take such a u, and for real numbers 0 let v(8) Then 0
= 'U'
uv - e . . 1 - u'iJe t8
v(8) is a continuous mapping from the real line into U, and v
v(8) dO.
This lemma, an immediate consequence of the Cauchy integral formula, was used in [BGM] to obtain easily the fact that if U generates A as a Banach algebra then the closed unit ball of A is the closed
convex hull of U. We shall use it in pretty much the same way to prove the second part of the following result: THEOREM 5. Let A be a uniform algebra on X, and suppose that U generates A as a Banach
algebra. Then A separates the points of X (and so is a uniform algebra on X), and is generated as a Banach algebra by indeed, the closed unit ball of A is the closed convex hull of
PROOF. The hypotheses imply that U separates the points of X, and then an easy argument (see for instance Lemma 4.12 in [GI]) shows that the absolute values of the functions in the algebra generated
algebraically by U, and so the absolute
values of the functions in A, are uniformly dense in the set of non-negative realvalued continuous functions on X. An argument used by Bernard in [Be] for the Dirichlet algebra case trivially works
here as well to give the fact that A separates the points of X. To tackle the second part of the statement, first note that if 0 < r < 1, then continuity of 0 1--+ v(9) and convergence of the
integral in the lemma are uniform over all v as in the lemma that satisfy Ilvll < r. Now suppose W = (Wk) is in the open unit ball of A, and take r < 1 such that Ilwll < r. By the hypotheses we may
approximate w as closely as we wish on N x X, and so on X, by a function ii = (Vk) E A such that for each k, Vk is in the subalgebra of A generated algebraically by U, and Ilvk II < r. For each k
choose Uk and define vi9) as in the lemma, so 0 1--+ vi9) is a continuous mapping of the real line into U, and Vk
1 211'
(9) Vk
the continuity and convergence being uniform in k. For each 0 we have ii(9) (vi9») E U, the mapping 0 1--+ ii(9) is continuous, and most important,
ii =
r 211' 10
ii(9) dO.
Thus ii lies in the closed convex hull of U.
4. Proof of the reduction theorem We first require a simple lemma, which will be applied to to A and U.
A and U rather than
LEMMA 6. Let A be a uniform algebra on X, and suppose A is generated as a Banach algebra by its subset U of unimodular (on X) functions. If c.p and 1jJ are distinct elements ofE(A) and c.p ¢. X, then
there exists u E U such that u(c.p) = 0 =Iu(1jJ). PROOF. c.p has a representing measure J.L on X which cannot be a point mass, so some v E U must be nonconstant on the support of J.L, hence Iv(c.p)1
< 1. Composing v with a Mobius transformation, we may suppose v(p) =I- O. If v(1jJ) =I- v(c.p), let W = v; if v(1jJ) = v(c.p), multiply v by an element of U that separates 1jJ and c.p to get w. In
either case, W E u, Iw(c.p) I < 1, and w(1jJ) =I- w(c.p). Composing w with an appropriate Mobius transformation produces the required u. 0 We now prove the Reduction Theorem. PROOF OF REDUCTION
THEOREM. We are given that A is a uniform algebra on X generated by its set U of unimodular elements, and that there are always constants C(n, 0) such that whenever Ul, ... ,Un are elements of U such
that maxj IUjl ?: 0 throughout E(A), it follows that there are gl,'" ,gn in A satisfying ~j Ujgj = 1 and Ilgj II ::; C(n,o). According to Proposition 3, we must deduce that N x E(A) is dense in E(A).
Suppose c.p E E(A). If c.p E X then there is nothing to prove, so assume c.p ¢. X. By Theorem 5 and Lemma 6, for each 1jJ E E(A) other than c.p there is an element of U that is zero at c.p and
nonzero at 1jJ. Thus if W is any neighborhood of c.p in
S. J. SIDNEY
I:(A), standard topological arguments provide finitely many functions U and a number 0 < 8 < 1 such that Uj (
E I:(A) :
ut. ... ,Un in
IUj (1/1) I < 8 \lj} C W. If W n (N x I:(A)) = 0
Let Uj = (Ujk)k where Ujk E U. then for each k, maxj IUjk(1/1) I ~ 8 for every 1/1 E I:(A), so there are 91k, ... ,9nk in A that satisfy Lj Ujk9jk = 1 and 1!9jkll ::; C(n,8). Letting [}J = (9jk)k E
A, we have Lj Uj9j = 1 on N x I:(A), so on I:(A); but this is impossible at the point <po Thus W n (N x I:(A)) i=- 0 after all, completing the proof of the theorem. 0
References [Be] A. Bernard, Espaces des parties rt~elles des elements d'une algebre de Banach de fonctions, J. Functional Anal. 10 (1972), 387-409. [BGM] A. Bernard, J. B. Garnett and D. E. Marshall,
Algebras generated by inner functions, J. Functional Anal. 25 (1977), 275-285. [Bi] F. Birtel, editor, Function algebras, Scott-Foresman, Fair Lawn, N. J., 1966. [e] L. Carleson, Interpolations by
bounded analytic functions and the corona problem, Ann. of Math. 76 (1962), 547-559. [Ga] T. W. Gamelin, Uniform algebras and Jensen measures, London Math. Soc. Lecture Note Series 32, Cambridge
Univ. Press, Cambridge, England, 1978. [GI] I. Glicksberg, Measures orthogonal to algebras and sets of antisymmetry, Trans. Amer. Math. Soc. 105 (1962), 415-435. [HS] S. Hwang and S. J. Sidney,
Sequence spaces of continuous functions, Rocky Mountain J. Math. 31 (2001), 641-659. [R] J.-P. Rosay, Sur un probleme pose par W. Rudin, C. R. Acad. Sci. Paris Ser. A-B 267 (1968), A922-A925.
DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CONNECTICUT, STORRS, CONNECTICUT 062693009
Contemporary Mathematics Volume 328, 2003
Analytic Functions on Compact Groups and their applications to almost periodic functions Thomas Tonev and S. A. Grigoryan ABSTRACT. This is a survey on some recent developments in the theory of
uniform algebras of continuous functions on compact groups, that are invariant under group shifts. Contents:
I. Analytic Functions on Groups 1. Almost periodic functions 2. Shift invariant algebras on groups II. Shift Invariant Algebras on Groups 1. Rad6's and Riemann's theorems for analytic functions on
groups 2. Extension of linear multiplicative functionals of shift invariant algebras on groups 3. Automorphisms of shift invariant algebras on groups 4. Primary ideals of algebras of analytic
functions on solenoidal groups 5. Asymptotic almost periodic functions III. Inductive limits and Shift Invariant Algebras on Solenoidal Groups 1. Inductive limits of disc algebras on G-discs 2.
Inductive limits of algebras on subsets of G-discs 3. Gleason parts of inductive limits of disc algebras on G-discs 4. Inductive limits of HOC spaces on G-discs 5. HOC spaces on solenoidal groups 6.
Bourgain algebras and inductive limits of algebras
I. Analytic functions on groups 1. Almost periodic functions. Almost periodic functions were introduced by H. Bohr [4] who has established their basic properties. Other results were obtained by
Besicovitch [2] and Jessen [26]. Bohr discovered almost periodicity in the course of his study of Dirichlet series of analytic functions. For a deeper insight on almost periodic functions we refer
the reader to the books of Loomis [29], and 1991 Mathematics Subject Classification. Primary 46J15; Secondary 30H05, 46J10. Key words and phrases. Uniform algebra, compact group, shift invariant
algebra. The authors acknowledge the support of a NSF Cooperative Research Grant in Modern Analysis @ 2003 American Mathematical Society
T. TONEV AND S. GRIGORYAN
Corduneanu [I1J. Unless otherwise said, all continuous functions in the sequel will be considered complex valued. A continuous function f on the real line lR. is said to be almost periodic if for
every E > 0 there is an L > 0 such that within every interval I c lR., III 2: L there is an x E I such that max If(t) - f(t + x)1 < E (H. Bohr [4]). According to the tEIR
famous theorem of Bochner [3J, f is almost periodic on lR. if and only if the set of all its translates h(x) = f(x + t), t E lR. is relatively uniformly compact in BC(lR.), the space of bounded
continuous functions on R Equivalently, f is almost periodic if it can be approximated uniformly on lR. by exponential polynomials, i.e. by functions n
akeiskX, where ak are complex, and Sk are real numbers. It is easy to k=l see that the set AP(lR.) of all almost periodic functions on lR. is an algebra over C. Actually, under the uniform norm AP
(lR.) is a commutative Banach algebra with unit. of type
Dirichlet coefficients a{, A E lR. of an almost periodic function f(x) on lR. are
the numbers a{ = lim -T T--+oo
Y T
+ f(x)e-i>,xdx, where the limit, and its value in the
right hand side exists independently on y E R Dirichlet coefficients a{ are nonzero for count ably many A'S at most, which are called Dirichlet exponents of f(x). The set sp (I) of all Dirichlet's
exponents of f(x) is called the spectrum of f. Hence, sp (I) = {A E lR. : a{ =f. O} is a countable set. It is customary to express the fact that Ak are the Dirichlet exponents of f(x) and the numbers
= are the Dirichlet coefficients of f(x) for any k = 1,2, ... by a power series notation, namely
A' at
f(x) '"
L A'ei>'k X. This series, not necessarily convergent, is called the Dirichlet
k=l series of f(x). If all Dirichlet coefficients of a f E AP(lR.) are zero, then, as it is easy to see, f == O. Consequently, the correspondence between almost periodic functions and their Dirichlet
series is injective. E~ery almost periodic function f on lR. can be extended as a continuous function f on the Bohr compactification J3lR. of R The Fourier coefficients c[ of the extended in this way
function 1 on J3lR. equal the Dirichlet coefficients A' of f. Moreover, the maximal ideal space MAP(IR) of the algebra of almost periodic functions on lR. is homeomorphic to the Bohr compactification
J3lR. of R For every A C lR., by APA(lR.) we denote the space of all almost periodic Ajunctions, namely, almost periodic functions on lR. with spectrum contained in the set A, i.e.
APA(lR.) = {f E AP(lR.) : sp (I) C A}. Note that every f E APA(lR.) can be approximated uniformly on lR. by exponential n
A-polynomials, i.e. by exponential polynomials of type
L akei8kX, Sk k=l
E A.
2. Shift invariant algebras on groups. Let G be a compact abelian group, and let S be an additive subsemigroup of its dual r = 8, containing the origin. Linear combinations over C of functions of
type Xa , a E S are called S -polynomials on G. Denote by As the set of all continuous functions on G whose Fourier coefficients c!
f(g)xa(g) da are zero for any a outside
Here a is the normalized Haar
measure on G. The functions in As are called S-functions on G. Any S-function on G can be approximated uniformly on G by S-polynomials, and vice versa. The set As is a uniform algebra on the group G.
A uniform algebra A on G is G-shift invariant if, given an f E A and 9 E G, the translated function fg(h) = f(gh) belongs to A. Every algebra of S-functions is invariant under shifts by elements of
G. Vice versa, every G-shift invariant uniform algebra on G is an algebra of S-functions for some uniquely defined subsemigroup S C 8 (Arens, Singer [1]). Algebras As of S-functions are natural
generalization of polydisc algebras A(']['n), n E N. With G = ']['n, = 8 = zn, and S = Z+.' the algebra As in fact coincides with the algebra Azn = A(']['n) on the torus ']['n, and Z+-functions
are traces on ,][,n of usual analytic functions in n variables in the polydisc continuous up to the boundary ']['n.
The maximal ideal space Ms of As is the set H(S) = Hom (S, ~), and the Shilov boundary BAs is the group G (Arens-Singer [1]). H(S) is a semigroup under the pointwise operation (cp1/1)(a) = cp(a)1/1
(a), a E S. The Gelfand transforms of elements f E As are continuous functions on Ms, and the space As = {i: f E As} is a uniform algebra on Ms.
As shown by Arens and Singer (e.g. Gamelin [14]), As is a maximal algebra if and only if the partial order generated by the semigroup S in 8 is Archimedean. Note that in this case 8 c JR and there is
a natural embedding of the real line JR into G so that the restrictions of S-functions on this embedding are almost periodic functions that admit analytic extension on the upper half-plane II over
JR. Moreover, an algebra of type As is antisymmetric if and only if the semigroup S does not contain nontrivial subgroups, i.e. if S n (-S) = {O} (Arens, Singer [1]). A compact group G is said to be
solenoidal, if there is an isomorphism of the group JR of real numbers into G with a dense range. Equivalently, a compact group is solenoidal if and only if there is an isomorphism from 8 into R Note
that the Stone-Chech compactification (3T = of T is a solenoidal group for every additive subgroup r of JR. If G is a solenoidal group, then its dual group r = 8 is isomorphic to a subgroup of R If r
is not dense in JR, then it is isomorphic to Z. In this case G is isomorphic to the unit circle '][', S c Z+, and therefore the elements of the algebra As can be approximated uniformly on '][' by
polynomials. Hence they can be extended on the unit disc lJ)) as analytic functions, and therefore Ms = ~, while As ~ A(lJ))). If r is dense in JR, then the maximal ideal space Ms has a more
complicated nature.
In the case when S c JR+ and S u (-S) = r, the S-functions in As, are called analytic, or generalized analytic functions in the sense of Arens-Singer on G. As
T. TONEV AND S. GRIGORYAN
mentioned before, if S = R+ the group G coincides with the Bohr compactification ,BR of R In this case the maximal ideal space of the algebra AIR+ is the set ll}a = ([0,1] x G)/( {O} x G), which is
called the G-disc, or big disc over G. The algebra Ar+ = Ar+ (lI}a) is called also the G-disc algebra, or the big disc algebra. The points in the G-disc ll}a are denoted by r· g, where r E [0,1] and
9 E G =,BR We identify the points of type 0 . g, 9 E G, and the resulting point we denote by w. Hence, w = O· 9 for every 9 E G. The points of type 1· g, 9 E G, we denote by g. Since R is dense in G,
the set (0,1] x R is dense in the G-disc ll}a. Equivalently, the upper half-plane n ~ (0,1] x R can be embedded as a dense subset of the G-disc ll}a. Below we summarize some of the basic properties
of the G-disc algebra Ar+ (lI}a), where r+ = r n [0,00) (cf. Gamelin [14]). (i) Mr+ = ll}a. (ii) 8A r +(lI}a) = G. (iii) A local maximum principle holds on Ar+(lI}a), namely, for every analytic
r+-function f(r . g) on ll}a, for every compact set U c ll}a, and for each ro . go E U we have If(ro . go)l:::; max If(r. g)l· r·gEbU
(iv) Every f E LP(G, da), 1 :::; p :::; 00 can be approximated in the LP(G, da)norm by sp (f)-polynomials. In particular, every f E As can be approximated uniformly on G by S-polynomials. (v) Ar+(lI}
a) is an analytic algebra, i.e. every analytic r+-function which vanishes on a non-void open subset of ll}a vanishes identically on ll}a. (vi) Any real-valued analytic r+-function is constant. (vii)
Ar+ (lI}a) is a Dirichlet algebra; (viii) A r + (lI}a) is a maximal algebra. (ix) The upper half-plane n can be embedded as a dense subset of the G-disc ll}a.
Examples 1. (a) Let G be a solenoidal group, and S is an additive subsemigroup ofR, containing the origin. Note that the restriction of a character Xa E Gon R is the function eiax , x E R. As an
algebra generated by the characters Xa, a E S on G, the algebra As of analytic S-functions is isometrically isomorphic to the algebra APs(R) of almost periodic S-functions on R, generated by the
functions eiax , x E R, a E S. (b) It is easy to see that As, S C R is isometrically isomorphic to the algebra on '][' \ {I} generated by the singular functions ea :~~, a E S via a Mobius
transformation. In the case when S C R+, As is isometrically isomorphic to the subalgebra .!.ll He; of H OO generated by the functions ea .- 1 , a E S on II} \ {I}. (c) The portion over Jij \ {O} of
the Riemann surface is densely embeddable into the G-disc ll}a.
of the function log z
Example 1 b) implies the following PROPOSITION 1. Let G be a solenoidal group, such that its dual group r = is a dense subgroup of R, and let S be an additive subsemigroup of r+ = r
G n
[0,00), containing the origin. Then the algebra As of analytic S-functions on G is isometrically isomorphic to the algebra of almost periodic S-functions on R DEFINITION 1. Let S be a semigroup of G.
The weak enhancement [S]8 of S is the set of elem ts a E G for which there is a rna E N such that na E S for every n ~ rna. The stron enhancement [S]8 of S is the set of elements a E G for which
there is a rna EN su that rnaa E S. S is weakly enhanced, or strongly enhanced if [S]w = S, or [S]8 = respectively. Note that S c [S]w C [S]8 C weakly and strongly enhanced.
If S
c G and S U (-S)
then S is both
PROPOSITION 2 [23]. For an a E G\ S by Sa denote the semigroup Sa = S +Na. Then MSa = Ms if and only if a E [S]w. As an immediate consequence we obtain that M[sJw = Ms for every semigroup S c G.
Also, if S, reG are two subsemigroups of G such that S + (-S) = r + (-r) = G, and if [S]w = [r]w then Ms = Mr· PROPOSITION 3 [23]. Let S e r e G be two subsemigroups of G such that S + (-S) = G and
Mr = Ms. IE ASa is analytic for some a E r \ S, then Ms = MSa (and therefore a E [S]w according to the previous proposition). In particular, [S]w = [r]w if Ms = Mr·
II. Shift invariant algebras on groups 1. Rad6's and Riemann's theorems for analytic functions on groups. Let U be an open set in the maximal ideal space MA of a uniform algebra A. A continuous
function on U is said to be A-holomorphic on U if for every x E U there is a neighborhood V of x so that can be approximated uniformly on V by Gelfand transforms of functions in A. A uniform algebra
A is said to be analytic on its maximal ideal space MA if whenever a function f E A vanishes on an open subset of MA \ 8A then f vanishes identically on MA. If a G-shift invariant algebra As is
analytic, then S does not contain subgroups other than {O}, i.e. Sn( -S) = {O}. Throughout this section we will consider all algebras to be analytic, and that S + (-S) = G.
DEFINITION 2. A uniform algebra A satisfies Rad6 's property, if every function continuous on MA and A-holomorphic on MA \ Z(f) belongs to A.
The classical theorem of Rad6 asserts that the disc algebra A(j())) possesses Rad6's property. However, it fails for the algebra Ao(j())) of functions f E A(j())) with vanishing at 0 derivatives.
Observe that this algebra is of type As with S = {O, 2, 3, 4 ... }, whose weak enhancenment is Z+ =/:. S. THEOREM 1 (Grigoryan, Ponkrateva, Tonev [23]). The algebra As possesses Rad6's property if
and only if the semigroup S is weakly enhanced.
T. TONEV AND S. GRIGORYAN
DEFINITION 3. A uniform algebra A C C(MA) is integrally closed in C(MA) if every continuous function on MA satisfying a polynomial equation of type xn + alX n - 1 + ... + an = 0, aj E A belongs to A.
Integrally closed in C(MA) uniform algebras were studied extensively by Glicksberg [15]. Examples of integrally closed in C(MA) algebras are the disc algebra, the polydisc algebra, the algebra of
analytic S-functions on a G-disc over a group G with ordered dual. THEOREM 2 [23]. The algebra As is integrally closed in C(MA) if and only if the semigroup S is weakly enhanced. DEFINITION 4. A
uniform algebra A possesses Riemann's property if, given a function 9 E A with Z(g) n 8A = 0, then every bounded A-holomorphic function on MA \ Z(g) belongs to A. The classical theorem of Riemann
asserts that the disc algebra A(lJ))) possesses Riemann's property. Note that single points in the complex plane are zeros of certain analytic functions. DEFINITION 5. The bounded enhancement [S]b of
S is the set of elements a E which there are b, c E S with a = b - c, such that Xb/X c is bounded on Ms \ Z(X C ), where Z(X) = {m E Ms : m(x) = o} is the zero-set of X. A semigroup S is said to be
boundedly enhanced if [S]b = S.
G for
THEOREM 3 (Grigoryan, Ponkrateva, Tonev [23]). The algebra As possesses Riemann's property if and only if the semigroup S is boundedly enhanced. A uniform algebra A possesses the weak Riemann
property if, given a function n 8A = 0, then every bounded A-holomorphic function on MA \ Z(g) can be extended continuously on MA. One can show in a similar way that a G-shift invariant algebra As
possesses the weak Riemann property if and only if the weak and the strong enhancements of S coincide [23].
9 E A with Z(g)
2. Extension of linear multiplicative functionals of shift invariant algebras on groups. Let S be an additive semigroup which contains 0, and possesses the cancellation property, i.e. a = c whenever
a + b = c + b for some bE S. In this case S is a subsemigroup of a group. Denote by r = S - S the group generated by S. Consider a subsemigroup P ::::> S of r such that P + (-P) = r, and Pn (-P) =
{O}. P generates a partial (pseudo-) order on r, by b»- a if and only if b - a E P. Note that every non-negative semicharacter e E Hom (P, [0, 1]) is monotone decreasing on P with respect to the
order generated by P. Indeed, if b »- a for some a, bE r+, then b = a + p for some pEP. Therefore, e(b) = e(a)e(p) ::; e(a) since e(c) ::; 1 on P. Consequently, if a non-negative semicharacter e is
extendable on + as an element in Hom (P, [0, 1]), then ~!ly is monotone decreasing onSCP. / .
PROPOSITION 4 (Grigoryan, Tonev [25]). A positive semicharacter e E H(S) is extendable on r+ as a positive semicharacter if and only if e is monotone decreasing on S with respect to the order
generated by P.
Proof. Let the positive semicharacter eon S be monotone decreasing. If bE r+, then b = a - c for some a,c E S, a >- c, and e(b) = e(a)je(c) is a well defined and natural homomorphic extension of (}
on r+. Clearly, e(a) ~ e(c) if and only if e(b) ~ 1, i.e. if and only if e is a positive semicharacter on r+. THEOREM 4 [25]. Non-vanishing semicharacters rp on S can be extended as (non-vanishing)
semicharacters on r+ if and only if eveIY positive semicharacter e E H+(S) is monotone decreasing on S with respect to the order generated by P.
Proof. Let rp E H (S), rp =I- O. The function ')'(a) = {rp(a)jlrp(a)1 ')'( -a) = Ih(a) can be extended naturally on the group ;Y(b) =
for a E S for a E (-S)
r as a character of r by
whenever b = a - c E r, a, c E S.
Thus, rp = Irpl~ = (!')' extends on r+ (as an element of H(r+)) if and only if e = Irpl does. By the above proposition this happens if and only if e is monotone decreasing on S with respect to the
order on S generated by P. Let S c JR, and P = JR+. Define eg = Lio} E H+(S) to be the characteristic function of {O} in S, namely eg (0) = 1, eg (a) = 0 for every a E S \ {O}. Note that e~+ is the
only vanishing semicharacter on r+. Consequently, if S is an additive subsemigroup of JR+ containing 0 and P = JR+, then a vanishing semicharacter e E H(S) is extendable on r+ if and only if e = eg.
Therefore not every vanishing semicharacter e E H (S) possesses a semicharacter extension on a larger semigroup. PROPOSITION 5 [25]. Let S c P = JR+. A non-negative semicharacter e E H(S) is uniquely
extendable on r+ as a non-negative semicharacter on r+ if and only if e is monotone decreasing on S with respect to the order generated by P.
Proof. Assume that a semicharacter e is monotone decreasing and e( a) = 0 for some a E S. Then e(na) = 0 for every n E N, and the monotonicity argument shows that e(a) = 0 for all a E S \ {O}. In
this case e = eg extends naturally on r + to the semicharacter e = e~+ . Recently by S. Grigoryan, and independently - Sherstnev [31], have generalized Proposition 5 for arbitrary semigroups S with
cancellation property. Namely, a nonnegative semicharacter r on S can be extended (non-uniquely) as a (non-negative) semicharacter on a supsemigroup E :::::> S if and only if r is monotone decreasing
with respect to the order on S generated by E.
T. TONEV AND S. GRIGORYAN
Example 2 (cf. Tonev [32]). Let v > 0 be a positive number. Consider the semigroup rv = {O} U [v,oo) c R Clearly, r = rv - rv = lR., and r+ = lR.+. Since x(a + b) = x(a)x(b) ::; x(a) for every a, b E
r v , every semicharacter X on rv is monotone decreasing. Therefore, it is extendable on lR.+, namely as the characteristic function l?~+ of the origin {O}. Example 3. Let a be an irrational number.
Consider the 2-dimensional semigroup 80. = {n+ma : n, m E Il+} C R Here the group generated by 80. is ro. = 80.-80. = {n+ma: n,m Ell}, while (ro.)+ = ro.nlR.+ = {n+ma ~ 0: n,m Ell}. Clearly, 80. -=I-
(ro.)+' For instance the positive number a - [a] E (ro.)+ \ 80.' For a fixed a E (0,1) the function I'(n + ma) = an, n + ma E 80. is a homomorphism from 80. to (0,1] C iTh". However, I' is not
monotone decreasing on 8. Indeed, I'(ma) = 0, while I'(n) = an> 0 for every n > ma. The natural (and only possible) homomorphic extension 1 of I' on (ro.)+ is given by 1(n + ma) = an, n,m E Il,n+ma ~
O. However, 1 ¢ H((ro.)+), since, for instance, 1(a- [a]) = a-[o.] > 1. PROPOSITION 6 (Grigoryan, Tonev [25]). The maximal ideal space Ms of the algebra As of analytic 8-functions on G = r = 8 - 8
with spectrum in 8 C lR.+ is homeomorphic to the maximal ideal space Mr+ = iTh"c of the algebra Ar+ of analytic r+-functions on G if and only if all positive semicharacters on 8 are monotone
As an immediate consequence we get the following PROPOSITION 7 [25]. The maximalideal space MAPs(JR) ofthe algebra APs(lR.) of almost periodic functions with spectrum in a semigroup 8 C lR.+ is
homeomorphic to the G-disc iTh"c, where G = if and only if all positive semicharacters on 8 are monotone decreasing.
Since the upper half plane II = {z E C : 1m Z ~ O} can be embedded densely in the maximal ideal space Ms of the algebra As (and, together, of APs(lR.)) if and only if MAs = iTh"c, then the upper half
plane II is densely embeddable in the maximal ideal space MAPs(JR) of the algebra APs(lR.) of almost periodic functions with spectrum in 8 if and only if all positive semicharacters on 8 are monotone
her [6], II is densely embedable in MAPs decreasing. Note that, as shown by B if and only if every additive posit' e function 0 on 8 is of type O(a) = yoa for some Yo E [0,00), or O(a) = 00, for a O.
. .!.±!.
For an a E 8 let 'Pa E Hoo be the singular function 'Pa(z) = eW 1-. on the unit disc j[)). Recall that HS' is the Banach algebra on j[)) generated by the functions 'Pa(z), a E 8 equipped by the
sup-norm on j[)). As mentioned in Example 1 b), HS' is a subalgebra of Hoo, which is isometrically isomorphic to the algebra As of analytic 8-functions on G = (8 - 8)~. PROPOSITION 8 (Grigoryan,
Tonev [25]). The unit disc j[)) is dense in the maximal ideal space of the algebra HS' if and only if all positive semicharacters on 8 are monotone decreasing.
Let P be a semigroup of r that generates a partial order on r, and suppose that 8 C E are additive subsemigroups of P that contain the origin, and such that
[S] .. :J E, i.e. Na nSf. 0 for every a E E. Then every non-negative semicharacter E H(S) can be extended naturally on E as a monotone decreasing semicharacter, namely by ~(a) = [(!(na)]l/n. (!
PROPOSITION 9 [25]. If SeE are subsemigroups of P such that E C [S]s, then every semicharacter c.p E H(S) on S is uniquely extendable on E as a semicharacter in H(E), and therefore, Ms = ME.
In particular, if S is a subsemigroup of IR such that [S]s :J r+, then the upper half plane II is densely embedable in the maximal ideal space M APs (lR) of the algebra APs(lR) of almost periodic
functions on IR with spectrum in S. PROPOSITION 10 [25]. If S is a subsemigroup of IR such that [S]s :J r+, then the algebra H'S does not have corona, i.e. the unit disc ID> is dense in its maximal
ideal space MH:;'. PROPOSITION 11 [25]. Let S be a subsemigroup of R Then ME = Mr+ = ll}c for every semigroup E with SeE c 1R+ if and only if [S]s = r+, i.e. for every a E r+ = r n [0,00) there is an
n E N such that na E S.
Note that under the hypotheses of this proposition, the semicharacters on all semigroups E with SeE c 1R+ are uniquely extendable on r+ as semicharacters on r+.
3. Automorphisms of shift invariant algebras on groups. Assume that = {O}, i.e. that S contains no non-trivial subgroups. Under this condition the algebra As is antisymmetric. An element £ E Ms = H
(S) is an idempotent homomorphism of S if £2 = £. Let Is be the set of all idempotents in H(S) that are not identically equal to 0 on S. It is easy to see that Is is a subsemigroup of H(S). Clearly,
an idempotent homomorphism can take values 0 or 1 only. Denote by Z. the zero set {a E S : £(a) = O}, and by E. - the support set {a E S: £(a) = I} of £ E Is. It is easy to see that if £ is an
idempotent homomorphism of S, then E. is a semigroup of S, Z. is a semigroup ideal in S, Z. U E. = S, and Z. n E. = 0.
Sn (-S)
12 [20]. Let As be a G-shift invariant algebra on G, where £ E Is possesses a representing measure supported on a subgroup of G. PROPOSITION
SeC. Every idempotent homomorphism
Note that every idempotent homomorphism of S can be extended un~ely to an idempotent homomorphism on the strong saturation [S]8 of S, i.e. I{, '= IIS]. for every subsemigroup SeC. An automorphism on
a shift-invariant algebra As is an isometric isomorphism c.p : As --+ As that maps As onto itself. The conjugate mapping c.p* of c.p defined by ( c.p* (m) ) (f) = m ( c.p(f) ), is a homeomorphism of
the maximal ideal space M s onto itself. For instance, the conjugate mapping c.p* of an automorphism c.p of the disc algebra A(ID» = Az+ is a Mobius transformation of the unit disc, i.e. c.p*(z)
z - Zo
1- ZoZ
IGI = 1, Izol <
T. TONEV AND S. GRIGORYAN
Note that if the origin 0 is a fixed point of a Mobius transformation cp*, then cp*(z) = Cz for some constant C with ICI = 1. It is easy to see that this is also the case with the automorphisms of
the subalgebra Ao(lD») = {J E A(lIJ» : 1'(0) = O} of the disc algebra A(lIJ», i.e. the conjugate mapping of any automorphism of the algebra Ao(lIJ» fixes the origin. Observe that the conjugate
mapping of an automorphism cp : As --+ As maps idempotent homomorphisms of S to idempotent homomorphisms of S, i.e. cp* : Is --+ Is. Indeed, (cp*(t))2(f) = (t(cp(f)))2 = t(cp(f))) = (cp*(t)) (f),
(cp*(t))2 = (cp*(t)). An automorphism cp of a G-shift invariant algebra As is said to be inner, if there is aTE Hom (S, S) and an element 90 E G such that cp(X a) = Xa(90) . XT(a) for every Xa E S.
Every automorphism cp of the disc algebra A(lIJ» with conjugate of type cp*(z) = Cz, ICI = 1 is inner. Indeed, for every z E IDi we have (cp(f))(z) = f(cp*(z)) = f(Cz). For Xn E Z+ : Xn(z) = zn, n ~
0 we get (cp(Xn))(z) = (cp*(z)t = (Cz)n = cnXn(z), hence cp(Xn) = cnXn = Xn(c)xn, i.e. cp is an inner automorphism. Arens and Singer [1] have shown that every automorphism cp of the algebra As is
inner in the case when G is a solenoidal group and S is a semigroup in JR with SU (-S) = G. 5 (Grigoryan, Pankrateva, Tonev [20]). If G is a solenoidal group, then either As ~ A(lIJ», or every
automorphism of the algebra As is inner. THEOREM
Proof. If the group S generated by S is not dense in JR, then the algebra As is a subalgebra of the disc algebra A(lIJ». If As =1= A(lIJ», then As C Ao(lIJ». In the same way as for the algebra Ao
(lIJ» one can see that in this case every automorphism is the composition by a Mobius transformation, fixing the origin, i.e. every automorphism is inner. If the group S generated by S is dense in
JR, then the algebra As is a subalgebra of the S-~gebra As. If cp is an automorphism of As then the bounded analytic
il == 1 on JR, since ~(j(z)) = ~(z) =
function cp(Xa)(z) does not have zeros in II. Moreover, Icp(x a) 0
Ixal == 1 on G = BAs. By the Besicovitch theorem [2], Ce is % = CXs(j(z)), where s ~ 0, C E C, ICI = 1. It is easy to see that s E S, and that the mapping
T :
r ----+ S : Xa
XS is a homomorphism from S to S.
4. Primary ideals of algebras of analytic functions on solenoidal groups. Characterizing various types of ideals is an important and interesting topic in uniform algebra theory. A proper ideal of an
algebra is said to be a primary ideal if it is contained in only one maximal ideal of the alg a. By f r .g below will be denoted the maximal ideal of functions in nish at the point r . 9 E IDi.
Recall that every primary ideal J of the disc algebra A(lIJ» which is contained in some maximal ideal of type f%O' Izol < 1, admits the representation J = un A(lIJ» , where u(z) is the unimodular
z -
_. ZOZ
6 (Grigoryan [18]). Let r = lR. and S = Il~+ If J is a primary ideal of the algebra As that is contained in I w , then either J = Xs(J) Iw. or, J = Xs(J) As; Every primary ideal I of As that is
contained in a maximal ideal of type I r .g , 1'· 9 E Jl))c has a finite codimension in As. THEOREM
Let M{3 = HJ (lR.) . exp( ij3C I ) • l!t t 2 '
13 2:: O.
Note that M{3 :) M{3' for
13 2::
13' 2:: O. 7 [18]. Let J be a primary ideal of As that is contained in Ie = Ije (0)' Then there exists a 13 2:: 0 such that J.l.. = (As).l.. + C80 + M{3. THEOREM
5. Asymptotic Almost. Periodic~Ftmctions. A function f E BC(lR.) is asymptotic almost periodic, i~an almost periodic function j(x) on lR., such that limn~oo If(x n ) - j(xn)1 = 0 for every sequence
{xn}~=l ---+ ±oo. Since h(x) = f(x) - j(x) E Co (lR.) , we have that for every asymptotic almost periodic function f 011 lR. there are unique E AP(lR.) and h E Co(lR.) such that f = + h. One can show
THEOREM 8. Let G = j3lR. be the the Bohr compactification ofR The maximal ideal space M APo (1R) of the algebra of asymptotic almost periodic functions APo(lR.) is homeomorphic to the Cartesian
product G x T.
Let r be an additive subgroup of lR., and let APr(lR.) be the set of almost periodic r-functions. Clearly, APr(lR.) EB Co(lR.) is a uniform sub algebra of APo(lR.) , containing Co (lR.). It is not
hard to see that every antisymmetric subalgebra of APo(lR.) that contains Co(lR.) is of this type. THEOREM 9. Let A be an uniform subalgebra of APo(lR.) which is invariant under lR.-shifts. Then
there is a subgroup r c lR., and a closed subalgebra Ao of Co (lR.) , such that (a) The algebra APr(lR.) of almost periodic r-functions is a closed subalgebra of A. (b) A = APr(lR.) EB Ao. (c) Ao is
an ideal in A. DEFINITION
E APo(lR.) and
6. A function f E BC(lR.) is analytic asymptotic almost periodic if f possesses a bounded analytic extension on the upper half-plane
II. Clearly, the set AAPo(lR.) of analytic asymptotic almost periodic functions on lR. is an antisymmetric uniform algebra under the sup-norm on lR., and AAPo(lR.) C APo(lR.). Note that AAP(lR.) ~
AIR+ ~ AFlR+ (lR.). Consequently, MIR+ is the G-disc K»c over the group G = j3R We have also the following results. THEOREM 10. The maximal ideal space phic to the Cartesian product K»{31R x K».
of AAPo(lR.) is homeomor-
T. TONEV AND S. GRIGORYAN
11. Let G be a solenoidal group, such that its dual group r = G is a dense subgroup oflR, and let S be an additive subsemigroup of r+ containing the origin, with [S]s = r+. Tlwn there is a continuous
projection from MH'X> onto the maximal ideal space M AAPo (lR) ~ ~G x~. THEOREM
12. The maximal ideal space of any subalgebra of AAPo(lR) of type AAPs(lR) EB B, where S c lR+ and B C Co(lR)n HoI (II), is the set M.4APs (IR)EBB THEOREM
X MB.
In particular, the upper half-plane II is not dense in the maximal ideal space of any subalgebra AAPs(lR) EB B of AAPo(lR) which contains properly AAPs(lR); The unit disc II} is not dense in the
maximal ideal space of any subalgebra of the algebra [ea~ ,a E S] EB B c Hoc n A(~ \ {1}), where S is an additive semigroup in lR, and B =f. {a} is a subalgebra of the space {J E C(1l') : 1(1) = a}.
A function 1 E BC(lR) is called weakly almost periodic, if the set of alllR-shifts, It(x) = I(x + t), t E lR is relatively weakly compact in BC(JR) (e.g. Eberline [13], Burckel [7]). If W AP(lR)
denotes the set of weakly almost periodic functions onlR, then AP(lR) C APo(lR) c W AP(lR). In fact, W AP(lR) = AP(lR) EB C([-oo, 00])11R. Similarly to Theorem 11, one can show the following THEOREM
13. The maximal ideal space MAWAP(IR) of AW AP(lR) of analytically extendable on II weakly almost periodic functions on lR is homeomorphic to tile Cartesian product ~~IR x {([a, 1] x [0,1])/([0,1] x
The space AW AP(lR) orp is isometrically isomorphic to the subalgebra of HOC n =.±! A(II} \ {I}) generated by the functions ea z-1, a E lR+ and the set of continuous functions on 1l' \ {I} that
possess both one sided limits at 1. -
III. Inductive limits and shift invariant algebras on solenoidal groups 1. Inductive limits of disc algebras on G-discs. Let A C lR+ be a basis in lR over Q, and lR = lim r h Il)' where --->h.n)EJ
Let P h .n ) = r(Y, n)+ = r h .n ) U [0,00). If Ap(-Y.n) is the algebra of analytic P h .n )functions on G, one can show that AIR+ = [ lim A p ("Y,n ) (II}G)]. A similar expression --+ h,n)EJ
holds for the algebra As, S c lR+. Uniform algebras that can be expressed as inductive limits of disc algebras A(II}) are of special interest. Consider the inverse sequence {~kH' T;H }k=I' ~k = ~ and
T;H(z) = Zdk on ~k' The limit lim {~k+l,T;+I} of the inverse sequence ......... k ..... oc
{~k+l,T;H}, is the GA-disc ~G,\ = ([0,1] x GA)/({O} x GA) over the group GA = TA . There arises a conjugate inductive sequence {A(~k)' i~H}f of algebras A(~) ~
A('Jl') with connecting homomorphisms iZ+l: A(Jijk)
A(Jijk+d defined by
(iZ+1(f))(Z) = (f(z))d k , i.e. iZ+ 1 = (7:+ 1)*. The elements of the component algebras A(Jijd can be interpreted as continuous functions on G J1. The uniform closure A(Jijc,\) = [ {A(Jijk), iZ+l}]
in C(JijCA) of
k ....... oo
the inductive limit of the system {A(Jijk), i~+1 }k=1 and the corresponding restricted {A('Jl' k), iZ+l }] are isometrically isomorphic to the GJ1-disc algebra algebra
k ....... oo
A rA +, i.e., to the algebra of analytic r,1+-functions on the GJ1-disc (e.g. [21]). Consider an inductive sequence of disc algebras
where the connecting homomorphisms iZ+ 1 : A('Jl'k) ---+ A('Jl'k+l) are embeddings with Mi~+l(A(ll'k)) = Jij and 8(iZ+ 1 (A('Jl'k))) = 'Jl'. There are finite Blaschke products
Bk :
JD) ---+ JD),
= eiIJk
IT (
~~~) )
1 - zl
, Izfk)1 < 1, such that iZ+l
Bie for
every kEN, i.e. i~+1(f) = ! 0 B k. Let B = {Bdk=l be the sequence of finite Blaschke products corresponding to iZ+ 1 , i.e. (Bk)*(Z) = iZ+1(z). Let A = {dk}~1 be the sequence of orders of Blaschke
products {Bdk=l and let rJ1 c IQl be the group generated by l/mk, k = 0,1,2, ... , where mk = I1~=1 dj, mo = 1. Consider the inverse sequence Jijl ~ Jij2 ~ Jij3 ~ Jij4 ~ ...
The inverse limit VB
= lim {Jijk+1, Bd is a Hausdorff compact space. The limit +k-oo
of the composition system {A(Jijk), ,B~+1 HO of disc algebras A(Jijk) and connecting homomorphisms ,B~+1 = Bie : A(Jijk) ---+ A(Jijk+1): (,BZ+l(f))(zk+d = !(Bk(Zk+d) is an algebra of functions on VB
whose closure [lim {A(Jijk),,B~+1}] ---t
= A(VB )
k ....... oo
in C(V B ) we call a Blaschke inductive limit algebra. It is isometrically isomorphic to the algebra [lim {A('Jl' k), ,B~+1 }]. ---t
k ....... oo
PROPOSITION 13 (Grigoryan, Tonev [21]). Let B = {Bdk=l be a sequence of finite Blaschke products and let A(V B ) = [lim {A(JD)k), Bn] be the corresponding ---t k ....... oo
inductive limit of disc algebras. Then (i) A(VB) is a uniform algebra on the compact set VB = lim {Jijk+l, Bd. +k ....... oo
(ii) The maximal ideal space of A(VB) is VB. (iii) A(VB ) is a Dirichlet algebra. (iv) A(VB ) is a maximal algebra.
T. TONEV AND S. GRIGORYAN
(v) The Shilov boundary of A('DB) is a group isomorphic to GA, and its dual 00
group is isometric to the group rA
U (l/mk)Z c
Q, where mk
THEOREM 14 [21]. Let G be a solenoidal group, i.e. G is a compact abelian group with dual group G isomorphic to a subgroup r of JR.. The G-disc algebra Ar+ is a Blaschke inductive limit of disc
algebras if and only if r is isomorphic to a subgroup ofQ. THEOREM 15 [21]. Let B = {Bdk"=1 be a sequence of finite Blaschke products on ~, each with at most one singular points zak ) and such that
Bk(Zak+ 1») = zak ). Then the algebra A('DB ) is isometrically isomorphic to the algebra A(rA)+ with A = {ddk"=1' where dk = ordBk.
In particular, if every Blaschke product Bk in the above theorem is a Mobius transformation, then the algebra A('DB) is isometrically isomorphic to the disc algebra A z = A(lI'). 2. Inductive limits
of algebras on subsets of G-discs. Let lDJ[I",1J = {z E C : r ~ JzJ ~ I}, and blDJ[r,1J = {z E C : JzJ = r or JzJ = I} is the topological boundary of lDJ[r,1J. Denote by A(lDJ[r,1J) the uniform
algebra of continuous functions on the set ~[r,1J that are analytic in its interior. Note that A(lDJ[r,1J) = R(lDJ[r,1J), the algebra of continuous rational functions on lDJ[r,1J. By a well known
result of Bishop, the Shilov boundary of A(lDJ[I·,1J) is blDJ[r,lJ, and the restriction of A(lDJ[r,1J) on blDJ[r,1J is a maximal algebra with codim (Re (A(lDJ[r,1 J)JbllJ)lr.!J)) = 1. These results
can be extended to the case of analytic r+-flllctions on solenoid groups (e.g. S. Grigoryan [19]). Let G be a solenoidal group, and its dual group is denoted as r c JR.. Let lDJ~,1J = [r,l] x G, 0 <
r < 1 be the [r,I]-annulus in the G-disc ~G, [I" 1J [I" 1J [r 1J and let A(lDJa' ) = R(lDJa' ) be the G-annulus algebra on lDJa' ,generated by the functions a E r. Let A = {d k } k"=1 be a sequence
of natural numbers and T~+1 (z) = zd k , and let r be a fixed number, 0 < r ~ 1. For every kEN consider the sets
where mk = I17=1 dl, mo = 1, and E1 = iij[r,1 J• There arises an inverse sequence
of compact subsets of iij. Consider the conjugate composition inductive sequence
where the embedings
iZ+ 1 : A(Ek)
A(Ek+d are the conjugates of zd k , namely,
(iZ+1 0 J)(z) = J(zd k ). Let GA denote the compact abelian group whose dual group
is the subgroup of Q generated by A. The algebra [lim {A(Ek)' iZ+1}] ---+
is isomorphic to A([J)~,11). THEOREM 16 [21]. Let Fn+1 = B:;;l(Fn), Fl = [J)~,11. If the Blaschke products Bn do not have singular points on the sets Fn for any n E N, then D~,11 ~ [J)~,11, and the
algebra A(D~,11) = [lim {A(Fn), B~}] is isometrically isomorphic to the ---+
G-annulus algebra A([J)~,11). Below we summarize some of the basic properties of the algebra A(D~,11) (see [21]). (a) The maximal ideal space of A(D~,11) is homeomorphic to the set [J)~,11. (b) The
Shilov boundary of A(D~,11) is the set b[J)~,11 = {r, I} x G. (c) A(D~,11) is a maximal algebra on its Shilov boundary. (d) co dim (Re(A(D~,11)lb]IJ)[r"I)) = 1. G
Let B = {B I , B 2, ... , B n , ... } be a sequence of finite Blaschke products on ii} and let 0 < r < 1. Let D n+ l = B:;;l (Dn), Dl = [J)[O,r1 = {z E [J) : Izl ::; r}. Consider Tn\[O,r1
D.........fu. 4
.•. VB
of subsets of [J). The inductive limit A(D~,r1) = [lim {A(Dn), B~}] is a uniform ---+
algebra on its maximal ideal space ~ {Dn, Bn-tlDn} = D~,r1
c DB.
PROPOSITION 14 [21]. LetB = {B l ,B2,B3, ... } be a sequenceoffinite Blaschke products on ii} and let 0 < r < 1. Suppose that the set Dn does not contain singular points of B n- l for every n E N.
Then (i) There is a compact set Y such that D~,r1 = ~ {D n+ l , BnIDn+l} n-+oo
M A(D~.rl) is homeomorphic to the Cartesian product [J)[O,r1 x Y. (ii) A(D~,r1) is isometrically isomorphic to an algebra of functions f(x,y) E C([J)[O,r1 x Y), such that f(· ,y) E A([J)[O,r1) for
every y E Y.
(iii) A(D~,r1) 1]IJ)[o.rl x {y} ~ A([J)[O,r]) for every y E Y. The proof makes use of the fact that every finite Blaschke product of order n generates an n-sheeted covering over any simply connected
domain V c [J) free of singular points of B. Proposition 14 implies that the one-point Gleason parts of the algebra A(D~,r]) are the points of the Shilov boundary bD~,r1 ~ 'll'r x Y. PROPOSITION 15
[21]. Let B = {B I , B 2, B3,"'} be a sequence offinite Blaschke products on ii}, and let 0 < r < 1. Suppose that (a) For every n E N the points of the set :F = (Bl 0 B2 0 ' " 0 Bn_d-l(O) are the
only singular points for B n- l in Dn (b) All points in (a) have one and the same order dn - l > 1.
T. TONEV AND S. GRIGORYAN
Then (i) There is a compact Y such that
V~·r] = ~ {Ir»n+1' BnID,,+!} = M A(V~.rl) k-+oo
is homeomorphic to the Cartesian product Ir»~:] x Y, where A = {dd~1 is the sequence of the orders of B k . (ii) The algebra A(V~,r]) on V~,r] is isometrically isomorphic to an algebra of functions f
(x, y) E C(Ir»~:] x Y), such that f( . ,y) E A(Ir»~:]) for every yEY. (iii) A(V~,r])IIIli[(l.rlx{y}
= A(Ir»~:]) for every y
E Y.
The set Y in Propositions 14 and 15 is homeomorphic to the set {{yn}~=I' Yn E (Bl 0 B2 0 ' " 0 Bn_t}-I(O)}. Proposition 15 implies that there are no single-point Gleason parts of the algebra A(V~,r])
within the set M A(V~.rl) \ bV~,r] U {w} x Y, where w is the origin of the G A-disc ll}a/l' As an immediate consequence we obtain that A(V~,r]) is isomorphic to a Gdisc algebra if and only if the set
Y consists of one point. In particular, in the above setting the algebra A(V~'1']) is isomorphic to a G-disc algebra if and only if every Blaschke product Bn has a single singular point z6n ) in D~)
such that B n (Z6 n») = z6n+1) for all n hig enough. 3. Gleason parts of inductive limits of disc algebras on G-discs. The celebrated theorem by Wermer [36] asserts that in every non-single-point
Gleason part of the maximal ideal space of a Dirichlet algebra can be embedded an analytic disc. Therefore it is of particular interest to locate single-point Gleason parts of an algebra, and
especially those of them that do not belong to the Shilov boundary. While every point in the Shilov boundary is a separate Gleason part (e.g. Gamelin [14]), the opposite is not always true, i.e.
there are single-point Gleason parts outside the Shilov boundary. For instance, if G is a solenoid group with a dense in lR. dual group, then the origin w = ({O} x G)/( {O} x G) E Ir»a of the G-disc
Ir»a is a single-point Gleason part for the G-disc algebra Ar+. Of course w (j. 8 Ar+ = G.
Given a sequence of Blaschke products B = {Bn}~=1 on ll}, consider the Blaschke inductive limit algebra A(V B ) = [lim {A(ll}k) , Bn] on the compact ---> k-+oo
set VB = lim {ll}k, Bk-d. Recall that the Shilov boundary of A(VB ) is the group f--
TB = lim {1l'k' Bk-d. Let Br be the set of all Blaschke products on ll} whose zeros f-k-+oo
are inside the disc Ir»[O,r] ones at O.
= {Izl
~ r}, and let B~
c Br
be the set of the vanishing
PROPOSITION 16 [21]. Let B be a finite Blaschke product with B(O) = O. Consider the sequence B = {B, B, ... }. If the Blaschke inductive limit algebra A(VB) = [lim {A(Ir»k) , Bd]' Ir»k = Ir», Bk = B
is isometrically isomorphic to a ---> k-+oo
G-disc algebra, then necessarily B(z) = cz", where c E Ir»,
lei =
1, and n E N.
THEOREM 17 (Grigoryan, Tonev [21]). Let B be a finite Blaschke product on IDJ. The Blaschke inductive limit algebra A(D B ) is isometrically isomorphic to a G-di8c algebra if and only if B(z) is
conjugate to a power z'" of z, i.e. if and only if there is an mEN and a Mobius transformation 7 : IDJ --t IDJ such that (7- 1 OB 0 7)(Z) = zm.
THEOREM 18 [21J. Suppose that Bn E and ordBn > 1 for every Then tllere is only one single-point Gleason part in the set DB \ TB.
E N.
In particular, if B E Sr, B(O) =f. 0, and Bk(z) = zd k BCk, dk > 1 then there is only one single-point Gleason part in the set DB \ TB. The proof of Theorem 18 involves a thorough study of one-point
Gleason parts of the algebras involved. 4. Inductive limits of Hoo-spaces on G-discs. Let I = {iZ+ 1}k'=1 be a sequence of homomorphisms iZ+l : HOO(IDJ) --t HOO(IDJ). Consider the inductive sequence
HOO(lDJd -iL H OO (1DJ 2) 2L HOO(IDJ3) -.fL ... of algebras HOO(lDJ k ) ~ HOO(IDJ). Every adjoint mapping (iZ+l)* : Mk f-- Mk+l maps the maximal ideal space Mk+l of HOO(IDJk+d into the maximal ideal
space Mk of HOO(lDJ k ). The inverse limit
1;2)" ~1_
1;3)" ~2_
1;4)* ~3_
1;5)" ~4_
••• f - -
is the maximal ideal space of the inductive limit algebra
HOO(DI) = [lim {HOO(lDJk),i~+I}J. ---+ k-+oo Recall that the open unit disc IDJ is a dense subset of every Mk. In general, the mappings (i~+l)* are not obliged to map IDJ k+1 onto itself. The most
interesting situations, though, are when they do. Here we suppose that the mappings (iZ+l)* are inner non-constant functions on IDJ. For instance, algebras of type H oo (D I ) are the algebras [lim
{H OO (lDJ k), (zdk)*}dkEAJ = HOO(D A) c H OO (IDJ CA ), and also ---+ k-+oo the algebras of type HOO(DB) = [~{HOO(IDJd, Bk}], where B = {Bdk'=1 is k-+oo a sequence of finite Blaschke products Bk :
IDJ --t IDJ. Note that HOO(DB) is a commutative Banach algebra of functions on DB. Let A = {dd~1 be the sequence of orders of Blaschke products {Bd~1 from the mentioned above example, and let rA C Q
be the group generated by l/mk, k = 0,1,2, ... , where mk = I1~=1 d/, mo = 1. THEOREM 19 [21J. Let B = {B k }k'=1 be a sequence of finite Blaschke products on ~, each with at most one singular points
zbk ), and such that Bk(zbk+ 1») = zbk ). Then the algebra HOO(D B ) is isometrically isomolphic to the algebra HOO(DA) for A = {ddk'=1 with dk = ordBk. For instance, if the Blaschke products Bk are
of type Bk(Z) = zd k 'Pdz), where 'Pk are Mobius transformations and dk > 1, then the algebra HOO(D B ) is isometrically isomorphic to the algebra HOO(D A), where A = {1/dd~I' If every Blaschke
T. TONEV AND S. GRIGORYAN
product Bk in Theorem 19 is a Mobius transformation, then the algebra HOO(DB) is isometrically isomorphic to the algebra H oo . Indeed, the last theorem implies that HOO(D B ) ~ HOO(DA) with A = {I,
I, ... }. Therefore rA = Z and GA = T. Let iP = {'P1, 'P2, ... ,'Pk, ... } be a sequence of non-constant inner functions on ][)l. Consider the inverse sequence ][)l1 +---'£l. ][)l2 where][)lk
Denote by Dq, its inverse limit. The inductive limit lim {HZ", 'PkH" ---t
of the adjoint composition inductive sequence
Hf" 1.4 H2'
... 10 'Pk,
of algebras HZ" = HOO(][)lk) ~ HOO(][)l), where 'PkU) = is a subalgebra of BC(Dq,), the algebra of bounded continuous functions on the set Dq,. The closure HOO(Dq,) of lim {H OO , 'Pk} in BC(Dq,) is
a commutative Banach subalgebra of ---t
HOO(][)le). Its elements are referred to as iP-hyper-analytic lunctions on Dq,. Recall that according to the classical corona theorem for the space HOO on the unit circle (Carleson [8]), given h, ...
, /k, functions in Hoo with L~=lllil :::: a > 0 on ][)l, there exist functions gl, ... ,gk in Hoo such that L~=lligj = 1 on ][)l; If IIIi 1100 :::; 1, then 9j can be chosen to satisfy the estimates
II 9j II :::; C (k, a) for some constant C(k, a) > O. Next theorem is the corona problem for the algebra HOO(Dq,). THEOREM 20 (Grigoryan, Tonev [21]). If h, 12, ... , In, IIIi II :::; 1, are
iP-hyperanalytic functions on Dq, for which Ih(x)1 + ... + I/n(x)1 :::: 8 > 0 for each x E Dq" then there is a constant K(n,8) and iP-hyper-analytic functions gl, ... , gn on Dq, with Ilgj II :::; K
(n, 8), such that the equality h (x )gl (x) + ... + In (x )gn (x) = 1 holds for every point x in the set Dq,. In the case when iP = {Z2, z3, ... , zn+1 ... } the corresponding set Dq, coincides with
the open big disc ][)le over the compact abelian group G = ij, and the algebra HOO(Dq,) coincides with the set He of hyper analytic functions. In this case Theorem 20 reduces to the corona theorem
for the algebra He of hyper-analytic functions on G with estimates (cf. Tonev [32]). 5. Hoo-spaces on solenoidal groups. Let G be a solenoidal abelian group, i.e. r = G c R Let HOO(][)le) be the
algebra of bounded functions in the open G-disc ][)le that can be approximated on compact subsets of][)le by functions 1, I E Ar+. For every I E HOO(][)le), the limits
f*(g) = lim I(r)(g), where I(r)(g) = r-+1
f(r. g)
exist for almost all 9 E G, and f* E Hoo (G, a). The space of functions f*, I E HOO(][)le) we denote again by HOO(][)le). The algebra HOO(][)le) we interpret as a subspace of the set of functions in
LOO(G,a) that are boundary values of continuous functions on ][)le, equipped with the norm 11/1100 = lim sup I/(r)(g)l. Denote
r-+1 gEe
by 'HOC(lD>a) the weak*-closure of Ar+ in LOC(G,a) (cf. Gamelin [14]). Clearly HOC(lD>a) is a closed subalgebra of 'HOC (lD>a). Let I be a directed set. We consider a family {rdiEI of subgroups of r
indexed by I, such that ril C r i2 whenever il -< i2. Let r = limri , and H~(lD>a) denotes --+ iEI the space of functions f E Hoc (lD>a) with sp (I) C ri . The family {Hr: (lD>a) hEI of subalgebras
in HOC(lD>a) is ordered by inclusion. Denote by Hr(lD>a) the closure of the set U H~(lD>a) = limH~(lD>a) with respect to the norm II . lIoc. Hr(lD>a) iEI ~ is the set of I-hyper-analytic functions on
lD>a. In a similar way we define the algebra 'Hf(lD>a) as the II . Iloc-closure of the inductive limit ~ 'H~(lD>a), where iEI 'H~(lD>a) = {f E 'HOC(lD>a) : sp (I) c rd· THEOREM 21 (Grigoryan, Tonev
[22]). Let G be a solenoidal group such that its dual group r = G is the inductive limit of a family {rdiEI of subgroups of r, i.e. r = lim r • Let Hr='• (lD>a) and 'Hr', (lD>a) be the spaces
offunctions in Hoc (lD>a) --+ i iEI [resp. in 'HOC(lD>a)] with spectra in ri , i E I. Then the following statements are equivalent. (a) HOO(lD>a) = Hr(lD>a) and 'HOC(lD>a) = 'Hf(lD>a). (b) HOC(lD>a)
= U Hr:(lD>a) and 'HOC(lD>a) = U 'H~(lD>a). iEI iEI (c) Every countable subgroup ro in r is contained in some group from the family {rdiEI. Example 4. Let r = Q be the group of rational numbers with
the discrete topology. Assume that {rdiEI is an inductive system of subgroups of Q such that Q = lim n. The last theorem implies that if Q itself is not one of the groups in the --+ iEI family
{rdiEI, then Hr(lD>a) =I- H""(lD>a). In the case when all subgroups ri , i E I are isomorphic to Z, the algebra Hr(lD>a) coincides with the algebra of hyper-analytic functions (e.g. [34]). As seen
above, in this case the space Hr (lD>a) differs from HOC (lD>a). The properties of subalgebras of HOC (lD>a) on general compact groups G are less known. In particular it is not known if they possess
a corona, and their maximal ideal spaces and Shilov boundaries lack a good description. Example 5. Let r = R and let A C R+ be a basis in R over the field Q of rational numbers. Then R = ~ r(-y,n),
where (-y,n)EJ
Given an (-y, n) E J, consider the set
T. TONEV AND S. GRIGORYAN
The closure HJ'(lJJJc) of the set
H0',n)(lJJJc) under the
II . lloo-norm, i.e.
inductive limit algebra
lim H(oo-y,n )(lJJJc) is a subalgebra of HOC (lJJJc). There arises _ (-y.n)EJ
the question of whether or not the algebra HJ'(lJJJ c ) coincides with HOO(lJJJ c ). THEOREM 22 (Grigoryan, Tonev [22]). The set HJ(lJJJ c ) =
lim H(oo-y,n ) (lJJJ c ) _ (-y,n)EJ
is a proper closed subalgebra of H OO (lJJJ c ).
c HOO(lJJJ c ) is easy (e.g. [12]). Assume that H0',n) (lJJJc). By the previous theorem, the countable
Proof. The inclusion HJ'(lJJJc)
HOO(lJJJ c ) = HJ(lJJJc) =
~ (-y,n)EJ
subgroup Q c IR is a group in the family {rh,n) h-y,n)EJ' which is impossible since r(-y,n) is isomorphic to -Z} for some kEN.
The algebra HOO(lJJJ c ) is isometrically isomorphic to the algebra HfPr+(IR)(IR) C
HOO (1R) of boundary values of almost periodic r+-functions on IR that are analytic in the upper half plane. Similarly, the algebra HJ'(lJJJ c ) is isomorphic to a subalgebra HJ'(IR) of HZ,r+ (R)
(1R). As the last theorem shows, these algebras are different. Algebras of type HI (lJJJ c ) were introduced in connection with the corona problem for algebras of analytic r-functions (Tonev [32]).
R. Curto, P. Muhly and J. Xia [12J have introduced similar algebras of this type in connection with their study of Wiener-Hopf operators with almost periodic symbols.
6. Bourgain algebras and inductive limits of algebras. Bourgain algebras of a Banach space were introduced in 1987 by J. Cima and R. Timoney [9J as a natural extension of a construction due to J.
Bourgain [5J. The concept of tightness of an algebra was introduced by Cole and Gamelin [lOJ. Let A c B be two commutative Banach algebras, and 11" A : B ----t B I A is the natural projection. The
mapping Sf: A ----t (f A+A)IA c BIA; Sf: 9 ~ 1I"A(fg) is called the Hankel type operator. DEFINITION 7. An element fEB is said to be (a) a Bourgain element, (b) a wc-element, (c) a c-element for A,
if the Hankel type operator Sf : A ----t BIA is correspondingly (a) completely continuous, (b) weakly compact, (c) compact. The Bourgain algebra of A relative to B is the space A~ of all Bourgain
elements for A in B [9J. PROPOSITION 17 [35J. If the range Sf(A) = 1I"A(f A) of the Hankel type operator Sf for an fEB is finite dimensional then f E A~ In particular, (As)f(C) = C(G) if As is a
maximal algebra and xS \ X is a finite set for a character X E O\S. Indeed, X E (As)f(C) by the above proposition. Since X ~ S, then X ~ As, and consequently (As)f(C) = C(G) by the maximality of A.
Example 6. Let H be a finite group, G = (H E& Zr and S So' H E& Z+. Then (As)f(G) = C(G).
r- r
Note that if = G and XS\ X is finite for every X E then every character X E G has finitely many predecessors in r. As it follows from Proposition 17, (Ar)f(G) = C(G), and therefore the corresponding
big disc algebra Ar possesses the Dunford-Pettis property. THEOREM 23 (Yale, Tonev) [35]. Let G =,BJR be the Bohr compactification of lR. The Bourgain algebra (AIR+ )f(G) of the big disc algebra AIR+
is isomorphic to AIR+' Proof. Clearly, JR is a subset of (AIR+ )f(G). Since, as one can easily see, the
seqnence of real valued functions 'Pn(x) = as n
00 for every x
11 +2ei ';i 12n
converges pointwisely to
2n . JR, then the real valued functIOns 'l/Jn(g) = 11 + X2-a-(g)1
converge pointwisely to 1 as n --+ 00 for every 9 E G. Suppose that X3 E (AIR+ )f(G). Consider the sequence ~n(g) = 'l/Jn(g) -1, where 'l/Jn is as before. {X1~n}n is a weakly null sequence in AR+
since it converges pointwisely to 0 on G. By the Bourgain algebra property there are functions hn E AIR+ such that IIX3X1~n - hnll < l/n for every n, where II . II is the sup norm on G. By
integrating over Ker(x~), if necessary, we can assume that hn
then (X',pn)(g)
1 • = qn(X") for some polynomIal qn'
~ (x«g))" ( 1 + ;~(g))
2n the j-th Cesaro mean
af" =
= max I(X1~n)(g) gE G = ~Eas IPn (x-a- (g)) ( z)
Fa, j
= (1 + z)2n + 2n(1 + z)2n-1 = (2n +
= max I(X 1'I/Jn)(g) gE G
- X1(g) - (x-a- (g)) 3n qn P (z)
~~(g))" ~ Pn(X" (g)).
= max 1(~X1~n)(g) - hn(g)1 gE G
= (1+Z)2n -2,
SO+S1+"'+S, j +1 J of Pn, where Sk is the k-th
partial sum of Pn, we have 4n(2n + l)a~~(z) 1 + z)(l + z)2n-1. Now II~X1~n - hnll
(1 +
If Pn(z)
X1(g) - X3(g)h n (g)1
(x~ (g)) I = ~tf IPn(Z)_zn_ z3nqn (z)l·
z3n q (z)
P Note that a" " (z) because the Cesaro mean a2n de2n - (z) = a 2nn - pends only on the first 2n terms of the Taylor series. Since the inequality max la~(z)1
f E A(1l') we see that
:::; max If(z)1 holds for every zE'lr
maxlaPn ( zE'lr
p (z) = maxla n zE'lr 2n
z3n q (z) n
T. TONEV AND S. GRIGORYAN
However, O"~~(z)_zn (z) = O"~~(z) (z) - zn(n + 1)/(2n + 1) and thus O"~~(z)_zn (-1)
1/2 as n --> 00 for odd n. Hence AIR+ by the maximality of AIR+'
-X3 f/. (AIR+)f(O) and consequently (AIR+)f(O) =
THEOREM 24 (Tonev [33]). Let {AO" }O"EL', {BO" }O"EE be two monotone increasing families of closed subspaces of a commutative Banach algebra B such that BO" are algebras, and AO" c BO" for every 0" E
E. Let A = [ U AO"] be a (linear) subO"EE
space, and let B = [ U BO"] be a subalgebra of B. Suppose that for every
E E
tllere is a bounded linear mapping r 0" : B --> BO", such that (i) rO"IB" = idB" (ii) rO"(fg) = frO"(g) for every f E BO", 9 E B (iii) sup Ilr0" II < 00. O"EE Then A~ c [ U (AO")~"]. O"EE Proof. Let
fEB be a Bourgain element for A. Fix a 0" E E, and consider a weakly null sequence {
O. Hence,
Consequently, r 0" (f) is a Bourgain element for AO", i.e. r 0" (f) E (AO")~" for every E E. Note that under the hypotheses every fEB is approximable by the elements rO"(f) in the norm of B. Indeed,
let fO"n E BO"n be such that fO"n --> f. Then Ilf-rO"n(f)II::; Ilf-fO"nll+llrO"n(fO"J-rO"..(f)II::; Ilf-fO""II+supllrO"..IlllfO"n -fll· Hence rO"n(f) --> f and, consequently, f E [ U (AO")~"]. O"EE
lim rO", let Hf? = {! E HOO(J]))o) : sp (f) c rO"}. Note that H't:.. is a --+ " O"EE closed sub algebra of HOO(J]))o), and H'f:. c Hr:. if and only if rO" err. Therefore, the family {H'f:. }O"EE of
subalgebras of HOO (J]))o) is ordered by the inclusion. Denote by H~ the closure of the set U H'f:. with respect to the norm II . 1100' Theorem O"EE 24 implies that if r = lim rO" and G = f, then the
Bourgain elements for H roo are --+ + O"EE approximable in the L 00 - norm on G by Bourgain elements for H'f:., 0" E E. Note that H~, HOO(J]))o), and the weak* closure HOO(G, dO") of Ar+ in LOO(G,
dO") are commutative Banach subalgebras of LOO(G,dO"), which are in principle different from each other, except in the case of G = 'll', when they coincide (Grigoryan [19]).
If r
The algebra HQ;/n = H OO 0 Xl/n = {! 0 Xl/n: f E HOO} is a closed subalgebra of HOO(J]))o). The closure HQ' of U HQ' with respect to the norm II . 1100 is the + nEN lin algebra of hyper-analytic
junctions on G = f3Q (cf. Tonev [34]). By Theorem 24 the Bourgain algebra of is contained in the algebra + C(G).
THEOREM 25 (Tonev [33]). If the hypotheses of Theorem 24 are satisfied. then A~c c [ U (AO')~~]; A~ c [ U (AO')~O']; O'EE
(H~J~:(G) .
c [U
(H~)~:(G)]; (HrJ~OO(G)
In particular, the algebra H~
c [U
+ C(,8Q)
contains the spaces (H~)~:({3Q) and
(H~ )~OO({3Q). A uniform algebra A C C(X) is said to be tight [strongly tight] if every f E C(X) is a we-element [resp. c-element] for A, i.e. if (A)~~G) = C(X) [resp. (A)f(G) = C(X)] (cf. Cole,
Gamelin [10], also Saccone [30]). Theorem 25 implies that the algebra H~ is neither tight nor strongly tight.
1. R. Arens and 1. Singer, Genemlized analytic functions, Trans. Amer. Math. Soc., 81(1956), 379-393. 2. A. Besicovitch, Almost Periodic Functions, Cambridge University Press, 1932. 3. S. Bochner,
Boundary values of analytic funct'ions in seveml variables and of almost periodic functions, Ann. Math., 45(1944), 708-722. 4. H. Bohr, Zur Theorie der fastperiodischen Funktionen, III.
Dirichletentwicklung analytischer Funktionen, Acta Math., 47(1926), 237-281. 5. J. Bourgain, The Dunford-Pettis property for the ball algebm, polydiscalgebm and the Sobolev spaces, Studia Math., 77
(1984), 245-253. 6. A. Boettcher, On the corona theorem for almost periodic functions, Integral Equations and Operator Theory, 33(1999), 253-272. 7. R. Burckel, Weakly almost periodic functions on
semigroups, Gordon and Breach, 1970. 8. L. Carleson, Interpolation by bounded analytic functions and the corona problem, Ann. of Math., 76(1962), 542-559. 9. J. Cima and R. Timoney, The
Dunford-Pettis property for certain planar uniform algebms, Michigan Math. J. 34(1987), 66-104. 10. B. Cole and T. W. Gamelin, Tight uniform algebms, J. Funct. Anal. 46(1982), 158-220. 11. C.
Corduneanu, Almost Periodic Functions, Interscience, N.Y., 1968. 12. R. Curto, P. Muhly and J. Xia, Wiener-Hopf opemtors and genemlized analytic functions, Integral Equations and Operator Theory, 8
(1985), 650-673. 13. W.F. Eberline, Abstmct ergodic theorems and weak almost periodic functions, Trans. Amer. Math. Soc., 67(1949), 217-240. 14. T.W. Gamelin, Uniform Algebms, 2nd ed., Chelsea, New
York, 1984. 15. 1. Glicksberg, Maximal algebms and a theorem of Rad6, Pacific J. Math. 14(1964), 919-941. 16. S. Grigoryan, Algebms of finite type on a compact group G, Izv. Akad. Nauk Armyan. SSR,
Ser. Mat. 14(1979), No.3, 168-183. 17. S. A. Grigorian, Maximal algebms of genemlized analytic functions (Russian), Izv. Akad. Nauk Armyan. SSR Ser. Mat. 16(1981), no. 5, 358-365.
T. TONEV AND S. GRIGORYAN
18. S. A. Grigoryan, Primary ideals of algebras of generalized analytic functions, J. Contemp. Math. Anal. 34, no. 3(1999), 26-43. 19. S. A. Grigorian, Generalized analytic functions, Uspekhi Mat.
Nauk 49(1994), 3-42. 20. S. Grigoryan, T. Ponkrateva, and T. Tonev, Inner Automorphisms of Shiftinvariant Algebras on Compact Groups, J. Contemp. Math. Anal., Armen. Acad. ScL, 5(1999), 57-62. 21. S.
Grigoryan and T. Tonev, Blaschke inductive limits of uniform algebras, International J. Math. and Math. Sci., 27, No. 10(2001), 599-620. 22. S. A. Grigoryan and T. V. Tonev, Inductive limits of
algebras of generalized analytic functions, Michigan Math. J., 42(1995), 613-619. 23. S. A. Grigoryan T. Ponkrateva, and T. V. Tonev, The validity range of two complex analysis theorems, Compl.
Variables, (2002). 24. S. Grigoryan and T. Tonev, Inductive limits of algebras of generalized analytic functions, Michigan Math. J., 42(1995), 613-619. 25. S. A. Grigoryan and T. V. Tonev, Linear
multiplicative functionals of algebras of S-analytic functions on groups, Lobachevsky Math. J., 9(2001), 29-35. 26. B. Jessen, Uber die Nullstellen einer analytischen fastperiodischen Funktion. Eine
Verallgemeinerung der Jensenschen Formel, Math. Ann. 108(1933),485-516. 27. E. R. van Kampen, On almost periodic functions of constant absolute values, J. Lond. Math. Soc. XII, No. 1(1937), 3-6. 28.
G. M. Leibowitz, Lectures on Complex Function Algebras, Scott, Foresman and Co., Glenview, IL, 1970. 29. L. Loomis, Introduction in Abstract Harmonic Analysis, Van Nostrand, Princeton, N.J., 1953.
30. S. Saccone, Banach space properties of strongly tight uniform algebras, Studia Math., 114(1995), 159-180. 31. A. Sherstnev, An analog of the Hahn-Banach theorem for commutative semigroups,
preprint. 32. T. Tonev, The Banach algebra of bounded hyper-analytic functions on the big disc has no corona, Analytic Functions, Lect. Notes in Math., Springer Verlag, 798(1980), 435-438. 33. T.
Tonev, Bourgain algebras and inductive limit algebras, In: Function Spaces (Ed. K. Jarosz), Contemporary Math., AMS, 232(1999), 339-344. 34. T. Tonev, Big-planes, Boundaries and Function Algebras,
Elsevier - NorthHolland, 1992. 35. K. Yale and T. Tonev, Bourgain algebras and the big disc algebra, Rocky Mountain J. Math., preprint. 36. J. Wermer, Dirichlet algebras, Duke Math. J., 27(1960),
DEPARTMENT OF MATHEMATICAL SCIENCES, UNIVERSITY OF MONTANA, MISSOULA, MONTANA 59812-1032 CHEBOTAREV INSTITUTE OF MATHEMATICS AND MECHANICS, KAZAN, SIA | {"url":"https://limebomber.com/article/function-spaces-pdf-free-download","timestamp":"2024-11-09T10:53:18Z","content_type":"text/html","content_length":"908963","record_id":"<urn:uuid:5b975990-c6ba-4f83-9e16-adb4f90fecde>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00462.warc.gz"} |
Playables.PlayableGraph - Unity 스크립팅 API
The PlayableGraph is also the link to different systems, through structs that implement IPlayableOutput. For example, AnimationPlayableOutput or AudioPlayableOutput.
Connect Connects two Playable instances.
Destroy Destroys the graph.
DestroyOutput Destroys the PlayableOutput.
DestroyPlayable Destroys the Playable.
DestroySubgraph Destroys the Playable and all its inputs, recursively.
Disconnect Disconnects the Playable. The connections determine the topology of the PlayableGraph and how it is evaluated.
Evaluate Evaluates all the PlayableOutputs in the graph, and updates all the connected Playables in the graph.
GetEditorName Returns the name of the PlayableGraph.
GetOutput Get PlayableOutput at the given index in the graph.
GetOutputByType Get PlayableOutput of the requested type at the given index in the graph.
GetOutputCount Returns the number of PlayableOutput in the graph.
GetOutputCountByType Get the number of PlayableOutput of the requested type in the graph.
GetPlayableCount Returns the number of Playable owned by the Graph.
GetResolver Returns the table used by the graph to resolve ExposedReferences.
GetRootPlayable Returns the Playable with no output connections at the given index.
GetRootPlayableCount Returns the number of Playable owned by the Graph that have no connected outputs.
GetTimeUpdateMode Returns how time is incremented when playing back.
IsDone Indicates that a graph has completed its operations.
IsPlaying Indicates that a graph is presently running.
IsValid Returns true if the PlayableGraph has been properly constructed using PlayableGraph.CreateGraph and is not deleted.
Play Plays the graph.
SetResolver Changes the table used by the graph to resolve ExposedReferences.
SetTimeUpdateMode Changes how time is incremented when playing back.
Stop Stops the graph, if it is playing.
"Unity"、Unity 徽标及其他 Unity 商标是 Unity Technologies 或其附属机构在美国及其他地区的商标或注册商标。其他名称或品牌是其各自所有者的商标。 | {"url":"https://docs.unity.cn/kr/2019.3/ScriptReference/Playables.PlayableGraph.html","timestamp":"2024-11-05T09:40:54Z","content_type":"text/html","content_length":"29580","record_id":"<urn:uuid:de1771bf-2cea-496b-b354-3a6c48c5ed4c>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00493.warc.gz"} |
Types of Ripple Frequency in context of ripple frequency
27 Aug 2024
Title: An Exploration of Types of Ripple Frequencies in the Context of Power Electronics
Abstract: Ripple frequencies are a crucial aspect of power electronics, influencing the performance and efficiency of various systems. This article delves into the different types of ripple
frequencies, their characteristics, and implications on system design.
In power electronic systems, ripple frequency refers to the periodic fluctuations in output voltage or current that occur due to the switching action of power devices. These fluctuations can be
categorized into several types based on their characteristics and effects on system performance.
Types of Ripple Frequencies:
1. Switching Frequency Ripple (fs)
The switching frequency ripple is the most fundamental type, occurring at the same frequency as the switching device’s switching cycle. It is characterized by a high-frequency oscillation in the
output voltage or current.
fs = 1 / Tsw
where Tsw is the switching period.
2. Resonant Frequency Ripple (fr)
The resonant frequency ripple occurs when the system’s resonant frequency coincides with the switching frequency, leading to a significant amplification of the ripple amplitude.
fr = sqrt(L/C)
where L is the inductance and C is the capacitance.
3. Harmonic Frequency Ripple (fh)
The harmonic frequency ripple arises from the non-linear behavior of power devices, resulting in the generation of harmonics at frequencies that are integer multiples of the switching frequency.
fh = n * fs
where n is an integer.
4. Intermodulation Frequency Ripple (fim)
The intermodulation frequency ripple occurs when two or more different frequencies interact with each other, producing new frequencies that are not present in the original signal.
fim = f1 ± f2
where f1 and f2 are the two interacting frequencies.
In conclusion, this article has explored the various types of ripple frequencies that occur in power electronic systems. Understanding these different types is essential for designing efficient and
reliable systems, as they can significantly impact system performance and efficiency.
Related articles for ‘ripple frequency’ :
• Reading: Types of Ripple Frequency in context of ripple frequency
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Central limit theorems for classical likelihood ratio tests for high-dimensional normal distributions
For random samples of size n obtained from p-variate normal distributions, we consider the classical likelihood ratio tests (LRT) for their means and covariance matrices in the high-dimensional
setting. These test statistics have been extensively studied in multivariate analysis, and their limiting distributions under the null hypothesis were proved to be chi-square distributions as n goes
to infinity and p remains fixed. In this paper, we consider the high-dimensional case where both p and n go to infinity with p/n → y ∈ (0, 1]. We prove that the likelihood ratio test statistics under
this assumption will converge in distribution to normal distributions with explicit means and variances. We perform the simulation study to show that the likelihood ratio tests using our central
limit theorems outperform those using the traditional chisquare approximations for analyzing high-dimensional data.
Bibliographical note
Publisher Copyright:
© Institute of Mathematical Statistics, 2013.
• Central limit theorem
• Covariance matrix
• High-dimensional data
• Hypothesis test
• Likelihood ratio test
• Mean vector
• Multivariate Gamma function
• Multivariate normal distribution
Dive into the research topics of 'Central limit theorems for classical likelihood ratio tests for high-dimensional normal distributions'. Together they form a unique fingerprint. | {"url":"https://experts.umn.edu/en/publications/central-limit-theorems-for-classical-likelihood-ratio-tests-for-h","timestamp":"2024-11-12T09:57:56Z","content_type":"text/html","content_length":"52085","record_id":"<urn:uuid:a702bc43-2c7c-4199-bd42-91bab53b8cc4>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00325.warc.gz"} |
proffuaad - SIPE_TUDelft
System Identification and Parameter Estimation
Lecture 1 - Introduction
The first lecture gives an overview of the course and a general introduction to system identification and parameter estimation. System identification, in this course, tries to elucidate the dynamic
relation between time-signals and to parameterize this relation in a mathematical model (where the model is based on differential equations). In this course emphasis is paid to system identification
in frequency domain. Key element of this approach is the Fourier transform. Major advantage of the frequency domain approach is that no a-priori knowledge is required of the type of the model (order
of the system). Every recorded time-signal will be contaminated with noise. Noise is, by nature, a random process and consequently measured signals are stochastic. In stochastic theory not the
individual realization is important but the statistical properties are, e.g. mean, standard deviation, and also probability density functions. With ergodicity it is thought that one, sufficiently
long, realization is representative for many realizations. This implicates that it is sufficient to capture one recording of a signal to assess its (statistical) properties (in stead of multiple
recordings). Cross-product and cross-covariance functions are measures to estimate the relation between two (stochastic) signals in time-domain.
A zipfile containing a Matlab model used for demonstrating noise removal in a second order system by averaging a number of realizations and the effect of a stochastic input signal:
Lecture 2 - Correlation functions in time and frequency domain
This lecture will first explain the correlation functions in time domain, then general properties of estimators and how to estimate these correlation functions will be explained. After that the
Fourier transform will be reviewed and the lecture ends with the explanation of the correlation functions in frequency domain.
Lecture 3 - Impulse and frequency response functions
This lecture will first discuss the estimation of an Impulse Response Function (IRF) and a Frequency Responce Function (FRF), then improving the estimation of spectral densities by using the "Welch"
method and frequency averaging will be explained. Other topics are open-loop versus closed-loop situations and coherency. A zipfile containing some examples of impulse and frequency response
functions in Matlab: Lec3_examples.zip
Lecture 4 - Assignment 1 & Perturbation Signal Design
This lecture will first explain the answers of Assignment 1 and then sources for error in estimators and ways to improve the estimation will be discussed. In most cases system identification is a
battle against noise. One should try to decrease the power of the noise and/or increase the power of the signal. Several methods exist to boost the power of the signal and such improve the
signal-to-noise ratio (SNR). However random signals always introduce leakage, an effect of the observation time and resulting discrete frequency resolution. Multisine signals are composed of multiple
sines. These deterministic signal do not introduce leakage and as the power is distribute over a limited number of frequencies the power per frequency can be high. With cresting, a technique to
minimize the ratio between the outliers of the time and the standard deviation of the signal, the power can even be further increased. And the effect of the input signal on the system identification
procedure is discussed.
Lecture 5 - Open- and Closed Loop Systems & Multivariable Systems
This lecture gives a little background on estimators in general and describes the ins and outs of identification of closed loop systems. An estimator gives an estimate for a certain ‘true’ variable
or function. All estimators have an error; this error can be divided in random errors (variance of the estimator) and structural errors (bias of the estimator). A good estimator has negligible, or
no, bias and low variance. With a consistent estimator the variance of the estimator reduces with the number of samples. The (raw = non-averaged) estimator for the spectral density based is not
consistent! With increasing observation time the resolution in frequency domain increases, but the variance of this estimator remains equal. Furthermore using the raw estimators of the spectral
density the estimator for the coherence is always 1. The variance of the estimator for the spectral density can be reduced by averaging over adjacent frequency bins. However at the cost of
resolution! Using averaged spectral densities results in a better estimate for the FRF and coherence. Note that the coherence is always overestimated.
Lecture 6 - Time Domain Models
ontinuous systems can be approximated well by discrete time models. Discrete time models are in fact the regression coefficients of a discrete impulse response function. Having N discrete signal
values, discrete models normally require far less regression coefficients (n<N) compared to FRFs (N). Major advantage of time domain models is that in the time domain noise can be separated from
signals. Compared to the frequency domain where noise is mixed with the signal that requires a posteriori averaging, in the discrete time domain noise models are estimated a priori. Immediately, this
a priori knowledge of the system’s structure which is often not known beforehand. In this lecture, different time domain models are presented. Besides the models are all linear input-output models,
they are not all linear in their parameters (i.e. regression coefficients). Linearity in the parameters means a linear contribution of the parameters to the model error, which is typically the
difference of the modeled output and the system’s output. Advantage of linearity in the parameters is that the model parameters can be obtained algebraically from the input and output signals only.
Linearity in the parameters depends on the chosen noise model. E.g. ARX is linear in its parameters and ARMAX is nonlinear in its parameters. Two discrete closed loop estimators are presented (two
stage and coprime factorization), both utilizing two open loop estimation steps but each in a different way.
Lecture 7 - Assignment 2 & Overview of System Identification
In the first part of this lecture Assingment 2 will be thouroughly explained. Then lectures 1-6 will be resumed which is the part of System Identification.
Lecture 8 - Optimization methods
This lecture gives an overview of different types of optimization methods for static and dynamic systems.
Lecture 9 - Physical Modeling, Model and Parameter Accuracy
This lecture deals with application of optimization routines in order to retrieve the parameters from the model. Parameterization in both time and frequency domain will be explained and basic steps
in an 'ideal' experiment will be given.
Lecture 10 - Assignment 3 & Nonlinear Models
The first hour of the lecture is spent on explaining Assignment 3. After explaining Assignment 3 the lecture will be about non-linear models, where non-linear behaviour, harmonics and the Volterra
series are discussed.
Lecture 11 - Final Assignment & Time Variant Identification
The first part of this lecture is an explanation of the final assignment. Then 'Nonlinear models' will be discussed, continued on the topic which was discussed in lecture 10. The third part of this
lecture will explain more on 'Time Variant Identification'.
Lecture 12 - Identification of Joint Impedance
In this final lecture, three case studies will be explained: the first two cases are linear SIPE and the third case is about nonlinear Parameter Estimation (PE). For the linear cases, one example is
given for 1DOF (case1) and one case study is about 3DOF (case 2). The non-linear case 3 is about intrinsic and reflexive torque of the ankle in stroke. | {"url":"https://fuaad.fke.utm.my/subjects/mkem1763-system-identification-and-estimation-module/sipe_tudelft","timestamp":"2024-11-09T09:50:55Z","content_type":"text/html","content_length":"134984","record_id":"<urn:uuid:89b6c472-5abe-4b0a-8ec7-ae4a425ddfa1>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00700.warc.gz"} |
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What is: Joint Bivariate Distribution
What is Joint Bivariate Distribution?
Joint Bivariate Distribution refers to the probability distribution that describes the likelihood of two random variables occurring simultaneously. In statistics, understanding the relationship
between two variables is crucial for data analysis and interpretation. This distribution provides a comprehensive framework for examining how two variables interact, allowing statisticians and data
scientists to model complex phenomena effectively. By analyzing the joint distribution, one can derive insights into the correlation, dependency, and potential causation between the variables in
Mathematical Representation
The joint bivariate distribution can be mathematically represented using a joint probability density function (PDF) for continuous variables or a joint probability mass function (PMF) for discrete
variables. For two continuous random variables X and Y, the joint PDF is denoted as f(x, y), which satisfies the condition that the integral of f(x, y) over the entire space equals one. This
representation allows for the calculation of probabilities associated with specific ranges of values for both variables, facilitating a deeper understanding of their relationship.
Marginal Distributions
Marginal distributions are derived from the joint bivariate distribution by integrating or summing over one of the variables. For instance, the marginal distribution of variable X can be obtained by
integrating the joint PDF over all possible values of Y. This process yields the marginal PDF f(x), which provides insights into the behavior of X independently of Y. Understanding marginal
distributions is essential for interpreting the joint distribution, as it highlights the individual characteristics of each variable while still acknowledging their interdependence.
Conditional Distributions
Conditional distributions are another critical aspect of joint bivariate distributions. They describe the probability of one variable given the value of another. For example, the conditional
distribution of Y given X is represented as f(y|x). This distribution is obtained by dividing the joint PDF by the marginal PDF of X. Conditional distributions are particularly useful in data
analysis, as they allow researchers to explore how the distribution of one variable changes in response to the values of another variable, providing insights into causal relationships and
Correlation and Covariance
Correlation and covariance are statistical measures that quantify the degree to which two random variables are related. In the context of joint bivariate distributions, the correlation coefficient
(often denoted as ρ) indicates the strength and direction of the linear relationship between the two variables. Covariance, on the other hand, measures the extent to which two variables change
together. Both metrics are derived from the joint distribution and are essential for understanding the nature of the relationship between the variables, guiding decisions in data analysis and
predictive modeling.
Applications in Data Science
Joint bivariate distributions have numerous applications in data science, particularly in fields such as finance, healthcare, and social sciences. For instance, in finance, analysts may use joint
distributions to model the relationship between asset returns, helping to assess risk and optimize investment portfolios. In healthcare, researchers might explore the relationship between patient
characteristics and treatment outcomes, allowing for more personalized medical interventions. By leveraging joint bivariate distributions, data scientists can uncover hidden patterns and
relationships within complex datasets, leading to more informed decision-making.
Visualization Techniques
Visualizing joint bivariate distributions is crucial for understanding the relationship between two variables. Common visualization techniques include scatter plots, contour plots, and heatmaps.
Scatter plots display individual data points, allowing for the identification of trends and correlations. Contour plots and heatmaps provide a more comprehensive view of the joint distribution,
illustrating the density of data points across different value combinations. These visual tools are invaluable for data exploration, enabling analysts to communicate findings effectively and derive
actionable insights from the data.
Limitations and Assumptions
While joint bivariate distributions are powerful tools for statistical analysis, they come with certain limitations and assumptions. One key assumption is that the relationship between the two
variables is adequately captured by the chosen distribution model. If the underlying relationship is more complex, a simple bivariate distribution may not suffice. Additionally, joint distributions
assume that the data is independent and identically distributed (i.i.d.), which may not hold true in real-world scenarios. Understanding these limitations is essential for accurate interpretation and
application of joint bivariate distributions in data analysis.
In summary, joint bivariate distributions are fundamental to understanding the relationship between two random variables in statistics and data science. By providing a framework for analyzing joint
probabilities, marginal and conditional distributions, and measures of correlation, they enable researchers to uncover insights and make informed decisions based on data. Their applications span
various fields, highlighting their importance in both theoretical and practical contexts. As data continues to grow in complexity, the relevance of joint bivariate distributions in statistical
analysis will only increase. | {"url":"https://statisticseasily.com/glossario/what-is-joint-bivariate-distribution/","timestamp":"2024-11-02T11:53:52Z","content_type":"text/html","content_length":"139838","record_id":"<urn:uuid:10df30d7-96a2-4d99-8ea9-eeb183802bf3>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00306.warc.gz"} |
Flutter Suppression of Long-Span Bridges Using Suboptimal Control
World Journal of Engineering and Technology Vol.06 No.02(2018), Article ID:84668,7 pages
Flutter Suppression of Long-Span Bridges Using Suboptimal Control
Lingjun Zhuo^1, Yunjin Dong^2, Xinyu Xu^3
^1Research Center for Wind Engineering, Southwest Jiaotong University, Chengdu, China
^2Xihua University, Chengdu, China
^3China Railway Eryuan Engineering Group Co., Ltd., Chengdu, China
Received: March 13, 2018; Accepted: May 19, 2018; Published: May 22, 2018
Based on the Theodorsen’s Theory of the aerodynamic forces on wing-aileron, the Scanlan’s Theory is expanded considering a deck-flap system. It is suggested that a new forced vibration method can
acquire aerodynamic derivatives of this deck-flap system theoretically. After obtaining the wind induced forces, a deck-flap equation of motion in time domain is established to investigate its
control law. Numerical simulation results indicate suboptimal control law of the deck-flap system can suppress the flutter effectively, and the flutter speed can be increased for desirable purpose.
Aerodynamic Force, Deck-Flap System, Flutter Suppression, Control Law
1. Introduction
Flutter is a self-excited motion, which eventually leads to catastrophic damage in bridge structures. Nowadays, with more and more long-span bridges to emerge, finding ways to suppress flutter and
increase stability can tackle severe wind-induced problems. Adding stiffness of a girder, application of mechanical dampers are common ways to improve a bridge aerodynamic property. Using active
control is a new way to solve these problems.
Some researchers tried to put flaps away from deck in order to omit interference of aerodynamic forces between deck and flaps and can easily apply Theodorsen’s Theory of aerodynamic forces [1].
However, this interference cannot be omitted and can improve aerodynamic property in a way [2] [3].
In this paper, the active control system composes of a deck and flaps symmetrically mounted adjacent to the deck. A deck-flap equation of motion in time domain is established. Along with a new forced
vibration method, aerodynamic forces can be calculated theoretically. In the end, a numerical simulation helps to investigate its control law.
2. Equation of Motion in Time Domain
Flutter analysis is usually done in the frequency domain. The frequency-dependent motion-induced forces should be transformed to time-dependent ones so that they can be applied in the active control
analysis. Based on the Scanlan’s Theory, the two dimensional equation of motion can be expressed as:
$I\stackrel{¨}{\alpha }+{c}_{\alpha }\stackrel{˙}{\alpha }+{k}_{\alpha }\alpha =M={M}_{d}+{M}_{f}$(1b)
where m = mass of the system, I = moment of inertia of the system, ${c}_{h}$ , ${c}_{\alpha }$ = damping of vertical and torsional motion respectively, ${k}_{h}$ , ${k}_{\alpha }$ = stiffness of
vertical and torsional motion respectively, L, M = total lift and moment respectively, ${L}_{d}$ , ${M}_{d}$ = motion-induced lift and moment of the deck respectively, ${L}_{f}$ , ${M}_{f}$ =
motion-induced lift and moment of the flaps respectively.
The motion-induced forces of the deck can be expressed as:
${L}_{d}=\frac{1}{2}\rho {U}^{2}\left(2B\right)\left({K}_{h}{H}_{1}^{*}\frac{\stackrel{˙}{h}}{U}+{K}_{h}{H}_{2}^{*}\frac{B\stackrel{˙}{\alpha }}{U}+{K}_{h}^{2}{H}_{3}^{*}\alpha +{K}_{h}^{2}{H}_{4}^
${M}_{d}=\frac{1}{2}\rho {U}^{2}\left(2{B}^{2}\right)\left({K}_{\alpha }{A}_{1}^{*}\frac{\stackrel{˙}{h}}{U}+{K}_{\alpha }{A}_{2}^{*}\frac{B\stackrel{˙}{\alpha }}{U}+{K}_{\alpha }^{2}{A}_{3}^{*}\
alpha +{K}_{\alpha }^{2}{A}_{4}^{*}\frac{h}{B}\right)$(2a)
where ρ = air density, U = wind velocity, B = deck width, h, α = vertical and torsional displacement respectively, ${K}_{h}={\omega }_{h}B/U$ , ${K}_{\alpha }={\omega }_{\alpha }B/U$ , ${H}_{i}^{*},
{A}_{i}^{*}\left(i=1~4\right)$ = aerodynamic derivatives of the deck.
The aerodynamic flaps can be driven on both sides. Figure 1 shows that when flutter of the system is detected, the trailing flap is motivated and the leading flap is locked along with the deck. In
this way, Theodorsen’s Theory on wing-aileron forces can be applied [4].
${L}_{f}=\frac{1}{2}\rho {U}^{2}\left(2B\right)\left({K}_{h}{H}_{5}^{*}\frac{B\stackrel{˙}{\beta }}{U}+{K}_{h}^{2}{H}_{6}^{*}\beta \right)$(3a)
${M}_{f}=\frac{1}{2}\rho {U}^{2}\left(2{B}^{2}\right)\left({K}_{\alpha }{A}_{5}^{*}\frac{B\stackrel{˙}{\beta }}{U}+{K}_{\alpha }^{2}{A}_{6}^{*}\beta \right)$(3b)
where β = torsional displacement of trailing flap, ${H}_{i}^{*},{A}_{i}^{*}\left(i=5,6\right)$ = aerodynamic derivatives of the trailing flap.
To obtain the aerodynamic derivatives for the deck-flap system, a forced
vibration method is proposed. When the system is forced to rotate sinusoidally, displacements of the system can be assumed as:
$\begin{array}{ccc}h=0& \alpha ={\alpha }_{0}{e}^{i\omega t}& \beta ={\beta }_{0}{e}^{i\left(\omega t+\phi \right)}\end{array}$
The total lift and moment are:
$\begin{array}{c}L=\frac{1}{2}\rho {U}^{2}\left(2B\right)\left[K\left({H}_{2}^{*}+{l}_{1}{H}_{5}^{*}+{l}_{2}{H}_{6}^{*}\right)\frac{B\stackrel{˙}{\alpha }}{U}+{K}^{2}\left({H}_{3}^{*}+{l}_{3}{H}_{5}^
{*}+{l}_{4}{H}_{6}^{*}\right)\alpha \right]\\ =\frac{1}{2}\rho {U}^{2}\left(2B\right)\left(K{\stackrel{^}{H}}_{2}\frac{B\stackrel{˙}{\alpha }}{U}+{K}^{2}{\stackrel{^}{H}}_{3}\alpha \right)\end{array}
$\begin{array}{c}M=\frac{1}{2}\rho {U}^{2}\left(2{B}^{2}\right)\left[K\left({A}_{2}^{*}+{m}_{1}{A}_{5}^{*}+{m}_{2}{A}_{6}^{*}\right)\frac{B\stackrel{˙}{\alpha }}{U}+{K}^{2}\left({A}_{3}^{*}+{m}_{3}
{A}_{5}^{*}+{m}_{4}{A}_{6}^{*}\right)\alpha \right]\\ =\frac{1}{2}\rho {U}^{2}\left(2{B}^{2}\right)\left(K{\stackrel{^}{A}}_{2}\frac{B\stackrel{˙}{\alpha }}{U}+{K}^{2}{\stackrel{^}{A}}_{3}\alpha \
where ${l}_{i},{m}_{i}\left(i=1,2,3,4\right)$ = a combination of flap’s amplitude and its phase angle, ${\stackrel{^}{H}}_{i},{\stackrel{^}{A}}_{i}\left(i=2,3\right)$ = new derivatives of the system
in the forced vibration method.
Using the same way in the regular forced vibration method, the new derivatives can be acquired from:
$\begin{array}{cc}{\stackrel{^}{H}}_{2}=\frac{2}{\rho {B}^{3}{\omega }^{2}{\alpha }_{0}}\mathrm{Im}\left[F\left(L\right)\right]& {\stackrel{^}{H}}_{3}=\frac{2}{\rho {B}^{3}{\omega }^{2}{\alpha }_{0}}
$\begin{array}{cc}{\stackrel{^}{A}}_{2}=\frac{2}{\rho {B}^{4}{\omega }^{2}{\alpha }_{0}}\mathrm{Im}\left[F\left(M\right)\right]& {\stackrel{^}{A}}_{3}=\frac{2}{\rho {B}^{4}{\omega }^{2}{\alpha }_{0}}
And their relations can be put as:
$\left[\begin{array}{c}{\stackrel{^}{H}}_{2}\\ {\stackrel{^}{H}}_{3}\end{array}\right]=\left[\begin{array}{cccc}1& 0& {l}_{1}& {l}_{2}\\ 0& 1& {l}_{3}& {l}_{4}\end{array}\right]{\left[\begin{array}
{cccc}{H}_{2}& {H}_{3}& {H}_{5}& {H}_{6}\end{array}\right]}^{T}$(5a)
$\left[\begin{array}{c}{\stackrel{^}{A}}_{2}\\ {\stackrel{^}{A}}_{3}\end{array}\right]=\left[\begin{array}{cccc}1& 0& {m}_{1}& {m}_{2}\\ 0& 1& {m}_{3}& {m}_{4}\end{array}\right]{\left[\begin{array}
{cccc}{A}_{2}& {A}_{3}& {A}_{5}& {A}_{6}\end{array}\right]}^{T}$(5b)
As seen from above, it needs only a few tests to figure out a statically indeterminate matrix for solving the aerodynamic derivatives. Thus, inserting Equations (2) and (3) into Equation (1), and
transforming it into Laplace domain with zero initial conditions gives:
$\stackrel{¯}{M}=\left[\begin{array}{cc}mB& \\ & I\end{array}\right]$ , $\stackrel{¯}{C}=\left[\begin{array}{cc}{c}_{h}B& \\ & {c}_{\alpha }\end{array}\right]$ , $\stackrel{¯}{K}=\left[\begin{array}
{cc}{k}_{h}B& \\ & {k}_{\alpha }\end{array}\right]$ , $\stackrel{˜}{q}=\left[\begin{array}{c}\stackrel{˜}{h}/B\\ \stackrel{˜}{\alpha }\end{array}\right]$ , $\stackrel{˜}{F}=\left[\begin{array}{c}{\
stackrel{˜}{Q}}_{L}\\ {\stackrel{˜}{Q}}_{M}\end{array}\right]$
A common way to transform wind induced forces into time domain is a rational function approximation by Roger [5]. The details can be found in any aeronautical textbook. Each coefficient ${Q}_{ij}$
can be expressed as:
${Q}_{ij}={A}_{0}^{ij}+{A}_{1}^{ij}\stackrel{¯}{s}+\underset{m=2}{\overset{N}{\sum }}{A}_{m}^{ij}\frac{\stackrel{¯}{s}}{\stackrel{¯}{s}+{\gamma }_{m-1}}$(7)
Combined with aerodynamic derivatives acquired from forced vibration method, each coefficient can be obtained through the rational function approximation.
3. Feedback Control
Equation of motion can be rewritten as:
$\stackrel{¯}{M}\stackrel{¨}{q}+\stackrel{¯}{C}\stackrel{˙}{q}+\stackrel{¯}{K}q-{q}_{d}\left({A}_{0}+{A}_{1}\stackrel{¯}{s}+\underset{m=2}{\overset{N}{\sum }}{A}_{m}\frac{\stackrel{¯}{s}}{\stackrel
{¯}{s}+{\gamma }_{m-1}}\right)q={q}_{d}\left({B}_{0}+{B}_{1}\stackrel{¯}{s}+\underset{m=2}{\overset{N}{\sum }}{B}_{m}\frac{\stackrel{¯}{s}}{\stackrel{¯}{s}+{\gamma }_{m-1}}\right)\beta$
In order to study control law for stabilization, the equation of motion can be put in the state space form:
$\left\{\begin{array}{c}\stackrel{˙}{x}=Ax+Bu\\ y=Cx\end{array}$(8)
$x=\left[\begin{array}{c}X\\ \stackrel{˙}{X}\\ {X}_{a3}\\ ⋮\\ {X}_{am}\end{array}\right]$ , $A=\left[\begin{array}{ccccc}0& I& 0& \cdots & 0\\ -{M}^{-1}K& -{M}^{-1}C& {q}_{d}{M}^{-1}{A}_{3}& \cdots &
{q}_{d}{M}^{-1}{A}_{m}\\ 0& I& -\frac{U{\gamma }_{1}}{B}I& \dots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& I& 0& \cdots & -\frac{U{\gamma }_{m-2}}{B}I\end{array}\right]$ ,
$B=\left[\begin{array}{c}0\\ {M}^{-1}G\\ 0\\ ⋮\\ 0\end{array}\right]$ , $u={\beta }_{c}$ , $y=\left[X\right]$ , $C=\left[\begin{array}{cccc}I& & & \\ & 0& & \\ & & \ddots & \\ & & & 0\end{array}\
right]$ ,
$M=\stackrel{¯}{M}-{q}_{d}\frac{{B}^{2}}{{U}^{2}}{A}_{2}$ , $C=\stackrel{¯}{C}-{q}_{d}\frac{B}{U}{A}_{1}$ , $K=\stackrel{¯}{K}-{q}_{d}{A}_{0}$ , $G=\left[\begin{array}{c}0\\ 0\\ {k}_{c}{k}_{\beta }\
end{array}\right]$ .
Optimal output control is a mature way [3] [6], but it needs all the state vector to be measured, which isn’t easy in the deck-flap system. On the other hand, suboptimal output control can be exerted
through only a few state vector. It’s more reliable in wind tunnel experiments [7]. Suppose that the control is generated via output linear feedback gains:
where ${K}_{con}$ = feedback gain matrix to be determined.
To solve the suboptimal output control problem is to figure out a optimization of the averaged performance index:
$J=\frac{1}{2}{\int }_{0}^{\infty }\left({x}^{T}Qx+{u}^{T}Ru\right)dt$(10)
where Q, R = appropriate weighting matrices.
Inserting the output equation from the state space form and performing a simple mathematical operation yields
$\begin{array}{l}QJ=\frac{1}{2}{\int }_{0}^{\infty }\left({x}^{T}Qx+{x}^{T}{C}^{T}{K}_{con}^{T}R{K}_{con}Cx\right)dt\\ =\frac{1}{2}{\int }_{0}^{\infty }{x}^{T}\left(Q+{C}^{T}{K}_{con}^{T}R{K}_{con}C\
right)xdt=\frac{1}{2}{\int }_{0}^{\infty }{x}^{T}{Q}_{1}xdt\end{array}$(11)
It appears that ${K}_{con}$ is the solution of three equations:
$\mathrm{min}J=\mathrm{min}\frac{1}{2}{\int }_{0}^{\infty }{x}^{T}{Q}_{1}xdt$(12a)
4. Numerical Simulation
A plate with decks on both sides is simulated to see the results of the application of active control. The deck is 40 meters wide, mass of the deck $m=20000kg/m$ , moment of inertia $I=4.5×{10}^{6}kg
\cdot {m}^{2}/m$ , ${\omega }_{h}=0.1788Hz$ , ${\omega }_{\alpha }=0.5028Hz$ , air density $\rho =1.225kg/{m}^{3}$ , the width of flap is 3 m.
First, its aerodynamic forces should be transformed through the Roger approach, as shown in Figure 2.
The aim of the active control is to suppress flutter up to 159 m/s. when using optimal control, the value of its gain is:
$\begin{array}{l}{K}_{op}=\left[\begin{array}{cccccc}-0.5373& 2.8299& 3.6154& -1.8600& 0.0069& 1.0017\end{array}\\ \begin{array}{cccccccc}& & 2.5807& -6.5471& -1.6502& 3.5028& -1.0229& -4.3174\end
Figure 3 shows the responses of the deck-flap system under optimal control in the wind speed of 159 m/s, when the system is exerted external forces.
The value of gain after applying suboptimal control is:
${K}_{con}=\left[\begin{array}{cc}-0.0098& 0.6779\end{array}\right]$
And its responses in the wind speed of 159 m/s is shown in Figure 4, when exerted with external forces.
The numerical example shows that both control laws can suppress flutter in the high wind speed. Despite the quick response under optimal control, it's hard to track all the parameters in state
vector. The suboptimal control provides a brief strategy to suppress flutter with few parameters in state vector. In this case, the control law depends on the vertical and torsional displacements.
For wind tunnel experiments, the two displacements are very common and easy to track.
5. Conclusions
The modeling of deck-flap system is studied in this paper. A state space form of equations of motion in time domain are obtained through Roger’s approach.
Considering the interference between the deck and flaps, a forced vibration method is proposed. Theoretically, this method can help researchers obtain
Figure 2. Rational approximation for motion-induced forces.
Figure 3. Time-history of system displacements under optimal control.
Figure 4. Time-history of system displacements under suboptimal control.
aerodynamic derivatives of the deck-flap system with only a few wind tunnel tests. But its accuracy need to be checked.
The numerical simulation shows that the aerodynamic flaps can suppress flutter on desired wind speed. Although feedback gain from the optimal control effectively stabilize the system in high wind
speed, its state vector is not that easy to track. The suboptimal control can greatly suppress the vibration and relies on a few state vector parameters.
Cite this paper
Zhuo, L.J., Dong, Y.J. and Xu, X.Y. (2018) Flutter Suppression of Long-Span Bridges Using Suboptimal Control. World Journal of Engineering and Technology, 6, 34-40. https://doi.org/10.4236/ | {"url":"https://file.scirp.org/Html/84668_84668.htm","timestamp":"2024-11-01T23:56:11Z","content_type":"application/xhtml+xml","content_length":"109109","record_id":"<urn:uuid:8b5669fe-41b2-49c4-ab2b-a0c62be05995>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00856.warc.gz"} |
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Export Reviews, Discussions, Author Feedback and Meta-Reviews
Submitted by Assigned_Reviewer_1
Q1: Comments to author(s). First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. (For detailed reviewing guidelines, see http:
The authors learn a convolution filter for preprocessing signals from calcium imaging, which led them to be similarly effective at predicting neural connectivity from calcium imaging data as a
winning algorithm in a recent competition on this problem. The winning algorithm used a grid search approach to preprocessing, but the approach described here will likely be more robust, and thus is
of particular interest to a subset of the NIPS community.
1) Please plot f, n, and y on figure 1c in a more easy-to-visualize manner, as the relation between these is the crucial result supplied by this paper. Currently f and y are plotted but they are
difficult to see due to the thick line-width and overlap.
page 6 typo “fluoresce” should be fluorescence
Q2: Please summarize your review in 1-2 sentences
This paper presents a useful approach to preprocessing calcium imaging data based on learning a convolution filter, which will likely be of interest to members of the NIPS community that work with
such data.
Submitted by Assigned_Reviewer_43
Q1: Comments to author(s). First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. (For detailed reviewing guidelines, see http:
The paper describes a method for estimating sparse connectivity graphs of firing neurons. An L_2 norm is used to obtain an penalised inverse covariance matrix as it improves the cost function.
Furthermore, a previously published hard thresholding is replaced by a soft one. A cost function is formulated as the X-entropy and optimised using BFGS.
The paper is self-contained and the references are extensive. The text is very easy to read albeit the structure of the paper at times emerges from the text rather than being explicitly presented to
the reader. The result is that the paper reads as if it is chronological and incremental rather than theoretically and scientifically motivated.
The paper compares favourably to other published methods, namely those who took part in the Kaggle Connectnomics competition. However, the heavily repeated reference to the competition makes the
paper read like a late submission which, with the benefit of hindsight, is the best. This style also adds to the incremental feel of the publication.
Comments following author rebuttal:
I have decided to increase my score. What convinced me the the case of improved computational speed ("parametrized in a differentiable way with a very simple, easy-to-implement formula") and a better
justification for the choices made, e.g. filter order and chi2 value.
Q2: Please summarize your review in 1-2 sentences
The paper is well written and theoretically complete. Nevertheless, it reads like an belated contribution to the Kaggle competition showing an (albeit not insignificant) improvement using mainly
logistic regression for preprocessing and an X-entropy formulation of the cost that allows the use efficient off-the-shelf solvers.
Submitted by Assigned_Reviewer_44
Q1: Comments to author(s). First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. (For detailed reviewing guidelines, see http:
This paper addresses the reconstruction of network topology from calcium imaging data, using inverse covariance matrix estimation. It is shown empirically that a simple convolution filter (to be
applied to the calcium traces) can be learned (once and for all) that substantially improves the reconstruction performance, and the time it takes to infer connectivity on new datasets.
Overall, what is done seems solid. There are a couple of wrinkles that I would like the authors to clarify.
First: the authors restricted the length of the convolution filter to 10 time steps. The learned filter peaks at time step 8, which left me to wonder whether there was enough room for the filter to
"converge" (in "time lag space"). For example, had it been restricted to 5 time steps, the authors would have missed this presumably crucial peak. Was that a computational restriction? If not, is
there a biophysical (or empirical?) reason why correlations would fall off anyway after 10 time steps? The authors should probably discuss this.
Second: although I understand that the study is focused on improving the inverse covariance method specifically, I would have liked to hear more of a discussion regarding the inherent limitations of
the method. For example, the fact that only an undirected graph can be extracted seems like a big restriction as far as neural circuits are concerned.
Third, how crucial is it to get the value of $\chi$ in Eq.7 in the right ballpark? In particular, how does reconstruction generalize to datasets of very different sparsity than the one assumed for
The paper is well-written and well structured.
It'd be great if the Kaggle dataset (e.g. the fact that the ground truth connectivity is known) could be described upfront (at the beginning of the methods) in 1/2 sentences.
I would also like the authors to clarify the timescales for the non-experts (e.g. what does "one time step" in the convolution filter mean? what does "time (20 ms)" mean exactly in the x-axis labels?
that 20ms are shown in total, or 1000*20ms = 20 sec?). It would also help the readers to appreciate the difficulty of the task.
Also, AUC is nowhere defined (not even spelled out -- Area Under the (ROC) Curve?).
I'm not an expert in this specific calcium imaging literature, but it seems surprising that nobody had tried (even heuristic) convolution filters on top of calcium data prior to covariance estimation
before... Anyhow, the work presented here is original, and clearly improves on the current leaderboard for Kaggle.
Estimating network topology has very important implications for the neurosciences, especially with the advent of whole-network imaging techniques. There definitely is a need for statistical methods.
While this study provides state-of-the-art performance and speed, I believe it remains essentially a simple (though important) addition to a known algorithm (L2-regularized inverse covariance
estimation) which unfortunately does not address its most inherent limitations (e.g. underlying, implicit Gaussian assumption, undirected graph recovery, ...).
Q2: Please summarize your review in 1-2 sentences
This paper is technically good, well written, and achieves state-of-the-art performance in (a specific case of) topology extraction from calcium data, an important problem in neuroscience. My
greatest concern is the lack of a proper understanding for why the convolution filter learned here improves performance, and why the same filter can be re-used on other datasets with good
generalization performance. It also looks like a rather incremental addition to a known algorithm.
Q1:Author rebuttal: Please respond to any concerns raised in the reviews. There are no constraints on how you want to argue your case, except for the fact that your text should be limited to a
maximum of 6000 characters. Note however, that reviewers and area chairs are busy and may not read long vague rebuttals. It is in your own interest to be concise and to the point.
Comment: The heavily repeated reference to the competition makes the paper read like a late submission which, with the benefit of hindsight, is the best. This style also adds to the incremental feel
of the publication.
Response: Inverse covariance estimation has been an active field of research in recent years (including several articles in NIPS), however to our knowledge no published articles develop methods for
preprocessing of the data prior to applying inverse covariance. To quantify this contribution, direct application of inverse covariance achieves an AUC score of approximately 0.88 on the networks we
consider, not significantly better than many other methods such as transfer entropy, discretized correlation, or granger causality. A dramatic improvement in accuracy is only achieved by using
supervised signal processing for inverse covariance estimation (AUC approximately 0.94). We believe this result alone is worthy of publication.
While we are taking advantage of “competition hindsight”, we would argue that a research competition is less a definitive conclusion on a particular research topic, but instead a way to spark new
research directions and to widen the research community in an area. In particular, neither the other solutions in the competition nor the methods developed in prior research articles made the leap
that the inverse covariance operation can be parametrized in a differentiable way with a very simple, easy-to-implement formula. This allows inverse covariance to be coupled with a convolution step
forming a “ConvOp” that can then be attached to a loss function in supervised learning. In real applications to inferring neural connectivity, we believe our method will be much more attractive to
researchers than a method that takes weeks to train on a cluster, even if it was only discovered after the conclusion of the competition.
Comment: the authors restricted the length of the convolution filter to 10 time steps. The learned filter peaks at time step 8, which left me to wonder whether there was enough room for the filter to
"converge" (in "time lag space"). For example, had it been restricted to 5 time steps, the authors would have missed this presumably crucial peak. Was that a computational restriction? If not, is
there a biophysical (or empirical?) reason why correlations would fall off anyway after 10 time steps?
Response: There are two ways in which we determined the size of the convolution filter. The first is through inspecting the decay of cross-correlation as a function of the time-lag. For the networks
we consider in the paper, this decay takes place over 10-15 time units. The second method is to add an additional time unit one at a time until cross-validated AUC scores no longer improve. This
happens for the networks we consider at 10 time units. Presumably for other networks one may need bigger or smaller filters.
Comment: Although I understand that the study is focused on improving the inverse covariance method specifically, I would have liked to hear more of a discussion regarding the inherent limitations of
the method. For example, the fact that only an undirected graph can be extracted seems like a big restriction as far as neural circuits are concerned.
Response: Empirically, using directed network estimation did not improve our methods, nor the methods of other top solutions. This may be due to the fact that the resolution of Calcium fluorescence
imaging is coarser than the timescale of network dynamics, so that directionality information is lost in the imaging process.
That being said, it is possible to adapt our method for estimation of directed networks. This can be accomplished by introducing two different filters $\alpha_i$ and $\alpha_j$ into Equations 5 and 6
to allow for an asymmetric covariance matrix $S$ in Equation 6. It would be interesting to assess the performance of such a method on networks with higher resolution imaging in future research.
Comment: How crucial is it to get the value of $\chi$ in Eq.7 in the right ballpark? In particular, how does reconstruction generalize to datasets of very different sparsity than the one assumed for
Response: For $\chi$ values within 10% of the true value, AUC scores are above .935. Without data balancing, the model achieves an AUC score of .925, so the introduction of $\chi$ is important.
Comment: While this study provides state-of-the-art performance and speed, I believe it remains essentially a simple (though important) addition to a known algorithm (L2-regularized inverse
covariance estimation) which unfortunately does not address its most inherent limitations (e.g. underlying, implicit Gaussian assumption, undirected graph recovery, ...).
Response: See response to comment 3 regarding directed graph recovery. Concerning the Gaussian assumption, if the input data $f_t^i$ was Gaussian and independent in $t$, then no signal processing
would be necessary prior to applying inverse covariance estimation. However, neural firing time series data is generated by a nonlinear, mutually-exciting point process. We believe that it is the
fact that the input data is non-Gaussian that the signal processing is so crucial. In this case $f_t^i$ and $f_s^j$ are highly dependent for $10>t-s>0$ and $j\rightarrow i$. Empirically, the learned
convolution filter compensates for the model mis-specification and allows for the “wrong” model to still achieve a high degree of accuracy. | {"url":"https://proceedings.neurips.cc/paper_files/paper/2014/file/8b16ebc056e613024c057be590b542eb-Reviews.html","timestamp":"2024-11-11T11:27:44Z","content_type":"application/xhtml+xml","content_length":"20231","record_id":"<urn:uuid:71955d6e-5465-45db-b83f-c7917e1c2809>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00839.warc.gz"} |
The Complete Guide: How to Report Odds Ratios | Online Tutorials Library List | Tutoraspire.com
The Complete Guide: How to Report Odds Ratios
by Tutor Aspire
In statistics, an odds ratio tells us the ratio of the odds of an event occurring in a treatment group compared to the odds of an event occurring in a control group.
When reporting an odds ratio, we typically include the following:
• The value of the odds ratio
• The confidence interval for the odds ratio
• How to interpret the odds ratio in the context of the problem
For example, we might report something like this:
There was no significant difference in the odds of contracting a disease between the smoking and non-smoking groups (OR = 1.44, 95% CI [0.91, 1.97] ).
Note: If a confidence interval for an odds ratio includes the number “1” then there is not a statistically difference in the odds of an event happening between the two groups. Read a full explanation
The following examples show how to report an odds ratio in different scenarios.
Example 1: Odds Ratio Between Training Programs
Suppose a basketball coach uses a new training program to see if it increases the number of players who are able to pass a certain skills test, compared to an old training program.
The coach recruits 50 players to use each program and records the number of players who pass using each program.
He finds that the odds ratio between the two programs is 0.599 and the 95% confidence interval for the odds ratio is [0.245, 1.467].
Here is how he may report the results:
There was no significant difference in the odds of passing the skills test between players who used the new program compared to players who use the old program (OR = 0.599, 95% CI [0.245,
Example 2: Odds Ratio Between Drugs
Suppose a doctor recruits 20 patients to try drug A and 20 patients to try drug B to determine if there is a difference in the odds of a patient being able to pass a breath-holding test.
He finds that the odds ratio between program A and program B is 1.78 and the 95% confidence interval for the odds ratio is [1.57, 1.99].
Here is how she may report the results:
There was a significant difference in the odds of passing the breath-holding test between patients who took drug A compared to patients who took drug B (OR = 1.78, 95% CI [1.57, 1.99]).
Example 3: Odds Ratio Between Studying Programs
Suppose a teacher recruits 30 students to use a weekly studying program and 30 students to use a daily studying program to determine if there is a difference in the odds of a student being able to
pass a specific exam.
She finds that the odds ratio between the weekly program and the daily program is 1.22 and the 95% confidence interval for the odds ratio is [0.91, 1.53].
Here is how she may report the results:
There was not a significant difference in the odds of passing the exam between the two studying programs (OR = 1.22, 95% CI [0.91, 1.53]).
Additional Resources
The following tutorials provide additional information on how to calculate and interpret odds ratios:
How to Interpret Odds Ratios
How to Calculate a Confidence Interval for an Odds Ratio
Odds Ratio vs. Relative Risk: What’s the Difference?
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How to Add Labels to Scatterplot Points in Excel
You may also like | {"url":"https://tutoraspire.com/how-to-report-odds-ratio/","timestamp":"2024-11-12T00:47:26Z","content_type":"text/html","content_length":"350226","record_id":"<urn:uuid:b4f2fd74-3158-4c0e-b543-2684cb37a223>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00573.warc.gz"} |
The Picard-Lindelöf Theorem
This is my rendition of the proof of the existence-uniqueness theorem for first-order differential equations, also known as the Picard-Lindelöf theorem. I tried to be as formal and explicit as
possible while also making the proof easy to read and comprehend.
You can read the proof here (PDF, ~200 KB, 7 pages). For more information about how it was created, see the About page. | {"url":"https://intuitiveexplanations.com/math/picard-lindelof-theorem","timestamp":"2024-11-11T04:38:57Z","content_type":"text/html","content_length":"4888","record_id":"<urn:uuid:b5f68c0d-2776-4c18-b3fa-91efdeeb02d5>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00388.warc.gz"} |
Multiple Regression R-square Confidence Interval Calculator
Compute the 90%, 95%, and 99% confidence intervals for an R-square value, given the R-square value, the number of predictor variables, and the total sample size. Knowing the confidence interval for
an R-square value can be very useful in analytics when considering the true degree of usefulness that a regression model might have in the overall population.
Please provide the necessary values, and then click 'Calculate'.
Predictor variables:
R^2 (observed):
Sample size: | {"url":"https://www.analyticscalculators.com/calculator.aspx?id=28","timestamp":"2024-11-04T23:15:44Z","content_type":"text/html","content_length":"32087","record_id":"<urn:uuid:377cb063-7ce6-4f62-ac68-63aa3b32a7eb>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00677.warc.gz"} |
CE Europa x Albacete Betting tips for October 30 in Spain Copa del Rey
Our betting tip for CE Europa x Albacete, Wednesday, 30/10/2024
📅 30/10/2024 CE Europa X Albacete
17:30 4.40 3.50 1.75
This is the bet made by our artificial intelligence for CE Europa x Albacete:
🔮 Albacete wins the match
💰 When you bet the bonus of $500 from Bet365 on Albacete, you can win up to $875.00!
Or access: https://www.bet365.com
Some important points for the tip for CE Europa x Albacete:
👉 If you had bet $100 on CE Europa in each of its last 5 matches as home, you would have had a profit of $187.0.
👉 If you had bet $100 on Albacete in each of its last 5 matches as away, you would have had a loss of $-180.0.
👉 In the last 8 matches as the home team, CE Europa scored at least 1 goal(s).
👉 In the last 3 matches as the away team, Albacete scored at least 1 goal(s).
👉 In the last 3 CE Europa matches as the home team, it finished over 2.5 goals.
👉 In the last 3 Albacete matches as the away team, it finished over 2.5 goals.
👉 In the last 7 matches as the away team, Albacete conceded at least 1 goal(s).
👉 CE Europa is good playing home: it has 3 wins in a row in its last matches at home.
Looking for more options of bookmakers for your bet on CE Europa x Albacete?
If you want other options of bookmakers to bet on CE Europa x Albacete, no problem. Right below you have the sports betting sites that we used the most in 2024. You will be in good hands on any of
them, just pick your favorite and start betting:
Analysis from CE Europa x Albacete for the Spain Copa del Rey – 30 of October
🏟️ CE Europa X Albacete – Spain Copa del Rey
📅 30 of October, 2024 – 17:30
🔵 CE Europa – Winning probability: 8.58% | Fair line: 11.66
⚪ Tied game – Probability of tied match: 15.27% | Fair line: 6.55
🔴 Albacete – Winning probability: 76.15% | Fair line: 1.31
⚖ Handicap 1×2: +0.5 CE Europa
⚽ Expected goals: 3.50 goals
⛳ Expected corner kicks: 9.25 corner kicks
A good bettor is not the one with a high hits percentage, but the one who bets on positive expected value events. Our AI model knows it very well and has looked for these opportunities for the match
between CE Europa and Albacete.
Therefore, to find the numbers, our artificial intelligence has colected information from around 1212778 matches on our betting database, in order to calculate the probabilities of this match ending
with a home team victory, a tied match or even an away team victory. Check out the results:
Tips for the Match Odds market for CE Europa x Albacete
Is it a good idea to bet on CE Europa?
🔵 CE Europa: the model predicted that the probability of this event happening is 8.58% and we know that avarage odds offered by the bookies are 4.40. In other words, this means that if you made 1000
bets of $1 like this, you:
• Would have hit 90 times – having a profit of $306.00;
• And would have lost other 910 times – with a loss of -$910.00 because of them.
So this bet does not have value. After all, you would have an expected loss of 💰-$604.00.
Is it worth betting on draw?
⚪ draw: the algorithm believes the chances of this event happening are 15.27% and we have noticed that the avarage odds offered by the bookmakers are 3.50. In other words, this means that if you made
1000 bets of $1 like this, you:
• Would hit 150 times – this would give you a profit of $375.00
• And would lose other 850 times – losing -$850.00 with them.
We are talking about a bet without profit, because after all you would have an expected final loss of 💰-$475.00.
Is betting on Albacete worth it?
🔴 Albacete: the model predicted that the probability of this event happening is 76.15% and we know that avarage odds offered by the bookies are 1.75. That is, when you make 1000 bets of $1 like this
one, you:
• Would hit 760 times – this would give you a profit of $570.00
• And would lose other 240 times – having a loss of -$240.00 with them.
Mathematically speaking, this is a value bet because you would have an expected final profit of 💰$330.00.
Handicaps analysis for the match CE Europa x Albacete
Another approach is trying to find the fair handicap for the main markets and our machine learning model will also give you a helping hand with that. If you have no idea what a handicap is, do not
worry! We have a complete article about the Asian handicap on sporting betting that will answer all your questions. You can click on it and leave it open on another tab so you can read later.
Take a look at what our artificial intelligence has set apart for these markets:
⚖ Handicap 1×2: +0.5 CE Europa
⚽ Expected goals: 3.50 goals
Tips for the Handicap 1×2 market for CE Europa x Albacete
⚖ Handicap 1×2: considering the estimates of our model, the main handicap should be +0.5 CE Europa, and the current handicap offered by bookies is +0.75 CE Europa.
We can notice that the difference between the handicap predicted by the model and the one available to bet at the bookmaker is -0.25. Therefore, the recommended bet for the Handicap 1×2 is on: +0.75
CE Europa.
Tips for the Goals Handicap market for CE Europa x Albacete
⚽ Goals Handicap: our model believes that the ideal handicap is 3.50 goals and the available handicap to bet at the moment is 2.25 goals.
We can notice that the difference between the handicap predicted by the model and the one available to bet at the bookmaker is 1.25. Therefore, the recommended bet for the Goals Handicap is on: +
2.25 goals. | {"url":"https://bets.co.ke/predictions/others/ce-europa-albacete-betting-tips-october-30-2024/","timestamp":"2024-11-05T15:25:34Z","content_type":"text/html","content_length":"175493","record_id":"<urn:uuid:4279815e-0117-468f-872e-f7f841f7b364>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00433.warc.gz"} |
Paths as homeomorphisms of the closed interval from 0 to 1
Previously, I defined a path in terms of a continuous function from a closed interval on the reals to a set of points in a topological space.
Because the function is continuous, by definition, the resultant image is homeomorphic to the closed interval on the reals. Because any closed interval on the reals is itself homeomorphic to the
specific closed interval [0, 1] then the image of a path can be said to be homeomorphic to the real interval [0, 1].
UPDATE (2005-06-01): As Michael Hudson points out in a comment, a path will only be homeomorphic to the closed interval [0, 1] if it doesn't cross over itself. Homeomorphisms require the function to
be bijective, continuous and have a continuous inverse. A path that crosses over itself doesn't meet these criteria.
UPDATE: next post
The original post was in the category: poincare_project but I'm still in the process of migrating categories over.
The original post had 5 comments I'm in the process of migrating over. | {"url":"https://jtauber.com/blog/2005/05/28/paths_as_homeomorphisms_of_the_closed_interval_from_0_to_1/","timestamp":"2024-11-13T02:58:17Z","content_type":"text/html","content_length":"13365","record_id":"<urn:uuid:42bd2384-1028-40af-9abf-02c06080f7a5>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00531.warc.gz"} |
Data structure -- Huffman tree and its application
Basic concepts of Huffman
Path: the branch from one node to another in the tree forms the path between the two nodes
Node path length: the number of branches on the path between two nodes
Path length of tree: the sum of the path length from the tree root to each node is recorded as TL
In the binary tree with the same number of nodes, the complete binary tree is the shortest binary tree
Weight: assign a node in the tree to a value with a certain meaning, then this value is called the weight of the node
Weighted path length of a node: the product of the path length from the root node to the node and the weight of the node
Weighted path length of tree (WPL): the sum of weighted path lengths of all leaf nodes in the tree
Record as: W P L = ∑ i = 0 k w k l k WPL=\sum_{i=0}^{k} w_ k l_k WPL=i=0∑kwklk
ω \omega ω—— Weight l k l_k lk -- path length from node to root
Huffman tree: optimal tree - the tree with the shortest weighted path length (WPL)
1. A full binary tree is not necessarily a Huffman tree
2. The more powerful the leaf in the Huffman tree, the closer it is to the root
3. Huffman trees with the same weighted nodes are not unique
Construction algorithm of Huffman tree
Greedy algorithm: to construct Huffman tree, first select the leaf node with small weight
Huffman algorithm (method of constructing Huffman tree)
Pithy formula:
1. Structural forests are all roots;
2. Select two small trees to build new trees;
3. Delete two new people
4. Repeat 2 and 3, leaving a single piece
The degree of the node of Huffman tree is 0 or 2, and there is no node with degree 1;
The Huffman tree with n leaf nodes has 2n-1 nodes;
The forest containing N trees must be merged n-1 times to form Huffman tree, with n-1 new nodes;
1. In Huffman algorithm, there are N binary trees at the beginning, which need to be merged n-1 times to finally form Huffman tree
2. After n-1 merging, n-1 new nodes are generated, and these n-1 new nodes are branch nodes with two children
It can be seen that the Huffman tree has n+n-1 = 2n-1 nodes, and the degree of all its branch nodes is not 1
Implementation of Huffman tree construction algorithm
Using sequential storage structure -- one-dimensional structure array
Node type definition:
typedef struct{
int weight;
int parent, lch, rch;
}HTNode, *HuffmanTree;
The Huffman tree has 2n-1 nodes in total, does not use 0 subscript, and the array size is 2n
For example, if the weight of the first node is 5, it can be expressed as H [i] weight = 5;
Example: n = 8, weight W = {7, 19, 2, 6, 32, 3, 21, 10}, construct Huffman tree
1. Initialize HT [1... 2n-1]: lch = rch = parent = 0;
2. Enter the initial n leaf nodes: set the weight value of HT[1... N]
3. Perform the following n-1 merges to generate n-1 nodes HT[i], i = n+1... 2n-1
a) in HT[1... i-1, select two nodes HT[s1] and HT[s2] with the smallest weight that have not been selected (from the nodes with parent==0), and s1 and s2 are the subscripts of the two smallest
b) modify the parent values of HT[s1] and HT [S2]: HT[s1] parent=i; HT[s2].parent = i;
c) modify the newly generated HT[i]:
1)HT[i].weight = HT[s1].weigth + HT[s2].weight;
2)HT[i].lch = s1;HT[i].rch = s2;
//Constructing Huffman tree -- Huffman algorithm
void CreatHuffmanTree(HuffmanTree &HT, int n){
if(n <= 1) return;
m = 2 * n - 1; //The array has 2n-1 elements in total
HT = new HTNode[m + 1]; //Unit 0 is not used, HT[m] represents the root node
for(i =1; i <= m; ++i){ //Set lch, rch and parent of 2n-1 elements to 0
HT[i].lch = 0;
HT[i].rch = 0;
HT[i].parent = 0;
for(i = 1; i <= n; ++i)
cin >> HT[i].ewight; //Enter the weight value of the first n elements
//After initialization, let's start building Huffman tree
//Merging to produce n-1 nodes -- Constructing Huffman tree
for(i = n + 1; i <= m; i++){
Select(HT, i - 1, s1, s2); //Select two in HT[k] (1 ≤ K ≤ i-1) whose parental domain is 0,
//And the node with the smallest weight, and return their sequence numbers s1 and s2 in HT
HT[s1].parent = i; //Indicates that s1 and s2 are deleted from F
HT[s2].parent = i;
HT[i].lch = s1; //s1 and s2 are the left and right children of i respectively
HT[i].rch = s2;
HT[i].weight = HT[s1].weight +HT[s2].weight; //The weight of i is the sum of the weight of left and right children
Example: let n = 8, w = {5, 29, 7, 8, 14, 23, 3, 11}, and try to design Huffman code
(m = 2*8-1 = 15)
Huffman coding
Question: what prefix code can make the total length of the message shortest—— Huffman coding
1. Count the average probability of each character in the character set appearing in the message (the greater the probability, the shorter the code is required)
2. Using the characteristics of Huffman tree: the greater the weight, the closer the leaf is to the root; Taking the probability value of each character as the weight to construct Huffman tree, the
node with higher probability will have shorter path
3. Mark 0 or 1 on each branch of Huffman tree:
node left branch mark 0, right branch mark 1
connect the labels on the path from the root to each leaf as the encoding of the characters represented by the leaf
Explanation of Huffman coding algorithm: https://www.bilibili.com/video/BV1nJ411V7bdp=106&spm_id_from=pageDriver
Implementation of Huffman coding algorithm:
//The Huffman code of each character is obtained from the leaf to the inverse root and stored in the coding table HC
void CreatHuffmanCode(HuffmanTree HT, HuffmanCode &HC, int n){
HC = new char *[n + 1]; //Allocate n character encoded header pointer vectors
cd = new char [n]; //Allocate dynamic array space for temporarily storing codes
cd[n - 1] = '\0' ; //Code Terminator
for(i = 1; i <= n; ++i){ //Huffman coding character by character
start = n - 1;
c = i;
f = HT[i].parent;
while(f != 0){ //Trace back from the leaf node to the root node
--start; //Backtracking once start points forward to a position
if(HT[f].lchild == c) //If node c is the left child of f, the production code is 0
cd[start] = '0' ;
else //If node c is the right child of f, code 1 is generated
cd[start] = '1' ;
c = f; //Keep going back up
f = HT[f].parent;
} //Find the coding of the ith character
HC[i] = new char [n - start]; //Allocate space for the ith string encoding
strcpy(HC[i], &cd[start]); //Copy the obtained code from the temporary space cd to the current line of HC
delete cd; //Free up temporary space
} //CreatHuffanCode
Encoding and decoding of documents
1, Code:
① input each character and its weight
② construct Huffman tree - HT[i]
③ Huffman coding - HC[i]
④ check HC[i] to get Huffman code of each character
2, Decoding:
① construct Huffman tree
② read in binary codes in sequence
③ read 0 and go to the left child; Read 1 and go to the right child
④ once a leaf is reached, characters can be translated
⑤ then continue decoding from the root until the end
Explanation: https://www.bilibili.com/video/BV1nJ411V7bd?p=107 | {"url":"https://programmer.ink/think/data-structure-huffman-tree-and-its-application.html","timestamp":"2024-11-10T11:29:38Z","content_type":"text/html","content_length":"15993","record_id":"<urn:uuid:9c1c9260-0b8f-4286-82bc-e3e6dd97b2f0>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00700.warc.gz"} |
Concrete Mix Calculator Sand Gravel Cement - CivilGang
What is a Concrete Mix Calculator (Sand, Gravel, Cement)?
A concrete mix calculator for sand, gravel, and cement is a web-based tool that allows users to calculate the total volume of a concrete mix based on the specified proportions of sand, gravel, and
This is useful for preparing accurate concrete mixtures in construction and building projects.
Why Use a Concrete Mix Calculator (Sand, Gravel, Cement)?
1. Material Planning: Ensures the correct proportion of sand, gravel, and cement in concrete mixtures, avoiding under or overuse of materials.
2. Cost Estimation: Helps in estimating costs for purchasing and mixing the right amounts of materials.
3. Project Efficiency: Aids in planning and executing construction projects by providing precise concrete mix proportions.
4. Resource Management: Efficiently manages material resources, reducing waste and costs.
Concrete Mix Calculator
Enter the volume proportions for sand, gravel, and cement to calculate the total volume of concrete mix:
Total Volume of Concrete Mix: 0 cubic units | {"url":"https://civildetail.com/concrete-mix-calculator-sand-gravel-cement/","timestamp":"2024-11-14T02:36:31Z","content_type":"text/html","content_length":"72874","record_id":"<urn:uuid:e5ecb451-37d7-4766-a544-a1bebc8af7f9>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00788.warc.gz"} |
Maximum Profit With Unlimited Stock Trading Transactions
Problem Statement:
Say you have an array prices for which the i
element is the price of a given stock on day i.
Design an algorithm to find the maximum profit. You may complete as many transactions as you like (i.e., buy one and sell one share of the stock multiple times).
Note: You may not engage in multiple transactions at the same time (i.e., you must sell the stock before you buy again).
Example 1:
Input: [7,1,5,3,6,4]
Output: 7
Explanation: Buy on day 2 (price = 1) and sell on day 3 (price = 5), profit = 5-1 = 4. Then buy on day 4 (price = 3) and sell on day 5 (price = 6), profit = 6-3 = 3.
Example 2:
Input: [1,2,3,4,5]
Output: 4
Explanation: Buy on day 1 (price = 1) and sell on day 5 (price = 5), profit = 5-1 = 4. Note that you cannot buy on day 1, buy on day 2 and sell them later, as you are engaging multiple transactions
at the same time. You must sell before buying again.
Example 3:
Input: [7,6,4,3,1]
Output: 0
Explanation: In this case, no transaction is done, i.e. max profit = 0.
• Finite state machines are a standard tool to model event-based control logic. If a given problem contains different events and depending on which event occurs we can transition from one state to
another then it is highly likely that the problem could be solved using Dynamic Programming State Machine approach.
Before going into detailed discussion I would like to bring to your attention that: even though the problem statement says that you can make as many transactions as you want it won't be logical to
make more than N number of transactions where N = total number of days. Since there is no gain in doing more than one transaction in a day, since if you buy and sell on the same day you are making 0
profit. Convince yourself.
Every day we can be one of these two states:
• either we have a stock with us. Let's call this state hasStock state.
• or, we have no stock with us. Let's call this state noStock state.
Transition: noStock
state by buying a stock.
state by selling the purchased stock.
is also the START state, because on day-1 (first day) we start with a clean slate (no stock with us). Since we can make as many transactions as we want, this
can go on happening as many times as we want, keeping in mind that
• we cannot buy a stock before selling the one we already have with us.
• our goal is to maximize profit.
This leads us to the
state machine diagram
State Machine Diagram:
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Recurrence Relation:
denote the maximum profit achievable at the end of day
if we have no stock with us on day
denote the maximum profit achievable at the end of day
if we have a stock with us on day
noStock[i] = Max(noStock[i - 1], hasStock[i - 1] + prices[i]);
hasStock[i] = Max(hasStock[i - 1], noStock[i - 1] - prices[i]);
Return value:
Since our goal is to maximize the profit we can never end the last day with a stock being held, because that would be waste of money that we bought a stock but did not sell it on or before the last
day. So the return value would always be
Base Condition(s):
noStock[0] = 0
hasStock[0] = -stockPrices[0]
We get the above base cases using the below logic:
• Ending the day with no stock being hold means we did not make any transaction. We started the day with no stock and ended the day with no stock. So net profit = 0;
noStock[0] = 0
• If we end the first day with a stock being held that would mean that we started the day with 0 profit and then spent stockPrices[0] to buy a stock where stockPrices[0] is the price of stock on
the first day.
hasStock[0] = -stockPrices[0]
Java Code:
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Python Code:
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Space Optimization:
In the above code, we have room for Space Optimization because we do NOT need the hasStock[] and noStock[] arrays due to our below observation:
• In the for loop notice that for every day = i we are relying on hasStock[i - 1] and hasStock[i - 1].
At no point of time we are using anything more than that and so having the whole array in memory is adding no value when we are dependent on just these two values
So what we can do is we can take two variables that would hold the value of noStock[i - 1] and hasStock[i - 1] when we are processing for day = i.
We would also need two variables to hold noStock and hasStock values for the current day we are computing for.
The above discussion leads us to the below space optimized code:
Java Code:
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Python Code:
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The below code shows how we got to the space optimized code from the original code (non-space-optimized one):
Java Code:
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Python Code:
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Find the Days to Buy and Sell the Stocks to Maximize Profit for the above problem:
Our goal now is to print the days we would be purchasing a stock and the days we would be selling the stock that would maximize the profit.
In almost all the DP problems, finding the path that resulted in the optimal result takes little bit more thinking than just computing the optimal value. For our case, it would be finding the
purchase days and selling days that led us to get the maximum profit.
It is always a good idea to first design the algorithm to compute the optimal value and then modify the code to find the optimal path. In most cases you would need the logic you implemented to
compute the optimal result, to compute the optimal path. We will see that in the below well-commented code.
Thought process involved in the solution:
• Using the same states we have been using so far, it is very easily understandable from the State Machine Diagram that if we reach noStock state on day i from hasStock state of day (i - 1), then
that would mean that we sold a stock on day i.
But a very important observation should be made here:
let's say the stock prices for 5 days look like this : [1, 2, 3, 4, 5] , i.e, when the stock price soars. Starting from day-2 everyday we would think that today is the best day to sell since
everyday we have better stock rate than day before. And starting from day-2 everyday we would transition to noStock state from hasStock (of day before). Try to convince this to yourself by
working on this example. So, we cannot just go on adding a day to the sellingDay list just because we transitioned to noStock state from hasStock state. A very easy way to know when this scenario
occurs is to see if all the already gotten purchase days and selling days are in pairs. Which means we would have more selling days in our list than the purchase days. But this is invalid since a
selling day would always have a corresponding purchase day.The solution here is: if we see we are breaking the constraint that number of selling days cannot be more than total number of purchase
days, we would know that this is the case where stock price is going up and we just got a better day to sell the last purchased stock. So remove the last stock selling day, and add the current
• If we transition to hasStock state from noStock on day i then day i is a purchase day. But if stock rate is declining like in [5, 4, 3, 2, 1] then everyday would prove to be a purchase day. In
such scenario we need to remove the last purchase day and add the current as updated purchase day. It is very easy to detect such scenario: If we see that we already have more purchase days than
selling days we would know that we are in a situation when stock prices is declining giving us better day for stock purchase.
Java Code:
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Python Code:
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Don't forget to look at the other problems in our
State Machine Approach
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Mathematical Analysis (BY-)
Offered by
General admission requirements.
Main purpose
The course aims to prepare the student for further studies in Civil engineering.
Furthermore, the purpose is to enable the student to read and interpret technical literature, which use mathematics.
The student will get knowledge about:
• Differentiation
• Trigonometric functions
• Exponential functions
• Integration
• Vectors in space
• Vector functions in space
After the completion of the course, the student will be able to:
• Identify and make simple calculation on selected transcendental functions
• Identify and make simple calculation on the branch of infinitesimal calculation, which deals with finding the derivative of functions with one variable, including different applications thereof
• Identify and make simple calculation on the branch of infinitesimal calculation, which deals with integration of functions with one variable and different applications thereof
• Analyze vectors and motion in space and perform calculations based on vector operations
IT is used in a pedagogical method in the course. The aim is that IT will support the learning process of the students and their understanding of the engineering professional possibilities in, for
example, the application of modelling, simulation, etc.
After completing the course, the student can:
• Perform a basic understanding for Calculus.
• Use their acquired skills and knowledge to study more advanced Calculus and Algebra courses on their Engineering programme.
• Use a commercial mathematical software to solve and perform serial technical calculations.
Teaching methods and study activities
The expected total workload for the student is 137.5 hours.
Every week a consulting class is offered. The consulting class is used to refresh math skills and is not mandatory.
The teaching will be lecturing and problem solving in class.
Students will prepare for class by reading sections of textbooks and solving problems initiated by the lecturer.
Homework assignment will be handed out.
• Thomas' Calculus in SI Units; Maurice D. Weir, Christopher E. Heil& Joel R. Hass, 15. edition
• Downloads from the Intra net.
• The teaching is based on the student having a PC with software able to perform operations corresponding to MathCad, and that the student independently acquaints themselves with the use of this.
The teaching is based on the use of MathCad.
Exam prerequisites:
Type of exam:
Individual written exam, 4 hours.
External assessment.
Allowed tools:
At 20-40% of the exam, the use of CAS programs is not allowed.
The exam set will state in which assignments CAS programs may not be used and in which assignments CAS programs may be used.
Equal to the ordinary exam.
Grading criteria
Grading based on the Danish 7-point scale.
Additional information
Morten Carter (mc)
Valid from
01-08-2023 00:00:00
Course type
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Poisson Process tutorial - Definition and examples | Prwatech
Poisson Process Tutorial
Poisson Process Tutorial, In this tutorial one, can learn about the importance of Poisson distribution & when to use Poisson distribution in data science. We Prwatech the Pioneers of Data Science
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Definition of a Poisson process
A Poisson Process is a model for a series of discrete events where the average time between events is known, but the exact timing of events is random.
The arrival of an event is independent of the event before (waiting time between events is memoryless).
Poisson Process Example 1
Suppose we an electricity connection, and the provider tells us that electric power goes down on an average of once per 60 days, but one failure doesn’t affect the
probability of the next. All we know is the average time between failures.
This is a Poisson process that looks like:
The important point is we know the average time between events but they are randomly spaced. We might have back-to-back failures, but we could also go years between failures due to the randomness of
the process.
Characteristics of Poisson distribution
A Poisson Process meets the following criteria:
• Events are independent of each other.
• The occurrence of one event does not affect the probability another event will occur.
• The average rate (events per time period) is constant.
• Two events cannot occur at the same time.
• The last point—events are not simultaneous
• From our Electric Power example, the entire interval maybe 6000 days, but each sub-interval (we can call it as one day) Electric Power either goes down or it doesn’t.
When to use Poisson distribution
A Poisson experiment is a statistical experiment that has the following properties:
• The experiment results in outcomes that can be classified as successes or failures.
• The average number of successes (μ) that occurs in a specified region is known.
• The probability that success will occur is proportional to the size of the region.
• The probability that success will occur in an extremely small region is virtually zero.
Importance of Poisson distribution
A Poisson random variable is the number of successes that result from a Poisson experiment. The probability distribution of a Poisson random variable is called a Poisson distribution.
Given the mean number of successes (μ) that occur in a specified region, we can compute the Poisson probability based on the following formula:
Suppose we conduct a Poisson experiment, in which the average number of successes within a given region is μ. Then, the Poisson probability is:
Poisson Process Formula
where x is the actual number of successes that result from the experiment, and e is approximately equal to 2.71828.
The Poisson distribution has the following properties:
The mean of the distribution is equal to μ.
The variance is also equal to μ.
Poisson Process Example 2
The average number of Projectors sold by some particular company is 5 Projectors per day. What is the probability that exactly 6 Projectors will be sold tomorrow?
Solution: This is a Poisson experiment in which we know the following:
μ = 5; since 5 Projectors are sold per day, on average.
x = 6; since we want to find the likelihood that 6 Projectors will be sold tomorrow.
e = 2.71828; since e is a constant equal to approximately 2.71828.
We plug these values into the Poisson formula as follows:
Thus, the probability of selling 6 Projectors tomorrow is 0.00059.
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Pool Area: Knowing how to calculate area is important for determining heat loss, sizing pool covers, bather loads in spas, and in some states, bather loads in pools.
Pool Depth: Depths are never even throughout a pool or spa. They may be sloped or if the depth is "constant", there is usually a coved or radius at the wall junctions.
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Gallons: Cubic feet or cubic meters are not useful in everyday discussions. So we take it one step further and convert to gallons or liters, things we know.
User Load: Usually user load is defined by local codes. Commonly, it is the number of bathers (or users) in the body at any specific time.
Heater Sizing: A British Thermal Unit (BTU) is the amount of heat that will raise the temperature of one pound of water, one degree Fahrenheit.
Turnover Rate: When the number of gallons in a pool goes through a filter, in 24 hours, that is one turnover a day. Most states require 3, 4 or even more turnovers a day.
Flow Rate: The way to achieve turnover rate is with Flow Rate (FR). The minimum Flow Rate is that which will realize the turnover rate (TR) requirements.
Filter Area: There are three basic filter types: sand; cartridge; and diatomaceous earth. For the same Flow Rate, each type of filter has a different filter area requirement.
Filter Media Rate: Filter Media Rate (FMR) is a NSFI standard that determines how fast water can flow through a given filter type. | {"url":"https://www.thepoolclass.com/support/math/tutorials/main.htm","timestamp":"2024-11-12T16:14:44Z","content_type":"text/html","content_length":"12211","record_id":"<urn:uuid:088b1889-1f39-4cf7-acee-2ca8f09a6c2d>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00819.warc.gz"} |
How much should I start investing in stocks?
“If you’re a typical working person or a beginning investor, you should know that it doesn’t take a lot of money to start,” IBD founder William O’Neil wrote in “How to Make Money in Stocks.” “You can
begin with as little as $500 to $1,000 and add to it as you earn and save more money,” he wrote.
How much money should I have before I start investing?
investments? You should aim to keep enough money in savings to cover three to six months of living expenses. You could consider investing money once you have at least $500 in emergency savings.
Is 1000 enough to buy stocks?
$1,000 is enough to make a single stock purchase through an online brokerage reasonable. You do lose some money in the transaction itself, but the right stock can return many times the transaction
How much should you invest in stocks per month?
Key Takeaways. Investing just $100 a month over a period of years can be a lucrative strategy to grow your wealth over time. Doing so allows for the benefit of compounding returns, where gains build
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• trader
The two best momentum indicators are seems to be Stochastic and RSI. I'd like to learn more about RSI trading please. How to combine those indicators. Also why momentum trading is so popular, what's
its importance exactly in. Thank you.
• FxIndicators
Momentum trading in Forex or any other market is all about being able to get in and out of a trade at the right moment - at the moment where market is gaining initial strength/momentum and runs first
few miles with great confidence and without looking back. When momentum is decelerating or lost, an uncertainty sets among investors as they try to re-evaluate their chances for further profitable
price chasing, and the big nice rallies or sell-offs are slowing down yielding no more profit potentials till the next momentum.
Momentum indicators help to catch that "Here and Now" time to place orders and lock in profits.
It is true that two most popular indicators in Forex remain to be Stochastic and RSI.
Although just one of them will be enough to identify periods of forming momentum in Forex market, you may combine two indicators in order to refer to either one or both of them when expectations for
new trading opportunities are high.
The best explanation on combining two momentum indicators could only be possible with a help of a specific Forex trading strategy, so we've found an example for you online from a very reputable site
RSI and Stochastic strategy combined
and hope you enjoy it!
• trader
Whats the exclusive use of momentum indicator,if all i need to see while investing is the trend and the trend indicator do that job very well. Explain the cons of investing without momentum
• FxIndicators
That's quite easy to explain.
Trend indicators tell which of the trends (up or down) is in place. Trend indicators also tell when the trend changes or is about to change. But, there is one thing trend indicators cannot show - and
this is the very moment at which all "powers of a trend" have gathered the momentum to produce a move.
In simple words, it's a situation when, for example, you know that a trend is up, but you don't know whether you should Buy it now or wait till you get a better price. That's where momentum
indicators come in handy. Momentum indicators show the very best moment when you can Buy it having the highest odds that the market will move into your favor immediately without looking back.
Also you can rely on momentum indicators to suggest when the "powers of a trend" (or simply the momentum) is weakening. Using this information you can take decisions on either taking quick profits
and be out of the trade or review and re-adjust your trading stops.
Without momentum indicators it would be difficult to time trades (specifically entries) precisely, besides there is a chance to enter a trade when the market has made quite a progress and as a result
used all its power (lost momentum) and is preparing to retrace to regain it, leaving late newcomers at the tip of a reversal.
• trader
how do you know whether a market is trending or not
• FxIndicators
When the market keeps making higher highs and higher lows - it is trending upwards and the trend is intact.
When the market keeps making lower highs and lower lows - it is trending downwards and the trend is intact.
Any changes in this sequence invites ranging market reaction.
This subject is further addressed in the comments at:
In addition, traders often use Moving averages.
Classic example, Alligator indicator: Alligator.mq4
When there is a prefect lineup: blue, red, green - there is a trend.
When lines mix up, there is no trend.
The first uptrend is marked with green circle - shows the beginning of a trend, when all 3 moving averages are lined up.
The red circle is the end of the downtrend - when the green moving average crosses over the red moving average, breaking the prefect sequence.
• trader
what is the 3 Moving average line? how many days?
• FxIndicators
3 Moving average line consists of 3 MAs with next settings: 13, 8 and 5.
See further details for Williams Alligator Indicator
• trader
what are the best moving average periods for trading 1 min or 5 mins chart?
• FxIndicators
If you want to see trends on 1 & 5 min, use 20 SMA. Also price would respect 200 EMA.
Smaller moving averages would only "dance" back and forth with price bars without providing any considerable use.
• trader
This is the advice I ever get for Forex so far
• trader
LOL at above comment!!!
• trader
I just came accross this site today and was amazed at what I saw. Hmm..., Thank you so much for your concern.
What is the best MAs combination for 15m time frame or 30mns?
• FxIndicators
Depends on your goals. Don't be surprised if someone tells you that there is no ideal Moving average or set of averages. Every one of them to some point hits the perfect trade one time and misses on
the other.
Check out this page about Moving averages, in particular the paragraph about most common moving averages. You can certainly use those on 15 min or 30 min time frame.
• trader
I need some enlightenment on the use of support and resistance points when trading. Please also clarify me on how to actually place your stop loss. For example you may place it at 10 pips away from
your position in line with your money management strategy but some times the price will swing to 12 pips thereby stopping you out only for it to swing back up and make much profit. How do you strike
a balance between your money management limit and the possibility of the price crossing your limit and swinging back in the opposite direction?
• Atar
Could you please send/write to me indicator:
Stochastic Momentum Indicator (Index).
I have been able to find its formula for metastock
100 * ( Mov( Mov(C - (.5 * ( HHV(H,13) + LLV(L,13))),25,E),2,E)/
(.5*Mov(Mov( HHV(H,13) - LLV(L,13),25,E),2,E)))
but i need it for
dot net sochastic momentum calculation
C# code or lib
Any help appreciated. Thank you
[email protected]
• trader
would you please telling me about which momentum indicators that has highest accuracy in time frame 5 or 15, and also the best parameter setting of it - absolutelly thanks before
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Problems & Exercises
11.2 Density
Gold is sold by the troy ounce (31.103 g). What is the volume of 1 troy ounce of pure gold?
Mercury is commonly supplied in flasks containing 34.5 kg (about 76 lb). What is the volume in liters of this much mercury?
(a) What is the mass of a deep breath of air having a volume of 2 L? (b) Discuss the effect taking such a breath has on your body's volume and density.
A straightforward method of finding the density of an object is to measure its mass and then measure its volume by submerging it in a graduated cylinder. What is the density of a 240-g rock that
displaces $89cm389cm3 size 12{"89" "." 0`"cm" rSup { size 8{3} } } {}$ of water? (Note that the accuracy and practical applications of this technique are more limited than a variety of others that
are based on Archimedes' principle.)
Suppose you have a coffee mug with a circular cross section and vertical sides (uniform radius). What is its inside radius if it holds 375 g of coffee when filled to a depth of 7.50 cm? Assume coffee
has the same density as water.
(a) A rectangular gasoline tank can hold 50 kg of gasoline when full. What is the depth of the tank if it is 0.500-m wide by 0.900-m long? (b) Discuss whether this gas tank has a reasonable volume
for a passenger car.
A trash compactor can reduce the volume of its contents to 0.350 their original value. Neglecting the mass of air expelled, by what factor is the density of the rubbish increased?
A 2.50-kg steel gasoline can holds 20 L of gasoline when full. What is the average density of the full gas can, taking into account the volume occupied by steel as well as by gasoline?
What is the density of 18.0-karat gold that is a mixture of 18 parts gold, five parts silver, and one part copper? (These values are parts by mass, not volume.) Assume that this is a simple mixture
having an average density equal to the weighted densities of its constituents.
There is relatively little empty space between atoms in solids and liquids, so that the average density of an atom is about the same as matter on a macroscopic scale—approximately $103kg/m3103kg/m3
size 12{"10" rSup { size 8{3} } `"kg/m" rSup { size 8{3} } } {}$. The nucleus of an atom has a radius about $10−510−5 size 12{"10" rSup { size 8{ - 5} } } {}$ that of the atom and contains nearly all
the mass of the entire atom. (a) What is the approximate density of a nucleus? (b) One remnant of a supernova, called a neutron star, can have the density of a nucleus. What would be the radius of a
neutron star with a mass 10 times that of our Sun (the radius of the Sun is $7×108m7×108m size 12{7 times "10" rSup { size 8{8} } `m} {}$)?
11.3 Pressure
As a woman walks, her entire weight is momentarily placed on one heel of her high-heeled shoes. Calculate the pressure exerted on the floor by the heel if it has an area of $1.50cm21.50cm2 size 12{1
"." "50"`"cm" rSup { size 8{2} } } {}$ and the woman's mass is 55 kg. Express the pressure in Pa. In the early days of commercial flight, women were not allowed to wear high-heeled shoes because
aircraft floors were too thin to withstand such large pressures.
The pressure exerted by a phonograph needle on a record is surprisingly large. If the equivalent of 1 g is supported by a needle, the tip of which is a circle 0.200 mm in radius, what pressure is
exerted on the record in $N/m2N/m2 size 12{"N/m" rSup { size 8{2} } } {}$?
Nail tips exert tremendous pressures when they are hit by hammers because they exert a large force over a small area. What force must be exerted on a nail with a circular tip of 1 mm diameter to
create a pressure of $3×109N/m2?3×109N/m2?$This high pressure is possible because the hammer striking the nail is brought to rest in such a short distance.
11.4 Variation of Pressure with Depth in a Fluid
What depth of mercury creates a pressure of 1 atm?
The greatest ocean depths on the Earth are found in the Marianas Trench near the Philippines. Calculate the pressure due to the ocean at the bottom of this trench, given its depth is 11 km and
assuming the density of seawater is constant all the way down.
Verify that the SI unit of $hρghρg size 12{hρg} {}$ is $N/m2N/m2 size 12{"N/m" rSup { size 8{2} } } {}$.
Water towers store water above the level of consumers for times of heavy use, eliminating the need for high-speed pumps. How high above a user must the water level be to create a gauge pressure of
$3×105 N/m23×105 N/m2 size 12{3 "." "00" times "10" rSup { size 8{5} } `"N/m" rSup { size 8{2} } } {}$?
The aqueous humor in a person's eye is exerting a force of 0.300 N on the $1.10-cm21.10-cm2 size 12{1 "." "10""-cm" rSup { size 8{2} } } {}$ area of the cornea. (a) What pressure is this in mm Hg?
(b) Is this value within the normal range for pressures in the eye?
How much force is exerted on one side of an 8.50 cm by 11 cm sheet of paper by the atmosphere? How can the paper withstand such a force?
What pressure is exerted on the bottom of a 0.500-m-wide by 0.900-m-long gas tank that can hold 50 kg of gasoline by the weight of the gasoline in it when it is full?
Calculate the average pressure exerted on the palm of a shot-putter's hand by the shot if the area of contact is $50 cm250 cm2 size 12{"50" "." 0`"cm" rSup { size 8{2} } } {}$ and he exerts a force
of 800 N on it. Express the pressure in $N/m2N/m2$ and compare it with the $1×106 Pa1×106 Pa$ pressures sometimes encountered in the skeletal system.
The left side of the heart creates a pressure of 120 mm Hg by exerting a force directly on the blood over an effective area of $15 cm2.15 cm2. size 12{"15" "." 0`"cm" rSup { size 8{2} } } {}$ What
force does it exert to accomplish this?
Show that the total force on a rectangular dam due to the water behind it increases with the square of the water depth. In particular, show that this force is given by $F=ρgh2L/2F=ρgh2L/2 size 12{F=ρ
ital "gh" rSup { size 8{2} } L/2} {}$, where $ρρ size 12{ρ} {}$ is the density of water, $hh size 12{h} {}$ is its depth at the dam, and $LL size 12{L} {}$ is the length of the dam. You may assume
the face of the dam is vertical. (Hint: Calculate the average pressure exerted and multiply this by the area in contact with the water. (See Figure 11.45.)
11.5 Pascal’s Principle
How much pressure is transmitted in the hydraulic system considered in Example 11.6? Express your answer in pascals and in atmospheres.
What force must be exerted on the master cylinder of a hydraulic lift to support the weight of a 2,000-kg car (a large car) resting on the slave cylinder? The master cylinder has a 2-cm diameter and
the slave has a 24-cm diameter.
A crass host pours the remnants of several bottles of wine into a jug after a party. He then inserts a cork with a 2-cm diameter into the bottle, placing it in direct contact with the wine. He is
amazed when he pounds the cork into place and the bottom of the jug (with a 14-cm diameter) breaks away. Calculate the extra force exerted against the bottom if he pounded the cork with a 120-N
A certain hydraulic system is designed to exert a force 100 times as large as the one put into it. (a) What must be the ratio of the area of the slave cylinder to the area of the master cylinder? (b)
What must be the ratio of their diameters? (c) By what factor is the distance through which the output force moves reduced relative to the distance through which the input force moves? Assume no
losses to friction.
(a) Verify that work input equals work output for a hydraulic system assuming no losses to friction. Do this by showing that the distance the output force moves is reduced by the same factor that the
output force is increased. Assume the volume of the fluid is constant. (b) What effect would friction within the fluid and between components in the system have on the output force? How would this
depend on whether or not the fluid is moving?
11.6 Gauge Pressure, Absolute Pressure, and Pressure Measurement
Find the gauge and absolute pressures in the balloon and peanut jar shown in Figure 11.19, assuming the manometer connected to the balloon uses water whereas the manometer connected to the jar
contains mercury. Express in units of centimeters of water for the balloon and millimeters of mercury for the jar, taking $h=0.0500 mh=0.0500 m size 12{h=0 "." "0500"`m} {}$ for each.
(a) Convert normal blood pressure readings of 120 over 80 mm Hg to newtons per meter squared using the relationship for pressure due to the weight of a fluid $(P=hρg)(P=hρg) size 12{ \( P=hρg \) } {}
$ rather than a conversion factor. (b) Discuss why blood pressures for an infant could be smaller than those for an adult. Specifically, consider the smaller height to which blood must be pumped.
How tall must a water-filled manometer be to measure blood pressures as high as 300 mm Hg?
Pressure cookers have been around for more than 300 years, although their use has strongly declined in recent years (early models had a nasty habit of exploding). How much force must the latches
holding the lid onto a pressure cooker be able to withstand if the circular lid is $25 cm25 cm size 12{"25" "." 0`"cm"} {}$ in diameter and the gauge pressure inside is 300 atm? Neglect the weight of
the lid.
Suppose you measure a standing person's blood pressure by placing the cuff on his leg 0.500 m below the heart. Calculate the pressure you would observe in units of mm Hg if the pressure at the heart
were 120 over 80 mm Hg. Assume that there is no loss of pressure due to resistance in the circulatory system; a reasonable assumption, since major arteries are large.
A submarine is stranded on the bottom of the ocean with its hatch 25 m below the surface. Calculate the force needed to open the hatch from the inside, given it is circular and 0.450 m in diameter.
Air pressure inside the submarine is 1 atm.
Assuming bicycle tires are perfectly flexible and support the weight of bicycle and rider by pressure alone, calculate the total area of the tires in contact with the ground. The bicycle plus rider
has a mass of 80 kg, and the gauge pressure in the tires is $3.50×105Pa3.50×105Pa size 12{3 "." "50" times "10" rSup { size 8{5} } `"Pa"} {}$.
11.7 Archimedes' Principle
What fraction of ice is submerged when it floats in freshwater, given the density of water at 0°C is very close to $1,000 kg/m31,000 kg/m3 size 12{"1000"`"kg/m" rSup { size 8{3} } } {}$?
Logs sometimes float vertically in a lake because one end has become water-logged and denser than the other. What is the average density of a uniform-diameter log that floats with $20 percent20
percent$ of its length above water?
Find the density of a fluid in which a hydrometer having a density of $0.750 g/mL0.750 g/mL size 12{0 "." "750"`"g/mL"} {}$ floats with $92 percent92 percent size 12{"92" "." 0%} {}$ of its volume
If your body has a density of $995 kg/m3995 kg/m3 size 12{"995"`"kg/m" rSup { size 8{3} } } {}$, what fraction of you will be submerged when floating gently in: (a) Freshwater? (b) Salt water, which
has a density of $1027 kg/m31027 kg/m3 size 12{"1027"`"kg/m" rSup { size 8{3} } } {}$?
Bird bones have air pockets in them to reduce their weight—this also gives them an average density significantly less than that of the bones of other animals. Suppose an ornithologist weighs a bird
bone in air and in water and finds its mass is $45 g45 g$ and its apparent mass when submerged is $3.60 g3.60 g size 12{3 "." "60"`g} {}$ (the bone is watertight). (a) What mass of water is
displaced? (b) What is the volume of the bone? (c) What is its average density?
A rock with a mass of 540 g in air is found to have an apparent mass of 342 g when submerged in water. (a) What mass of water is displaced? (b) What is the volume of the rock? (c) What is its average
density? Is this consistent with the value for granite?
Archimedes' principle can be used to calculate the density of a fluid as well as that of a solid. Suppose a chunk of iron with a mass of 390 g in air is found to have an apparent mass of 350.5 g when
completely submerged in an unknown liquid. (a) What mass of fluid does the iron displace? (b) What is the volume of iron, using its density as given in Table 11.1 (c) Calculate the fluid's density
and identify it.
In an immersion measurement of a woman's density, she is found to have a mass of 62 kg in air and an apparent mass of 0.0850 kg when completely submerged with lungs empty. (a) What mass of water does
she displace? (b) What is her volume? (c) Calculate her density. (d) If her lung capacity is 1.75 L, is she able to float without treading water with her lungs filled with air?
Some fish have a density slightly less than that of water and must exert a force (swim) to stay submerged. What force must an 85-kg grouper exert to stay submerged in salt water if its body density
is $1015kg/m31015kg/m3 size 12{"1015" `"kg/m" rSup { size 8{3} } } {}$?
(a) Calculate the buoyant force on a 2-L helium balloon. (b) Given the mass of the rubber in the balloon is 1.50 g, what is the net vertical force on the balloon if it is let go? You can neglect the
volume of the rubber.
(a) What is the density of a woman who floats in freshwater with four percent of her volume above the surface? This could be measured by placing her in a tank with marks on the side to measure how
much water she displaces when floating and when briefly held under water. (b) What percent of her volume is above the surface when she floats in seawater?
A certain man has a mass of 80 kg and a density of $955kg/m3955kg/m3 size 12{"955" `"kg/m" rSup { size 8{3} } } {}$ (excluding the air in his lungs). (a) Calculate his volume. (b) Find the buoyant
force air exerts on him. (c) What is the ratio of the buoyant force to his weight?
A simple compass can be made by placing a small bar magnet on a cork floating in water. (a) What fraction of a plain cork will be submerged when floating in water? (b) If the cork has a mass of 10 g
and a 20-g magnet is placed on it, what fraction of the cork will be submerged? (c) Will the bar magnet and cork float in ethyl alcohol?
What fraction of an iron anchor's weight will be supported by buoyant force when submerged in saltwater?
Scurrilous con artists have been known to represent gold-plated tungsten ingots as pure gold and sell them to the greedy at prices much below gold value but deservedly far above the cost of tungsten.
With what accuracy must you be able to measure the mass of such an ingot in and out of water to tell that it is almost pure tungsten rather than pure gold?
A twin-sized air mattress used for camping has dimensions of 100 cm by 200 cm by 15 cm when blown up. The weight of the mattress is 2 kg. How heavy a person could the air mattress hold if it is
placed in freshwater?
Referring to Figure 11.24, prove that the buoyant force on the cylinder is equal to the weight of the fluid displaced (Archimedes' principle). You may assume that the buoyant force is $F2−F1F2−F1
size 12{F rSub { size 8{2} } - F rSub { size 8{1} } } {}$ and that the ends of the cylinder have equal areas $AA size 12{A} {}$. Note that the volume of the cylinder and that of the fluid it
displaces equals $(h2−h1)A(h2−h1)A size 12{ \( h rSub { size 8{2} } - h rSub { size 8{1} } \) A} {}$.
(a) A 75-kg man floats in freshwater with three percent of his volume above water when his lungs are empty, and five percent of his volume above water when his lungs are full. Calculate the volume of
air he inhales—called his lung capacity—in liters. (b) Does this lung volume seem reasonable?
11.8 Cohesion and Adhesion in Liquids: Surface Tension and Capillary Action
What is the pressure inside an alveolus having a radius of $2.50×10−4m2.50×10−4m size 12{2 "." "50" times "10" rSup { size 8{ - 4} } `m} {}$ if the surface tension of the fluid-lined wall is the same
as for soapy water? You may assume the pressure is the same as that created by a spherical bubble.
(a) The pressure inside an alveolus with a $2×10−42×10−4 size 12{2 "." "00" times "10" rSup { size 8{ - 4} } } {}$-m radius is $1.40×103Pa1.40×103Pa size 12{1 "." "40" times "10" rSup { size 8{3} }
`"Pa"} {}$, due to its fluid-lined walls. Assuming the alveolus acts like a spherical bubble, what is the surface tension of the fluid? (b) Identify the likely fluid. You may need to extrapolate
between values in Table 11.3.
What is the gauge pressure in millimeters of mercury inside a soap bubble 0.100 m in diameter?
Calculate the force on the slide wire in Figure 11.32 if it is 3.50 cm long and the fluid is ethyl alcohol.
Figure 11.38(a) shows the effect of tube radius on the height to which capillary action can raise a fluid. (a) Calculate the height $hh size 12{h} {}$ for water in a glass tube with a radius of 0.900
cm—a rather large tube like the one on the left. (b) What is the radius of the glass tube on the right if it raises water to 4 cm?
We stated in Example 11.12 that a xylem tube is of radius $2.50×10−5m2.50×10−5m$. Verify that such a tube raises sap less than a meter by finding $hh$ for it, making the same assumptions that sap's
density is $1050kg/m31050kg/m3 size 12{"1050"`"kg/m" rSup { size 8{3} } } {}$, its contact angle is zero, and its surface tension is the same as that of water at $20º C20º C$.
What fluid is in the device shown in Figure 11.32 if the force is $3.16×10−3N3.16×10−3N size 12{3 "." "16" times "10" rSup { size 8{ - 3} } `N} {}$ and the length of the wire is 2.50 cm? Calculate
the surface tension $γγ size 12{g} {}$ and find a likely match from Table 11.3.
If the gauge pressure inside a rubber balloon with a 10-cm radius is 1.50 cm of water, what is the effective surface tension of the balloon?
Calculate the gauge pressures inside 2-cm-radius bubbles of water, alcohol, and soapy water. Which liquid forms the most stable bubbles, neglecting any effects of evaporation?
Suppose water is raised by capillary action to a height of 5 cm in a glass tube. (a) To what height will it be raised in a paraffin tube of the same radius? (b) In a silver tube of the same radius?
Calculate the contact angle $θθ size 12{θ} {}$ for olive oil if capillary action raises it to a height of 7.07 cm in a glass tube with a radius of 0.100 mm. Is this value consistent with that for
most organic liquids?
When two soap bubbles touch, the larger is inflated by the smaller until they form a single bubble. (a) What is the gauge pressure inside a soap bubble with a 1.50-cm radius? (b) Inside a 4-cm-radius
soap bubble? (c) Inside the single bubble they form if no air is lost when they touch?
Calculate the ratio of the heights to which water and mercury are raised by capillary action in the same glass tube.
What is the ratio of heights to which ethyl alcohol and water are raised by capillary action in the same glass tube?
11.9 Pressures in the Body
During forced exhalation, such as when blowing up a balloon, the diaphragm and chest muscles create a pressure of 60 mm Hg between the lungs and chest wall. What force in newtons does this pressure
create on the $600cm2600cm2 size 12{"600"`"cm" rSup { size 8{2} } } {}$ surface area of the diaphragm?
You can chew through very tough objects with your incisors because they exert a large force on the small area of a pointed tooth. What pressure in pascals can you create by exerting a force of $500
N500 N size 12{"500"``N} {}$ with your tooth on an area of $1mm21mm2 size 12{1 "." "00"`"mm" rSup { size 8{2} } } {}$?
One way to force air into an unconscious person's lungs is to squeeze on a balloon appropriately connected to the subject. What force must you exert on the balloon with your hands to create a gauge
pressure of 4 cm water, assuming you squeeze on an effective area of $50cm250cm2 size 12{"50" "." 0`"cm" rSup { size 8{2} } } {}$?
Heroes in movies hide beneath water and breathe through a hollow reed (villains never catch on to this trick). In practice, you cannot inhale in this manner if your lungs are more than 60 cm below
the surface. What is the maximum negative gauge pressure you can create in your lungs on dry land, assuming you can achieve $−3 cm−3 cm size 12{ - 3 "." "00"`"cm"} {}$ water pressure with your lungs
60 cm below the surface?
Gauge pressure in the fluid surrounding an infant's brain may rise as high as 85 mm Hg (5 to 12 mm Hg is normal), creating an outward force large enough to make the skull grow abnormally large. (a)
Calculate this outward force in newtons on each side of an infant's skull if the effective area of each side is $70cm270cm2 size 12{"70" "." 0`"cm" rSup { size 8{2} } } {}$. (b) What is the net force
acting on the skull?
A full-term fetus typically has a mass of 3.50 kg. (a) What pressure does the weight of such a fetus create if it rests on the mother's bladder, supported on an area of $90cm290cm2 size 12{"90" "." 0
`"cm" rSup { size 8{2} } } {}$? (b) Convert this pressure to millimeters of mercury and determine if it alone is great enough to trigger the micturition reflex (it will add to any pressure already
existing in the bladder).
If the pressure in the esophagus is $−2 mm Hg−2 mm Hg size 12{ - 2 "." "00"`"mm"`"Hg"} {}$ while that in the stomach is $+20 mm Hg+20 mm Hg size 12{+"20" "." 0`"mm"`"Hg"} {}$, to what height could
stomach fluid rise in the esophagus, assuming a density of 1.10 g/mL? This movement will not occur if the muscle closing the lower end of the esophagus is working properly.
Pressure in the spinal fluid is measured as shown in Figure 11.46. If the pressure in the spinal fluid is 10 mm Hg: (a) What is the reading of the water manometer in cm water? (b) What is the reading
if the person sits up, placing the top of the fluid 60 cm above the tap? The fluid density is 1.05 g/mL.
Calculate the maximum force in newtons exerted by the blood on an aneurysm, or ballooning, in a major artery, given the maximum blood pressure for this person is 150 mm Hg and the effective area of
the aneurysm is $20cm220cm2 size 12{"20" "." 0`"cm" rSup { size 8{2} } } {}$. Note that this force is great enough to cause further enlargement and subsequently greater force on the ever-thinner
vessel wall.
During heavy lifting, a disk between spinal vertebrae is subjected to a 5,000-N compressional force. (a) What pressure is created, assuming that the disk has a uniform circular cross section 2 cm in
radius? (b) What deformation is produced if the disk is 0.800 cm thick and has a Young's modulus of $1.5×109N/m21.5×109N/m2 size 12{1 "." 5 times "10" rSup { size 8{9} } `"N/m" rSup { size 8{2} } }
When a person sits erect, increasing the vertical position of their brain by 36 cm, the heart must continue to pump blood to the brain at the same rate. (a) What is the gain in gravitational
potential energy for 100 mL of blood raised 36 cm? (b) What is the drop in pressure, neglecting any losses due to friction? (c) Discuss how the gain in gravitational potential energy and the decrease
in pressure are related.
(a) How high will water rise in a glass capillary tube with a 0.500-mm radius? (b) How much gravitational potential energy does the water gain? (c) Discuss possible sources of this energy.
A negative pressure of 25 atm can sometimes be achieved with the device in Figure 11.47 before the water separates. (a) To what height could such a negative gauge pressure raise water? (b) How much
would a steel wire of the same diameter and length as this capillary stretch if suspended from above?
Suppose you hit a steel nail with a 0.500-kg hammer, initially moving at $15 m/s15 m/s size 12{"15" "." 0`"m/s"} {}$ and brought to rest in 2.80 mm. (a) What average force is exerted on the nail? (b)
How much is the nail compressed if it is 2.50 mm in diameter and 6-cm long? (c) What pressure is created on the 1 mm-diameter tip of the nail?
Calculate the pressure due to the ocean at the bottom of the Marianas Trench near the Philippines, given its depth is $11 km11 km size 12{"11" "." 0`"km"} {}$ and assuming the density of sea water is
constant all the way down. (b) Calculate the percent decrease in volume of sea water due to such a pressure, assuming its bulk modulus is the same as water and is constant. (c) What would be the
percent increase in its density? Is the assumption of constant density valid? Will the actual pressure be greater or smaller than that calculated under this assumption?
The hydraulic system of a backhoe is used to lift a load as shown in Figure 11.48. (a) Calculate the force $FF size 12{F} {}$ the slave cylinder must exert to support the 400-kg load and the 150-kg
brace and shovel. (b) What is the pressure in the hydraulic fluid if the slave cylinder is 2.50 cm in diameter? (c) What force would you have to exert on a lever with a mechanical advantage of five
acting on a master cylinder 0.800 cm in diameter to create this pressure?
Some miners wish to remove water from a mine shaft. A pipe is lowered to the water 90 m below, and a negative pressure is applied to raise the water. (a) Calculate the pressure needed to raise the
water. (b) What is unreasonable about this pressure? (c) What is unreasonable about the premise?
You are pumping up a bicycle tire with a hand pump, the piston of which has a 2-cm radius.
(a) What force in newtons must you exert to create a pressure of $6.90×105Pa6.90×105Pa size 12{6 "." "90" times "10" rSup { size 8{5} } `"Pa"} {}$ (b) What is unreasonable about this (a) result? (c)
Which premises are unreasonable or inconsistent?
Consider a group of people trying to stay afloat after their boat strikes a log in a lake. Construct a problem in which you calculate the number of people that can cling to the log and keep their
heads out of the water. Among the variables to be considered are the size and density of the log, and what is needed to keep a person's head and arms above water without swimming or treading water.
The alveoli in emphysema victims are damaged and effectively form larger sacs. Construct a problem in which you calculate the loss of pressure due to surface tension in the alveoli because of their
larger average diameters. Part of the lung's ability to expel air results from pressure created by surface tension in the alveoli. Among the things to consider are the normal surface tension of the
fluid lining the alveoli, the average alveolar radius in normal individuals and its average in emphysema sufferers. | {"url":"https://texasgateway.org/resource/problems-exercises-9?book=79096&binder_id=78561","timestamp":"2024-11-04T18:00:07Z","content_type":"text/html","content_length":"120481","record_id":"<urn:uuid:c293dba4-2684-46b9-8348-e98acded6646>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00421.warc.gz"} |
Quine Mccluskey Method Calculator - GEGCalculators
Quine Mccluskey Method Calculator
The Quine-McCluskey Method simplifies Boolean expressions. Start by listing minterms and their binary forms. Count the ‘1’s in each minterm and group them accordingly. Merge adjacent groups, treating
differing bits as “don’t cares.” Identify prime implicants and essential prime implicants. Construct the simplified expression using essential prime implicants for efficient digital logic design.
Quine-McCluskey Calculator
To apply the Quine-McCluskey Method for simplifying a Boolean expression, you need to create a table that lists all the minterms and their corresponding binary representations. Here’s how you can
create this table step by step:
Step 1: List the Minterms Start by listing all the minterms for the given Boolean function. Minterms are unique combinations of inputs that result in a ‘1’ output in the truth table. For a function
with ‘n’ variables, there will be 2^n minterms.
Step 2: Represent Minterms in Binary Represent each minterm in binary notation. If there are ‘n’ variables, you’ll need ‘n’ columns in your table for each variable, and an additional column for the
minterm number. For example, if you have a 3-variable function, the table might look like this:
Minterm Variable A Variable B Variable C
This table represents a 3-variable function with eight minterms (0 to 7), where each row corresponds to a unique combination of input variables.
Step 3: Determine the Number of 1s in Each Minterm In the next step, count the number of ‘1’s in each minterm’s binary representation. Add a new column to the table to record this count. This count
is essential for grouping minterms with the same number of ‘1’s.
Minterm Variable A Variable B Variable C Number of 1s
Step 4: Group Minterms by Number of 1s Group the minterms based on the number of ‘1’s they contain. Create separate groups for minterms with the same number of ‘1’s. In your table, add a column to
specify the group number for each minterm.
Minterm Variable A Variable B Variable C Number of 1s Group
Now, you have grouped the minterms based on the number of ‘1’s they contain, and each group is assigned a unique group number.
Step 5: Compare and Merge Adjacent Groups Compare adjacent groups to identify pairs of minterms that differ by only one bit. Merge these pairs into new groups, treating the differing bit as a “don’t
care.” Continue this process until no more merging is possible. Add another column to your table to indicate the merged groups.
Minterm Variable A Variable B Variable C Number of 1s Group Merged Group
0 0 0 0 0 0 –
3 0 1 1 2 2 –
5 1 0 1 2 2 –
6 1 1 0 2 2 –
In this example, groups 1 and 2 were merged into a new group, and the merged group is assigned the value 1.
Step 6: Identify Prime Implicants Now, identify the prime implicants from the merged groups. Prime implicants are groups that cannot be further merged with any other group. Add a column to your table
to indicate which groups are prime implicants.
Minterm Variable A Variable B Variable C Number of 1s Group Merged Group Prime Implicant
0 0 0 0 0 0 – X
1 0 0 1 1 1 1 X
2 0 1 0 1 1 1 X
3 0 1 1 2 2 – X
4 1 0 0 1 1 1 X
5 1 0 1 2 2 – X
6 1 1 0 2 2 – X
7 1 1 1 3 3 3 X
In this table, ‘X’ indicates that a group is a prime implicant.
Step 7: Determine Essential Prime Implicants Identify essential prime implicants by checking which minterms can only be covered by one prime implicant. Add a column to your table to indicate
essential prime implicants.
Minterm Variable A Variable B Variable C Number of 1s Group Merged Group Prime Implicant Essential PI
0 0 0 0 0 0 – X X
1 0 0 1 1 1 1 X X
2 0 1 0 1 1 1 X X
3 0 1 1 2 2 – X X
4 1 0 0 1 1 1 X X
5 1 0 1 2 2 – X X
6 1 1 0 2 2 – X X
7 1 1 1 3 3 3 X –
In this table, ‘X’ indicates essential prime implicants.
Step 8: Construct the Simplified Expression Finally, construct the simplified Boolean expression by using the essential prime implicants. You can use the table to identify which prime implicants to
include in the expression.
Once you’ve completed these steps, you will have successfully applied the Quine-McCluskey Method to simplify the given Boolean expression.
1. What is the Quine-McCluskey method? The Quine-McCluskey method is a technique used in digital logic design to simplify Boolean algebra expressions. It is used to minimize the number of terms and
literals in a Boolean function, resulting in a simplified expression that represents the same logical function but with fewer variables.
2. What is the difference between K-map and Quine McCluskey? The main differences between Karnaugh Maps (K-maps) and the Quine-McCluskey method are:
• Representation: K-maps use a graphical representation with cells arranged in a grid, while Quine-McCluskey uses a tabular method.
• Use of Don’t Cares: K-maps can easily handle don’t care conditions, while Quine-McCluskey requires additional steps to deal with them.
• Ease of Use: K-maps are generally considered more intuitive and easier for simplifying smaller functions, while Quine-McCluskey is preferred for larger functions with many variables.
3. What are the disadvantages of K-map, and how does the QM method overcome these disadvantages? Disadvantages of K-maps include:
• Inefficiency for larger functions with many variables.
• Difficulty in handling don’t care conditions.
The Quine-McCluskey method overcomes these disadvantages by providing a systematic, algorithmic approach that can handle larger functions efficiently and allows for explicit treatment of don’t care
4. What is the process of tabulation method? The tabulation method, also known as the Quine-McCluskey method, involves the following steps:
1. List all the minterms (or maxterms) of the given Boolean function.
2. Group the minterms with the same number of 1s (or 0s) in their binary representations.
3. Compare adjacent groups to find pairs that differ in only one bit position, creating a new group with that bit as a “don’t care.”
4. Repeat the grouping and comparison until no more combinations can be made.
5. Create prime implicants (minimal terms) from the groups.
6. Use prime implicants to find the essential prime implicants and construct the simplified Boolean expression.
5. Why do we need the Quine-McCluskey method? We need the Quine-McCluskey method to simplify Boolean expressions in digital logic design. It helps reduce the complexity of logical functions, which in
turn leads to more efficient and cost-effective digital circuits.
6. What is the Duhem-Quine principle? The Duhem-Quine principle, also known as the underdetermination of theory by evidence, is a philosophy of science concept that suggests that it’s impossible to
test a scientific hypothesis or theory in isolation. Instead, scientific theories are interconnected, and when experimental results conflict with a particular hypothesis, it may not necessarily be
the hypothesis itself that is incorrect; other parts of the theory or background assumptions might also be at fault.
7. Why is K-mapping better than Boolean algebra? Karnaugh mapping (K-mapping) is not necessarily “better” than Boolean algebra; rather, they are complementary methods for simplifying Boolean
expressions. K-maps are often considered more intuitive for simplifying smaller functions, while Boolean algebra and the Quine-McCluskey method are preferred for larger and more complex functions.
8. Why use K-map instead of Boolean theorems? K-maps are advantageous for simplifying Boolean expressions because they provide a visual and systematic approach that many find easier to use,
especially for smaller functions. Boolean theorems are still important and are used alongside K-maps and other methods, particularly for larger and more complex functions.
9. What is K-mapping, and what is the use of K-mapping? K-mapping is a graphical method used to simplify Boolean expressions. It is primarily used to minimize the number of terms and literals in a
Boolean function, making digital logic design more efficient and cost-effective. K-maps are particularly useful for simplifying functions with a small number of variables.
10. What is the drawback of the Karnaugh map method? One drawback of the Karnaugh map method is that it becomes less practical and more time-consuming as the number of variables in the Boolean
function increases. It is less efficient for functions with a large number of variables.
11. Why is Gray code used in K-map? Gray code is used in K-maps to ensure that adjacent cells in the map represent values that differ by only one bit. This property simplifies the grouping and
combination of terms in the map, making it easier to identify prime implicants.
12. Which binary code is used in K-maps? Gray code is typically used in K-maps to simplify the process of identifying adjacent cells that differ by a single bit.
13. What are the four types of tabulation? There are various tabulation methods in different fields, but in the context of the Quine-McCluskey method for Boolean function simplification, there are
typically four types of tabulation:
1. Minterm tabulation (for sum-of-products expressions).
2. Maxterm tabulation (for product-of-sums expressions).
3. Prime implicant chart/tabulation.
4. Essential prime implicant chart/tabulation.
14. What is the difference between Memoization and tabulation method? Memoization and tabulation are two techniques used in dynamic programming:
• Memoization is a top-down approach that stores computed results for subproblems in a cache (e.g., a dictionary) and recursively uses these results to avoid redundant computations.
• Tabulation is a bottom-up approach that starts with the simplest subproblems and builds up to the desired solution using an iterative process and a table (array) to store intermediate results.
15. What are the four parts of tabulation? In the context of dynamic programming, tabulation typically involves the following four parts:
1. Defining the table structure (array dimensions).
2. Initializing the base cases (values for the simplest subproblems).
3. Iterating through the subproblems and filling in the table entries.
4. Extracting the final result from the completed table.
16. What are the advantages and disadvantages of the Quine-McCluskey method? Advantages:
• Systematic and algorithmic approach.
• Works well for functions with a large number of variables.
• Can handle don’t care conditions.
• Produces minimal expressions.
• Can be time-consuming for small functions.
• Complex implementation.
• May require additional steps to identify essential prime implicants.
17. What is “don’t cares” in the Quine-McCluskey method? “Don’t cares” are conditions or combinations of inputs in a Boolean function for which the output value is not specified or irrelevant. In the
Quine-McCluskey method, don’t care conditions can be used to further simplify the Boolean expression by allowing certain terms to be included or excluded as needed to minimize the expression.
18. Who invented Quine-McCluskey? The Quine-McCluskey method is named after its inventors, Willard Van Orman Quine and Edward J. McCluskey. Quine was a philosopher and logician, while McCluskey was a
computer scientist.
19. Does Quine believe in a priori knowledge? Yes, Quine was known for his criticism of the concept of a priori knowledge. He argued that there is no clear distinction between a priori (knowledge
independent of experience) and a posteriori (knowledge derived from experience) knowledge and that all knowledge is ultimately based on empirical evidence.
20. What is holistic epistemology? Holistic epistemology is a philosophical approach to knowledge that emphasizes the interconnectedness of beliefs and the idea that the acceptance or rejection of a
particular belief is influenced by the entire web of beliefs within a person’s cognitive framework. It contrasts with foundationalism, which seeks to establish knowledge on a set of foundational or
self-evident beliefs.
21. What is the holism of Duhem-Quine thesis? The holism of the Duhem-Quine thesis refers to the idea that scientific theories are interconnected, and when testing a particular hypothesis or theory,
it’s not done in isolation but within the context of a larger theoretical framework. Thus, the acceptance or rejection of a hypothesis depends on the entire network of beliefs and assumptions within
a scientific theory.
22. What is the difference between a truth table and a K-map? A truth table is a tabular representation of all possible input combinations and their corresponding output values for a Boolean
function. In contrast, a Karnaugh map (K-map) is a graphical representation used for simplifying Boolean expressions by visually grouping adjacent cells with similar output values to identify
simplified terms.
23. What is the use of K-map in real life? K-maps are primarily used in digital logic design and circuitry to simplify Boolean expressions, leading to more efficient and cost-effective electronic
devices. Real-life applications include the design of computer processors, memory circuits, control systems, and more.
24. What is the application of K-map in real life? The applications of K-maps in real life include:
• Designing digital circuits and microprocessors.
• Implementing control systems for various devices.
• Reducing power consumption in electronic devices.
• Optimizing memory layouts for data storage.
25. What is a Karnaugh map in simple terms? A Karnaugh map (K-map) is a visual tool used in digital logic design to simplify Boolean expressions. It consists of a grid of cells, each representing a
unique combination of input values. By grouping adjacent cells with similar output values, engineers can find simplified expressions for digital logic functions.
26. What is the difference between canonical form and standard form? In Boolean algebra, the canonical form represents a Boolean expression in its most general and standard representation, without
any simplifications. Standard form, on the other hand, is a more simplified representation that may not necessarily be in its most general form. Canonical forms include the Sum of Products (SOP) and
Product of Sums (POS) forms, while standard forms are simpler expressions derived from the canonical forms.
27. How do you make a K-map for 4 variables? To create a K-map for 4 variables, you would arrange the cells in a 4×4 grid. The variables and their complements are typically labeled on the rows and
columns. Each cell represents a unique combination of input values, ranging from 0000 to 1111 in binary. You would then fill in the cells with the corresponding output values and use grouping to
simplify the Boolean expression.
28. What is the difference between SOP and POS? SOP (Sum of Products) and POS (Product of Sums) are two standard forms for representing Boolean expressions:
• SOP represents a Boolean function as the logical OR (sum) of multiple terms, where each term is the logical AND (product) of literals.
• POS represents a Boolean function as the logical AND (product) of multiple terms, where each term is the logical OR (sum) of literals.
29. How many cells are in a 4-variable K-map? A 4-variable K-map would have 16 cells arranged in a 4×4 grid. Each cell corresponds to a unique combination of the four variables.
30. Why is it called a “minterm”? A “minterm” is called so because it represents the smallest unit of a Boolean expression. It is a product term where all the variables in the expression appear once
in their either normal or complemented form. Minterms are used in the Sum of Products (SOP) form to represent all possible combinations of input variables.
31. What are two advantages of K-map? Two advantages of Karnaugh maps (K-maps) are:
1. Visual Representation: K-maps provide a visual representation of Boolean expressions, making it easier to identify patterns and simplify expressions.
2. Systematic Grouping: K-maps allow for systematic grouping of adjacent cells with similar output values, leading to more efficient simplifications.
32. What is a valid rule when working with Karnaugh maps? A valid rule when working with Karnaugh maps is that adjacent cells with a difference of only one variable (either 0 to 1 or 1 to 0) can be
grouped together to create simplified terms in the Boolean expression.
33. Do Karnaugh maps always work? Karnaugh maps work effectively for simplifying Boolean expressions in many cases, especially when dealing with a small number of variables. However, their efficiency
decreases as the number of variables increases, and for very large functions, other methods like the Quine-McCluskey algorithm may be preferred.
34. What is the difference between BCD and Gray code? BCD (Binary Coded Decimal) is a binary representation of decimal numbers, where each decimal digit is represented by a 4-bit binary code. Gray
code, on the other hand, is a binary numeral system in which two consecutive values differ in only one bit position. Gray code is used in Karnaugh maps to simplify grouping of adjacent cells.
35. How many cells are there in a 6-variable K-map? A 6-variable Karnaugh map would have 64 cells arranged in an 8×8 grid. Each cell corresponds to a unique combination of the six variables.
36. Why is Gray code used instead of binary? Gray code is used in Karnaugh maps and other applications because it ensures that adjacent values differ by only one bit, simplifying the process of
identifying adjacent cells with similar output values. This property is valuable for minimizing Boolean expressions efficiently.
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Present Value Calculator, Basic - Twin Pines Christian Church
Present Value Calculator, Basic
The future value interest factor is also based on the concept of TVM . And is useful in real-life scenarios by determining the future value today itself of amount to be received on a future date.
Overall, investors can use this type of discount factor template to translate future investment returns into net present value. Lender will calculate the prepayment premium using an Assumed
Reinvestment Rate of one basis point (+0.01%) in Section 10 and in the calculation of the Present Value Factor. Analogous to the future value and present value of a dollar, which is the future value
and present value of a lump-sum payment, the future value of an annuity is the value of equally spaced payments at some point in the future.
Note that n is the number of time periods that equal series of payments occur. Intuitively, the discount factor, which is always calculated by one divided by a figure, decreases the cash flow values.
This also ties back to what we discussed in the beginning, where receiving $1 today is more valuable than receiving $1 in the future.
How To Calculate Discount Factor
As shown in the future value case, the general formula is useful for solving other variations as long as we know two of the three variables. This is because at 12% the $15,000 is actually worth
$8,511.45 today, but you would need to make an outlay of only $8,000. According to these results, the amount of $8,000, which will be received after 5 years, has a present value of $4,540. As opposed
to decreasing over time, the factor increases in this case – thereby, the downward adjustment on the present value becomes more apparent in later years.
By letting the borrower have access to the money, the lender has sacrificed the exchange value of this money, and is compensated for it in the form of interest. The initial amount of the borrowed
funds is less than the total amount of money paid to the lender. Factoring companies, or companies that will buy your annuity or structured settlement, use discount rates to account for market risks
such as inflation and to make a small profit for granting you early access to your payments.
Formula For Present Value Of A Single Amount
The rate used to calculate the present value of future cash flows. It helps determines the effective future value of cash flows based on the compounding concept of interest calculation. It is useful
in the decision-making process or capital budgeting decisions. Any discount factor equation uses the assumption that today’s money will be worth less in the future due to factors like inflation,
which gives the discount factor a value between zero and one.
Recall how this time around, the cash flow will be divided by the discount factor to get the present value. Discount Factor is used to calculate what the value of receiving $1 at some point in the
future would be (the present value, or “PV”) based on the implied date of receipt and the discount rate assumption. Where APR is the annual nominal percentage rate, m is the number of compounding
periods per year and n is the total number of years. An annuity table is a tool for determining the present value of an annuity or other structured series of payments. Another advantage of the net
present value method is its ability to compare investments. As long as the NPV of each investment alternative is calculated back to the same point in time, the investor can accurately compare the
relative value in today’s terms of each investment. Imagine someone owes you $10,000 and that person promises to pay you back after five years.
Cost Accounting
A dollar today is worth more than a dollar tomorrow because the dollar can be invested and earn a day’s worth of interest, making the total accumulate to a value more than a dollar by tomorrow. Just
as https://accountingcoaching.online/ rent is paid to a landlord by a tenant without the ownership of the asset being transferred, interest is paid to a lender by a borrower who gains access to the
money for a time before paying it back.
This can be applied to goods, services, or investments, and is frequently used in corporate budgeting to determine whether a proposal will add future value. Think of the present value of a lump sum
in the future as the money you would need to invest today at a rate of interest that would accumulate to the desired amount in the future. In the example above, the amount of money you need to invest
today that will accumulate to $1,020 a year in the future at 2% is $1,000. Note that in using the present value or future HOW TO COMPUTE FOR PRESENT VALUE FACTOR AND FUTURE VALUE FACTOR value
formula, either the payment or the present value or future value could be blank, or they can both have values, depending on the investment. To learn more about or do calculations on future value
instead, feel free to pop on over to our Future Value Calculator. For a brief, educational introduction to finance and the time value of money, please visit our Finance Calculator. Beginning with the
future value equation and given a fixed time period, one can solve for the required interest rate as follows.
What Is The Formula For Calculating The Present Value Of An Annuity?
If offered a choice between $100 today or $100 in one year, and there is a positive real interest rate throughout the year, ceteris paribus, a rational person will choose $100 today. Time preference
can be measured by auctioning off a risk free security—like a US Treasury bill. If a $100 note with a zero coupon, payable in one year, sells for $80 now, then $80 is the present value of the note
that will be worth $100 a year from now. This is because money can be put in a bank account or any other investment that will return interest in the future.
• These calculations are used to make comparisons between cash flows that don’t occur at simultaneous times, since time and dates must be consistent in order to make comparisons between values.
• The concept is that a dollar today is not worth the same amount as a dollar tomorrow.
• Suppose we have invested $5,000 in the bank for 4 years at an interest rate of 8%.
• Continuously compounded interest, the mathematical limit of an interest rate with a period of zero time.
So, in this case, you’d divide $2,000 by (1 + 0.12), Which is 2.24%. When you start working with time value of money problems, you need to pay attention to distinguish between present value and
future value problems. Similar to future value tables, present value tables are based on the mathematical formula used to determine present value. Due to the relationship between future and present
values, the present value table is the inverse of the future value table. Many times in business and life, we want to determine the value today of receiving a specific single amount at some time in
the future. For 2021, the discount rate of 10% is added to 1, which is raised to the exponent of 1, as that is the first projected year. To arrive at the present value using the first approach, the
factor would then be multiplied by the cash flow to get the present value (“PV”).
2 Present And Future Value
For example, suppose you want to know the value today of receiving $15,000 at the end of 5 years if a rate of return of 12% is earned. The value of a future promise to pay or receive a single amount
at a specified interest rate is called the present value of a single amount. For more advanced future value calculations see our other future value calculators. See the Future Value of a Dollar
calculator to create a table of FVIF values. FREE INVESTMENT BANKING COURSELearn the foundation of Investment banking, financial modeling, valuations and more. Discounted cash flow is a valuation
method used to estimate the attractiveness of an investment opportunity. In addition, there is an implied interest value to the money over time that increases its value in the future and decreases
its value today relative to any future payment.
• Problems and questions like this are known as “present value of a single amount problems.” This is because we are interested in finding the present value, or the value today, of receiving a set
sum in the future.
• So, we need to multiply that with the factor P/ F i,n and discount it to the present time .
• If the opportunity cost of funds is 10% over next year, the factor is [1/(1 + 0.10)].
• The present value interest factor for a dollar on hand today is 0.
• And above each year, we have $2,000, starting from year one to year 10.
Discount rate depends on the risk-free rate and risk premium of an investment. Even, each cash flow stream can be discounted at a different discount rate, because of variation in expected inflation
rate and risk premium, but for simplicity purpose, we generally prefer to use single discounting rate.
Then, the 1st year cash flow of $100 is divided by 1.10 to get $91 for the PV of the cash flow. The 0.91 is subsequently multiplied by the cash flow of $100 to get $91 as the PV of the 1st year cash
flow. The formula for the second approach is virtually identical except for the absence of the negative sign in front of the period number exponent. This has been a guide to the Present Value Factor
and its meaning. Here we calculate the PV factor along with its examples and uses. Here we also provide you with a Present Value Calculator with a downloadable excel template.
• Present value calculations are influenced by when annuity payments are disbursed — either at the beginning or the end of a period.
• By factoring out future value, the 2nd portion of the formula is the present value factor which can be used to create a table to simplify the calculation.
• Free Financial Modeling Guide A Complete Guide to Financial Modeling This resource is designed to be the best free guide to financial modeling!
• In the future value example illustrated above, the interest rate was applied once because the investment was compounded annually.
• 2) Calculate end of year five lump sum settlement “F”, that is equivalent to receiving the end of the period payments.
The overall approximation is accurate to within ±6% (for all n≥1) for interest rates 0≤i≤0.20 and within ±10% for interest rates 0.20≤i≤0.40. The interpretation is that for an effective annual
interest rate of 10%, an individual would be indifferent to receiving $1000 in five years, or $620.92 today. Connect with a financial expert to find out how an annuity can offer you guaranteed
monthly income for life. Use your estimate as a starting point for conversation with a financial professional. Discuss your quote with one of our trusted partners, who can explain the present value
of your payments in more detail. Annuity.org carefully selects partners who share a common goal of educating consumers and helping them select the most appropriate product for their unique financial
and lifestyle goals. Our network of advisors will never recommend products that are not right for the consumer, nor will Annuity.org.
How Is The Present Value Factor Formula Derived?
Hence, it is important for those who are involved in decision making based on capital budgeting, calculating valuations of investments, companies, etc. Hence, the discounting rate of a risky
investment will be higher, as it denotes that the investor expects a higher return on the risky investment. Present Value Factor Formula is used to calculate a present value of all the future value
to be received. Time value of money is the concept that says an amount received today is more valuable than the same amount received at a future date. The formula for the present value factor is used
to calculate the present value per dollar that is received in the future. Below is more information about present value calculations so you understand the factors that affect your money and how to
use this calculator properly. Although you save yourself the cost of a financial advisor, you take complete responsibility for your financial future.
The offers that appear in this table are from partnerships from which Investopedia receives compensation. Investopedia does not include all offers available in the marketplace. Present value interest
factors are available in table form for reference. Present value interest factors are commonly used in analyzing annuities. For all questions in this set, interest compounds annually and there are no
transaction fees, defaults, etc. The price of borrowing money as it is usually stated, unadjusted for inflation.
Harold Averkamp has worked as a university accounting instructor, accountant, and consultant for more than 25 years. He is the sole author of all the materials on AccountingCoach.com. The future
value equals $14,185 (i.e. $10,000 multiplied by 1.4185). Will Kenton is an expert on the economy and investing laws and regulations. He previously held senior editorial roles at Investopedia and
Kapitall Wire and holds a MA in Economics from The New School for Social Research and Doctor of Philosophy in English literature from NYU. The amount of time that passes before interest begins to
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Noise contrastive estimation
In natural language we often wish to model the conditional distribution over the next word \(w\) in a sentence given the previous words \(c\), which stands for context. We can model the conditional
density using a real-valued function \(S_{\theta} (w, c)\):
\(\displaystyle p_{\theta} (w|c) = \frac{\exp (S_{\theta} (w, c))}{Z_{\theta} (c)} \qquad Z_{\theta} (c) = \int \exp (S_{\theta} (w', c)) d w'\) (1)
Intuitively, a particular value of \(w\) is exponentially more likely (higher density) if \(S_{\theta} (w, c)\) is larger. The denominator \(Z_{\theta} (c)\) is there to ensure that the probability
density integrates to \(1\):
\(\displaystyle \int p_{\theta} (w|c) d w = \frac{1}{Z_{\theta} (c)} \int \exp (S_{\theta} (w, c)) d w = 1\)
Notice that we were able to pull \(Z_{\theta} (c)\) out of the integral because it is not a function of \(w\).
1Maximum Likelihood
Suppose that for each context \(c\) there is a true data distribution over next words \(p_d (w|c)\). For each \(c\) we would like to maximize the log likelihood of the observed \(w\) under our model:
\(\displaystyle \max_{\theta} E_{p_d (\cdot |c)} [\log p_{\theta} (w|c)]\) (2)
To optimize this objective we would like to differentiate the expected log-likelihood:
\begin{eqnarray*} \frac{d}{d \theta} \mathbb{E}_{p_d (\cdot |c)} [\log p_{\theta} (w|c)] & = & \mathbb{E}_{p_d (\cdot |c)} \left[ \frac{d}{d \theta} \log (\exp (S_{\theta} (w, c))) \right] - \frac{d}
{d \theta} \log Z_{\theta} (c)\\ & = & \mathbb{E}_{p_d (\cdot |c)} \left[ \frac{d}{d \theta} S_{\theta} (w, c) \right] - E_{p_{\theta} (\cdot |c)} \left[ \frac{d}{d \theta} S_{\theta} (w, c) \right]
\hspace{3cm} \text{(3)} \end{eqnarray*}
For the second equality we are using:
\begin{eqnarray*} \frac{d}{d \theta} \log Z_{\theta} (c) & = & \frac{1}{Z_{\theta} (c)} \frac{d}{d \theta} \int \exp (S_{\theta} (w, c)) d w\\ & = & \frac{1}{Z_{\theta} (c)} \int \exp (S_{\theta} (w,
c)) \frac{d}{d \theta} S_{\theta} (w, c) d w\\ & = & \int \frac{\exp (S_{\theta} (w, c))}{Z_{\theta} (c)} \frac{d}{d \theta} S_{\theta} (w, c) d w \hspace{3cm} \text{(4)}\\ & = & E_{p_{\theta} (\cdot
|c)} \left[ \frac{d}{d \theta} S_{\theta} (w, c) \right] \end{eqnarray*}
We see from Eq. 4 that the second term of Eq. 3 involves an integral or sum over all possible values of \(w\), which can be difficult to compute.
2Classifying reals vs fakes
Suppose we have a batch of samples of \(w\) from two different distributions: one of the samples is from the real distribution \(p_d (w|c)\), and \(k\) additional samples are from a “noise”
distribution \(p_n (w|c)\). For each sample we attach a label \(D = 1\) if it is from the real distribution, or \(D = 0\) if it is from the noise distribution. We want to use our model \(p_{\theta}\)
to predict this label (a binary classification problem). Essentially, the idea behind noise contrastive estimation (NCE) is to use the model to differentiate samples from the true data distribution
versus those from the noise distribution.
Since we want our model distribution \(p_{\theta} (w|c)\) to closely approximate \(p_d (w|c)\), we use \(p_{\theta}\) to write down the likelihood of each label given \((w, c)\):
\(\displaystyle p (D = 1| w, c) = \frac{p_{\theta} (w|c)}{p_{\theta} (w|c) + kp_n (w|c)}\)
\(\displaystyle p (D = 0| w, c) = \frac{kp_n (w|c)}{p_{\theta} (w|c) + kp_n (w|c)}\)
In practice, NCE implementations often assume the partition function \(Z \equiv 1\), so for the rest of the section we assume
\(\displaystyle p_{\theta} (w|c) = \exp (S_{\theta} (w, c))\)
Then we can rewrite the likelihoods using the sigmoid \(\sigma (x) = \frac{1}{1 + e^{- x}}\):
\begin{eqnarray*} p (D = 1| w, c) & = & \frac{1}{1 + \frac{kp_n (w|c)}{\exp (S_{\theta} (w, c))}}\\ & = & \frac{1}{1 + \exp (- (S_{\theta} (w, c) - k \log p_n (w|c)))}\\ & = & \sigma (S_{\theta} (w,
c) - k \log p_n (w|c))\\ & = & \sigma (\Delta (w, c))\\ p (D = 0| w, c) & = & 1 - \sigma (\Delta (w, c)) \end{eqnarray*}
where we have defined
\(\displaystyle \Delta (w, c) = S_{\theta} (w, c) - k \log p_n (w|c)\)
So we can write our maximum likelihood objective for this binary classification task:
\(\displaystyle \max_{\theta} G (\theta) =\mathbb{E}_{p_d (\cdot |c)} [\log \sigma (\Delta)] + k\mathbb{E}_{p_n (\cdot |c)} [\log (1 - \sigma (\Delta))]\)
As before we differentiate the objective making use of the fact that \(\sigma' (x) = \sigma (x) (1 - \sigma (x))\) and that \(\frac{d \Delta (w, c)}{d \theta} = \frac{d}{d \theta} S_{\theta} (w, c)
\begin{eqnarray*} \frac{d G}{d \theta} & = & \mathbb{E}_{p_d (\cdot |c)} \left[ \frac{d}{d \theta} \log \sigma (\Delta) \right] + k\mathbb{E}_{p_n (\cdot |c)} \left[ \frac{d}{d \theta} \log (1 - \
sigma (\Delta)) \right]\\ & = & \mathbb{E}_{p_d (\cdot |c)} \left[ \frac{\sigma (\Delta) (1 - \sigma (\Delta))}{\sigma (\Delta)} \frac{d S_{\theta}}{d \theta} \right] - k\mathbb{E}_{p_n (\cdot |c)} \
left[ \frac{\sigma (\Delta) (1 - \sigma (\Delta))}{(1 - \sigma (\Delta))} \frac{d S_{\theta}}{d \theta} \right]\\ & = & \mathbb{E}_{p_d (\cdot |c)} \left[ (1 - \sigma (\Delta)) \frac{d S_{\theta}}{d
\theta} \right] - k\mathbb{E}_{p_n (\cdot |c)} \left[ \sigma (\Delta) \frac{d S_{\theta}}{d \theta} \right]\\ & = & \mathbb{E}_{p_d (\cdot |c)} \left[ (1 - \sigma (\Delta)) \frac{d S_{\theta}}{d \
theta} \right] -\mathbb{E}_{p_{\theta} (\cdot |c)} \left[ (1 - \sigma (\Delta)) \frac{d S_{\theta}}{d \theta} \right] \end{eqnarray*}
At the last equality we are using the fact that:
\begin{eqnarray*} k\mathbb{E}_{p_n (\cdot |c)} \left[ \sigma (\Delta) \frac{d S_{\theta}}{d \theta} \right] & = & k\mathbb{E}_{p_n (\cdot |c)} \left[ \frac{p_{\theta} (w|c)}{p_{\theta} (w|c) + kp_n
(w|c)} \frac{d S_{\theta}}{d \theta} \right]\\ & = & \mathbb{E}_{p_n (\cdot |c)} \left[ \frac{kp_n (w|c)}{p_{\theta} (w|c) + kp_n (w|c)} \frac{d S_{\theta}}{d \theta} \right]\\ & = & \mathbb{E}_{p_{\
theta} (\cdot |c)} \left[ (1 - \sigma (\Delta)) \frac{d S_{\theta}}{d \theta} \right] \end{eqnarray*}
Finally, since by inspecting the definition we can see \(\sigma (\Delta) \xrightarrow{k \rightarrow \infty} 0\):
\(\displaystyle \frac{d G}{d \theta} \xrightarrow{k \rightarrow \infty} \mathbb{E}_{p_d (\cdot |c)} \left[ \frac{d}{d \theta} S_{\theta} (w, c) \right] - E_{p_{\theta} (\cdot |c)} \left[ \frac{d}{d \
theta} S_{\theta} (w, c) \right]\)
So if we increase the number of noise samples \(k\) in the batch, our gradient approaches the original maximum likelihood gradient of Eq 3 asymptotically. Hence we see the motivation for why NCE's
binary classification task is a good approximation for the original objective in Eq 2.
3Further reading
This blog post and this paper contain good explanations of noise contrastive estimation, and have some more detailed derivations for some results. | {"url":"https://bland.website/notes/noise_contrastive_estimation/","timestamp":"2024-11-05T21:53:15Z","content_type":"text/html","content_length":"10844","record_id":"<urn:uuid:c529ac2f-f74f-4b25-a348-09c3be4e5451>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00607.warc.gz"} |
Capture the Flag
Is there a technical way to add this to the Server? Instead of voting for next Map when its stucked we could add this Option as it was used on st0rm..
That would be nice,lets do it!
We can host a server where it switches through game mods. Example: It randomly does aow, snipe, infantry only, CTF, and demolition. I could see doing it on a secondary server, but not on our
marathon. That our main player-base and we can't switch things up in there. Too risky.
• 1
I understand the risk of switching main things. My idea would have been to add it,not completly change the gameplay,if some of you remember it,im thinking of the way st0rm ran it back the days....?!
There is already a ctf server, not a rencorner server though.
• 1
We had planned to put up a sniper server which is up now. That is as much as we are going to change.
We do NOT want to be anything like St0rm.
Thanks for understanding.
• 1
Actually, I like the idea, it is a way to end stalemated games. All of St0rms ideas weren't bad, in fact this was good.
• 1
We had planned to put up a sniper server which is up now. That is as much as we are going to change.
We do NOT want to be anything like St0rm.
Thanks for understanding.
This, completely.
Actually, I like the idea, it is a way to end stalemated games. All of St0rms ideas weren't bad, in fact this was good.
!poll gameover works wonders Joe.
If I remember correctly, Wyld, we had a discussion about !pollgameover, don't know the outcome...It doesn't really change gameplay like being able to buy tanks and weapons does.
If you guys don't want to be like st0rm, hell, St0rm doesn't have Ren anymore! Plain CTF is boring, the AOW-CTF was a good fit.
• 1
Guest Animoski
I think alternative ways to end games are great.
I remember suggesting something like, after an hour of gametime on a specific map, people could !vote suddendeath or something, which would make whatever buildings are alive become non-repairable.
From that point on, it's up to however the teams are positioned and whatnot, and could make for some surprise endings imo.
• 2
Many Players leave after a gameover vote these days, basically cuz they feel they wasted their time, i dunno why ppl think it changes so much on rencorner. Its an alternative option and wont change
anything with the game itself. In fact ppl feel they ended a map the regular way and therefore its less ppl that rq and less ppl that leave due to stucked games.... Changes arent always a bad
decision, give it a tryout for 3-4 weeks and see how its going?!
Adding CTF to the gameplay is a major add. It takes the special gameplay of Renegade away: destroying the enemy base. A tryout for 3/4 weeks can give a disastrous outcome.
I personally don't like it.
• 2
I think alternative ways to end games are great.
I remember suggesting something like, after an hour of gametime on a specific map, people could !vote suddendeath or something, which would make whatever buildings are alive become
non-repairable. From that point on, it's up to however the teams are positioned and whatnot, and could make for some surprise endings imo.
Yeah I remember us talking about this and everyone seemed to agree but who can code to change to no-rep buildings after X amount of time? Can it be done based on a vote result? Sir Mr.
Adding CTF to the gameplay is a major add. It takes the special gameplay of Renegade away: destroying the enemy base. A tryout for 3/4 weeks can give a disastrous outcome.
I personally don't like it.
I agree. It usually ends up being two games within one from the start. I know several that would just spend the entire game in a apc going back and forth. Very frustating when the rest of the team is
planning a rush. Also with our average players count being 20 to 25, I can see frustated players because 3 or 4 on one team are CTF'ing and all the players on the other team are fully engaged in
attacking the base. It ends up like an uneven game.
• 1
StJohn Gumby
I'm not a big fan of Marathon games anyways, I much prefer 30 min AOW.
Hence my involvement in commwars here.
So, why not set a, say, 60 min time limit or something?
Put up a vote!
Do it before Wyld "The Closer" locks this topic.
• 1
Put up a vote!
Do it before Wyld "The Closer" locks this topic.
Joe, little do you know that the admins talk before we close any thread. It is never a "single" decision unless it is Ani. The two of the three recently closed were all discussed ahead of time to
close when the thread reached a conclusion. The 3rd one was closed as it was a duplicate made even when Shai had said not too. It's like the mod promotions, I didn't do those on my own, we discussed
it. I just happened to be the admin that posted it. We all share duties equally.
On point with this thread, you all can discuss something that will never happen all you want. RC will not have a CTF mod to help end games. Its a marathon server, learn to team play to win.
So far Joe in 3 or 4 of your recent posts your perception on the topic has been 100% wrong. If you are not sure which ones, PM me if you want to discuss further.
Hope this helps your "closed thread" syndrome. In the future as I said in the other thread, you have any questions you can always PM me.
• 1
For the stalemated games that usually end up in a !poll gameover, that's not a bad idea(they do not happen a lot mind you). There's also that ssgm feature(much like Campa made) for sudden death.
That'd be nice if it were configurable to where your shit goes offline with the last building on one team(right now it's just ref I think).
It could in theory even swap game modes while the game's running. Even funnier would be if it kicked it into that Asteroids game inside Renegade. lmao that'd be fucking funny...for testing purposes
of course. XD
• 1
How about "Capture the FAG"?
Basically someone preferably sean gets a stealth suit and we all go find and beat the shit outta them?
JK but I am a fan of long games. I can look back now and remember epic 7 hour field map years ago.
It never seems to happen these days, theres nothing more satisfying than uber teamwork when your on the back foot and then winning,
Just my opinion, it is a marathon after all.
• 2
How about "Capture the FAG"?
Basically someone preferably sean gets a stealth suit and we all go find and beat the shit outta them?
IS THIS POSSIBLE?!!!!
I'll be constantly moaning in ts until you find me
it'd need to be a big map like glacier_flying though. ^^
• 1
So in the end, what i read out of this thread is that *you can all discuss what u want,but nothing will change*. Ok then.
I think we are waiting to hear from Blacky if he can add what Ani had come up with or what Shai mentioned earlier. It's not CTF but it s something.
that sudden death sounds fun i would add that
• 1
Joe, little do you know that the admins talk before we close any thread. It is never a "single" decision unless it is Ani. The two of the three recently closed were all discussed ahead of time to
close when the thread reached a conclusion. The 3rd one was closed as it was a duplicate made even when Shai had said not too. It's like the mod promotions, I didn't do those on my own, we
discussed it. I just happened to be the admin that posted it. We all share duties equally.
On point with this thread, you all can discuss something that will never happen all you want. RC will not have a CTF mod to help end games. Its a marathon server, learn to team play to win.
So far Joe in 3 or 4 of your recent posts your perception on the topic has been 100% wrong. If you are not sure which ones, PM me if you want to discuss further.
Hope this helps your "closed thread" syndrome. In the future as I said in the other thread, you have any questions you can always PM me.
Frankly, Wyld, I don't know why a thread should be closed at all! Unless of course we have posts like mademan made asking for money, or something that is abusive. But a discussion is just that, a
discussion. The fact the you admins have access to the controls doesn't mean you should use it to stifle a dialog. Who knows? There probably was more to say...
Never mind the community's discussion, do as you will. You all know better. A poll would be a better choice.
Wyld, it's ok if you tell me what points I got wrong in public, I wouldn't mind.
Sorry to be so obstinate, I hope to be more co-operative in the future. Oh yeah Wyld, I apologize for creating drama in my RC Metemorphous post. Ii was created for another reason other than something
so trivial, as drama.
I may not be on tonight or tomorrow.
• 1
I talked to Blacky and he said he could do something. I also asked him how long it would take...I got a giggle. Not saying we doing it, not saying we aren't. We want to talk more about it, talk to
the players in the server, and Shai had an idea we may try. Stay tuned. | {"url":"https://rencorner.co/topic/952-capture-the-flag/","timestamp":"2024-11-04T01:07:54Z","content_type":"text/html","content_length":"290161","record_id":"<urn:uuid:e0f9c319-c734-4ffd-a096-b5b70f5ee9ee>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00788.warc.gz"} |
Integer Arithmetic (GNU Octave (version 6.2.0))
4.4.1 Integer Arithmetic
While many numerical computations can’t be carried out in integers, Octave does support basic operations like addition and multiplication on integers. The operators +, -, .*, and ./ work on integers
of the same type. So, it is possible to add two 32 bit integers, but not to add a 32 bit integer and a 16 bit integer.
When doing integer arithmetic one should consider the possibility of underflow and overflow. This happens when the result of the computation can’t be represented using the chosen integer type. As an
example it is not possible to represent the result of 10 - 20 when using unsigned integers. Octave makes sure that the result of integer computations is the integer that is closest to the true
result. So, the result of 10 - 20 when using unsigned integers is zero.
When doing integer division Octave will round the result to the nearest integer. This is different from most programming languages, where the result is often floored to the nearest integer. So, the
result of int32 (5) ./ int32 (8) is 1.
: idivide (x, y, op)
Integer division with different rounding rules.
The standard behavior of integer division such as a ./ b is to round the result to the nearest integer. This is not always the desired behavior and idivide permits integer element-by-element
division to be performed with different treatment for the fractional part of the division as determined by the op flag. op is a string with one of the values:
Calculate a ./ b with the fractional part rounded towards zero.
Calculate a ./ b with the fractional part rounded towards the nearest integer.
Calculate a ./ b with the fractional part rounded towards negative infinity.
Calculate a ./ b with the fractional part rounded towards positive infinity.
If op is not given it defaults to "fix". An example demonstrating these rounding rules is
idivide (int8 ([-3, 3]), int8 (4), "fix")
⇒ 0 0
idivide (int8 ([-3, 3]), int8 (4), "round")
⇒ -1 1
idivide (int8 ([-3, 3]), int8 (4), "floor")
⇒ -1 0
idivide (int8 ([-3, 3]), int8 (4), "ceil")
⇒ 0 1
See also: ldivide, rdivide. | {"url":"https://docs.octave.org/v6.2.0/Integer-Arithmetic.html","timestamp":"2024-11-11T14:19:36Z","content_type":"text/html","content_length":"6013","record_id":"<urn:uuid:bf361eb7-05e9-49c3-86e6-3180a2b2efe9>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00669.warc.gz"} |
Bi-Lipschitz arcs in metric spaces with controlled geometry (Journal Article) | NSF PAGES
We introduce and study the notion of *an outer bi-Lipschitz extension* of a map between Euclidean spaces. The notion is a natural analogue of the notion of *a Lipschitz extension* of a Lipschitz map.
We show that for every map f there exists an outer bi-Lipschitz extension f′ whose distortion is greater than that of f by at most a constant factor. This result can be seen as a counterpart of the
classic Kirszbraun theorem for outer bi-Lipschitz extensions. We also study outer bi-Lipschitz extensions of near-isometric maps and show upper and lower bounds for them. Then, we present
applications of our results to prioritized and terminal dimension reduction problems, described next. We prove a *prioritized* variant of the Johnson–Lindenstrauss lemma: given a set of points X⊂ ℝd
of size N and a permutation (”priority ranking”) of X, there exists an embedding f of X into ℝO(logN) with distortion O(loglogN) such that the point of rank j has only O(log3 + ε j) non-zero
coordinates – more specifically, all but the first O(log3+ε j) coordinates are equal to 0; the distortion of f restricted to the first j points (according to the ranking) is at most O(loglogj). The
result makes a progress towards answering an open question by Elkin, Filtser, and Neiman about prioritized dimension reductions. We prove that given a set X of N points in ℜd, there exists a
*terminal* dimension reduction embedding of ℝd into ℝd′, where d′ = O(logN/ε4), which preserves distances ||x−y|| between points x∈ X and y ∈ ℝd, up to a multiplicative factor of 1 ± ε. This improves
a recent result by Elkin, Filtser, and Neiman. The dimension reductions that we obtain are nonlinear, and this nonlinearity is necessary.
more » « less | {"url":"https://par.nsf.gov/biblio/10533579-bi-lipschitz-arcs-metric-spaces-controlled-geometry","timestamp":"2024-11-08T18:13:55Z","content_type":"text/html","content_length":"243096","record_id":"<urn:uuid:ef85294d-6152-4d6a-a88a-e39c0c558031>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00148.warc.gz"} |
<algorithm> Members
adjacent_find Searches for two adjacent elements that are either equal or satisfy a specified condition.
all_of Returns true when a condition is present at each element in the given range.
any_of Returns true when a condition is present at least once in the specified range of elements.
binary_search Tests whether there is an element in a sorted range that is equal to a specified value or that is equivalent to it in a sense specified by a binary predicate.
copy Assigns the values of elements from a source range to a destination range, iterating through the source sequence of elements and assigning them new positions in a forward
copy_backward Assigns the values of elements from a source range to a destination range, iterating through the source sequence of elements and assigning them new positions in a backward
copy_if Copy all elements in a given range that test true for a specified condition
copy_n Copies a specified number of elements.
count Returns the number of elements in a range whose values match a specified value.
count_if Returns the number of elements in a range whose values match a specified condition.
equal Compares two ranges element by element either for equality or equivalence in a sense specified by a binary predicate.
equal_range Finds a pair of positions in an ordered range, the first less than or equivalent to the position of a specified element and the second greater than the element's position,
where the sense of equivalence or ordering used to establish the positions in the sequence may be specified by a binary predicate.
fill Assigns the same new value to every element in a specified range.
fill_n Assigns a new value to a specified number of elements in a range starting with a particular element.
find Locates the position of the first occurrence of an element in a range that has a specified value.
find_end Looks in a range for the last subsequence that is identical to a specified sequence or that is equivalent in a sense specified by a binary predicate.
find_first_of Searches for the first occurrence of any of several values within a target range or for the first occurrence of any of several elements that are equivalent in a sense
specified by a binary predicate to a specified set of the elements.
find_if Locates the position of the first occurrence of an element in a range that satisfies a specified condition.
find_if_not Returns the first element in the indicated range that does not satisfy a condition.
for_each Applies a specified function object to each element in a forward order within a range and returns the function object.
generate Assigns the values generated by a function object to each element in a range.
generate_n Assigns the values generated by a function object to a specified number of element is a range and returns to the position one past the last assigned value.
includes Tests whether one sorted range contains all the elements contained in a second sorted range, where the ordering or equivalence criterion between elements may be specified
by a binary predicate.
inplace_merge Combines the elements from two consecutive sorted ranges into a single sorted range, where the ordering criterion may be specified by a binary predicate.
is_heap Returns true if the elements in the specified range form a heap.
is_heap_until Returns true if the specified range forms a heap until the last element.
is_partitioned Returns true if all the elements in the given range that test true for a condition come before any elements that test false.
is_sorted Returns true if the elements in the specified range are in sorted order.
is_sorted_until Returns true if the elements in the specified range are in sorted order.
iter_swap Exchanges two values referred to by a pair of specified iterators.
lexicographical_compare Compares element by element between two sequences to determine which is lesser of the two.
lower_bound Finds the position of the first element in an ordered range that has a value greater than or equivalent to a specified value, where the ordering criterion may be specified
by a binary predicate.
make_checked_array_iterator Creates a checked_array_iterator that can be used by other algorithms.
make_heap Converts elements from a specified range into a heap in which the first element is the largest and for which a sorting criterion may be specified with a binary predicate.
max Compares two objects and returns the larger of the two, where the ordering criterion may be specified by a binary predicate.
max_element Finds the first occurrence of largest element in a specified range where the ordering criterion may be specified by a binary predicate.
merge Combines all the elements from two sorted source ranges into a single, sorted destination range, where the ordering criterion may be specified by a binary predicate.
min Compares two objects and returns the lesser of the two, where the ordering criterion may be specified by a binary predicate.
min_element Finds the first occurrence of smallest element in a specified range where the ordering criterion may be specified by a binary predicate.
minmax Compares two input parameters and returns them as a pair, in order of least to greatest.
minmax_element Performs the work performed by min_element and max_element in one call.
mismatch Compares two ranges element by element either for equality or equivalent in a sense specified by a binary predicate and locates the first position where a difference
<alg> move Move elements associated with a specified range.
move_backward Moves the elements of one iterator to another. The move starts with the last element in a specified range, and ends with the first element in that range.
next_permutation Reorders the elements in a range so that the original ordering is replaced by the lexicographically next greater permutation if it exists, where the sense of next may be
specified with a binary predicate.
none_of Returns true when a condition is never present among elements in the given range.
nth_element Partitions a range of elements, correctly locating the nth element of the sequence in the range so that all the elements in front of it are less than or equal to it and
all the elements that follow it in the sequence are greater than or equal to it.
partial_sort Arranges a specified number of the smaller elements in a range into a nondescending order or according to an ordering criterion specified by a binary predicate.
partial_sort_copy Copies elements from a source range into a destination range where the source elements are ordered by either less than or another specified binary predicate.
partition Classifies elements in a range into two disjoint sets, with those elements satisfying a unary predicate preceding those that fail to satisfy it.
partition_copy Copies elements for which a condition is true to one destination, and for which the condition is false to another. The elements must come from a specified range.
partition_point Returns the first element in the given range that does not satisfy the condition. The elements are sorted so that those that satisfy the condition come before those that
do not.
pop_heap Removes the largest element from the front of a heap to the next-to-last position in the range and then forms a new heap from the remaining elements.
prev_permutation Reorders the elements in a range so that the original ordering is replaced by the lexicographically next greater permutation if it exists, where the sense of next may be
specified with a binary predicate.
push_heap Adds an element that is at the end of a range to an existing heap consisting of the prior elements in the range.
random_shuffle Rearranges a sequence of N elements in a range into one of N! possible arrangements selected at random.
remove Eliminates a specified value from a given range without disturbing the order of the remaining elements and returning the end of a new range free of the specified value.
remove_copy Copies elements from a source range to a destination range, except that elements of a specified value are not copied, without disturbing the order of the remaining
elements and returning the end of a new destination range.
remove_copy_if Copies elements from a source range to a destination range, except that satisfying a predicate are not copied, without disturbing the order of the remaining elements and
returning the end of a new destination range.
remove_if Eliminates elements that satisfy a predicate from a given range without disturbing the order of the remaining elements and returning the end of a new range free of the
specified value.
replace Examines each element in a range and replaces it if it matches a specified value.
replace_copy Examines each element in a source range and replaces it if it matches a specified value while copying the result into a new destination range.
replace_copy_if Examines each element in a source range and replaces it if it satisfies a specified predicate while copying the result into a new destination range.
replace_if Examines each element in a range and replaces it if it satisfies a specified predicate.
reverse Reverses the order of the elements within a range.
reverse_copy Reverses the order of the elements within a source range while copying them into a destination range
rotate Exchanges the elements in two adjacent ranges.
rotate_copy Exchanges the elements in two adjacent ranges within a source range and copies the result to a destination range.
search Searches for the first occurrence of a sequence within a target range whose elements are equal to those in a given sequence of elements or whose elements are equivalent in
a sense specified by a binary predicate to the elements in the given sequence.
search_n Searches for the first subsequence in a range that of a specified number of elements having a particular value or a relation to that value as specified by a binary
set_difference Unites all of the elements that belong to one sorted source range, but not to a second sorted source range, into a single, sorted destination range, where the ordering
criterion may be specified by a binary predicate.
set_intersection Unites all of the elements that belong to both sorted source ranges into a single, sorted destination range, where the ordering criterion may be specified by a binary
set_symmetric_difference Unites all of the elements that belong to one, but not both, of the sorted source ranges into a single, sorted destination range, where the ordering criterion may be
specified by a binary predicate.
set_union Unites all of the elements that belong to at least one of two sorted source ranges into a single, sorted destination range, where the ordering criterion may be specified
by a binary predicate.
sort Arranges the elements in a specified range into a nondescending order or according to an ordering criterion specified by a binary predicate.
sort_heap Converts a heap into a sorted range.
stable_partition Classifies elements in a range into two disjoint sets, with those elements satisfying a unary predicate preceding those that fail to satisfy it, preserving the relative
order of equivalent elements.
stable_sort Arranges the elements in a specified range into a nondescending order or according to an ordering criterion specified by a binary predicate and preserves the relative
ordering of equivalent elements.
swap Exchanges the values of the elements between two types of objects, assigning the contents of the first object to the second object and the contents of the second to the
swap_ranges Exchanges the elements of one range with the elements of another, equal sized range.
transform Applies a specified function object to each element in a source range or to a pair of elements from two source ranges and copies the return values of the function object
into a destination range.
unique Removes duplicate elements that are adjacent to each other in a specified range.
unique_copy Copies elements from a source range into a destination range except for the duplicate elements that are adjacent to each other.
upper_bound Finds the position of the first element in an ordered range that has a value that is greater than a specified value, where the ordering criterion may be specified by a
binary predicate.
See Also
Thread Safety in the Standard C++ Library | {"url":"https://learn.microsoft.com/en-us/previous-versions/visualstudio/visual-studio-2010/chzkfc23(v=vs.100)?redirectedfrom=MSDN","timestamp":"2024-11-04T16:39:05Z","content_type":"text/html","content_length":"60881","record_id":"<urn:uuid:8d03b12f-8464-4b2b-b3cf-fbd1d2d4880c>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00809.warc.gz"} |
Understanding Mathematical Functions: How To Divide Functions Long Div
Mathematical functions are a fundamental part of algebra and calculus, representing relationships between variables. Understanding and manipulating functions is essential for solving complex
mathematical problems and real-world applications. In this blog post, we will focus on the long division of functions, an important technique in simplifying and analyzing functions.
Long division of functions is a crucial skill for students and professionals alike, as it allows for the simplification of complex functions and the identification of key characteristics such as
roots, asymptotes, and behavior. Whether you're a student looking to ace your math exams or a professional needing to solve intricate engineering problems, mastering the long division of functions is
essential for success.
Key Takeaways
• Understanding and manipulating functions is essential for solving complex mathematical problems and real-world applications.
• Long division of functions is crucial for simplifying and analyzing complex functions, identifying key characteristics such as roots, asymptotes, and behavior.
• Mastering the long division of functions is essential for success in academic and professional settings.
• The steps to perform long division of functions involve dividing the leading terms, multiplying, and repeating the process until the remainder has a lower degree than the denominator.
• Addressing common mistakes and implementing tips and techniques can help improve accuracy in performing long division of functions.
Understanding Mathematical Functions: How to Divide Functions Long Division
Mathematical functions are a fundamental concept in mathematics, used to describe the relationship between inputs and outputs. Understanding functions is essential for solving mathematical problems
and real-world applications.
A. Define mathematical functions
A mathematical function is a rule that assigns each input from a set (the domain) to exactly one output from another set (the range). It can be represented using a variety of mathematical notation,
including algebraic expressions, tables, graphs, and verbal descriptions.
B. Discuss the basic elements of a function (input, output, rule)
• Input:
The input, also known as the independent variable, is the value that is fed into the function. It is the value we use to evaluate the function.
• Output:
The output, also known as the dependent variable, is the value that the function produces as a result of applying the rule to the input.
• Rule:
The rule of a function determines how the input is transformed into the output. It is the relationship or operation that defines the function's behavior.
The concept of dividing functions
When it comes to understanding mathematical functions, dividing one function by another is a fundamental concept that is often used in calculus and algebra. Dividing functions involves finding a new
function that represents the result of dividing one function by another.
A. Explain the concept of dividing one function by another
• Dividing one function by another involves finding the quotient function that represents the result of the division. This can be done by performing long division or using other methods such as
synthetic division.
• Just like dividing numbers, dividing functions requires identifying the quotient and the remainder. The quotient function represents the result of the division, while the remainder function
represents any leftover terms that cannot be divided evenly.
B. Discuss the application of long division in dividing functions
• Long division is a method used to divide functions, especially when the degree of the dividend function is higher than the degree of the divisor function. This method involves dividing the terms
of the dividend function by the leading term of the divisor function, and then subtracting the result from the dividend function.
• Long division is a systematic approach that allows for the step-by-step division of functions, making it easier to identify the quotient and the remainder. This method is especially useful in
finding the partial fraction decomposition of rational functions.
Steps to perform long division of functions
Long division of functions involves a series of steps to divide one function by another. By following these steps, you can efficiently divide functions and find the quotient and remainder. The
following steps outline the process of long division of functions:
A. Divide the leading term of the numerator by the leading term of the denominator
• Step 1: Identify the leading terms of the numerator and the denominator. Divide the leading term of the numerator by the leading term of the denominator to obtain the first term of the quotient.
B. Multiply the entire denominator by the quotient from Step 1 and subtract it from the numerator
• Step 2: Multiply the entire denominator by the quotient obtained in Step 1. Subtract this product from the numerator to obtain a new polynomial.
C. Repeat the process until the degree of the remainder is less than the degree of the denominator
• Step 3: Repeat Steps 1 and 2 until the degree of the remainder is less than the degree of the denominator. This ensures that the division process is complete, and the quotient and remainder are
accurately determined.
By following these steps, you can successfully perform long division of functions and obtain the quotient and remainder. This process is essential in understanding mathematical functions and their
Examples of long division of functions
Understanding how to divide functions using long division is a fundamental concept in mathematics, particularly in calculus and algebra. Let's explore some simple examples to demonstrate how long
division of functions works.
A. Provide a simple example to demonstrate the long division of functions
To illustrate the long division of functions, let's consider the following example:
f(x) = 2x^3 - 5x^2 + 3x - 7
g(x) = x - 2
B. Walk through the steps of long division with detailed explanations
When performing long division of functions, the dividend (f(x)) is divided by the divisor (g(x)) to obtain the quotient and the remainder.
Step 1: We start by dividing the highest degree term of the dividend by the highest degree term of the divisor. In this example, we divide 2x^3 by x, which gives us 2x^2.
Step 2: Next, we multiply the entire divisor (x - 2) by the result from Step 1 (2x^2), which gives us 2x^3 - 4x^2.
Step 3: We then subtract the result from Step 2 from the original dividend (f(x)), which gives us 2x^3 - 5x^2 + 3x - 7 - (2x^3 - 4x^2), resulting in -x^2 + 3x - 7.
Step 4: We repeat the process by dividing the highest degree term of the result from Step 3 (-x^2) by the highest degree term of the divisor (x), which gives us -x.
Step 5: We multiply the entire divisor (x - 2) by the result from Step 4 (-x), which gives us -x^2 + 2x.
Step 6: We subtract the result from Step 5 from the result of Step 3, which gives us -x^2 + 3x - 7 - (-x^2 + 2x), resulting in x - 7.
Step 7: At this point, we have no more terms to divide, and the degree of the result is less than the degree of the divisor. Therefore, the quotient is 2x^2 - x + 3 with a remainder of (x - 7).
By following these steps, we have successfully performed long division of functions to obtain the quotient and the remainder.
Understanding Mathematical Functions: How to Divide Functions Long Division
Common challenges and how to overcome them
Long division of functions can be a challenging concept for many students and even experienced mathematicians. Let's address some common mistakes and provide tips and techniques to avoid errors and
improve accuracy.
A. Address common mistakes in performing long division of functions
• Forgetting to consider all terms: One common mistake in long division of functions is forgetting to consider all terms in the divisor and dividend. This can lead to incorrect results.
• Incorrect placement of terms: Another common mistake is placing the terms in the wrong position, leading to confusion and errors in the calculation.
• Not simplifying before dividing: Failing to simplify the functions before dividing can make the process more complex and prone to errors.
B. Provide tips and techniques to avoid errors and improve accuracy
• Tip 1: Always check for all terms: When performing long division of functions, double-check that you have considered all terms in both the divisor and dividend before starting the calculation.
• Tip 2: Align terms correctly: Properly align the terms of the divisor and dividend to ensure clarity and accuracy in the calculation.
• Tip 3: Simplify before dividing: Simplify the functions as much as possible before beginning the long division process. This will make the calculation easier and reduce the chances of errors.
• Tip 4: Practice and review: Long division of functions requires practice to master. Regularly review the techniques and principles to reinforce your understanding and improve accuracy.
In conclusion, we have discussed the key points of how to perform long division of functions in mathematics. We have learned about the importance of understanding the process and the steps involved
in dividing functions using long division method.
Understanding long division of functions is essential for solving complex mathematical problems and developing a deeper understanding of mathematical functions. It allows us to simplify and analyze
functions, making it an invaluable tool for students and professionals in the field of mathematics.
Keep practicing and exploring the world of mathematical functions!
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Succinct filters for sets of unknown sizes
The membership problem asks to maintain a set S ⊆ [u], supporting insertions and membership queries, i.e., testing if a given element is in the set. A data structure that computes exact answers is
called a dictionary. When a (small) false positive rate ε is allowed, the data structure is called a filter. The space usages of the standard dictionaries or filters usually depend on the upper bound
on the size of S, while the actual set can be much smaller. Pagh, Segev and Wieder [28] were the first to study filters with varying space usage based on the current |S|. They showed in order to
match the space with the current set size n = |S|, any filter data structure must use (1 − o(1))n(log(1/ε) + (1 − O(ε)) log log n) bits, in contrast to the well-known lower bound of N log(1/ε) bits,
where N is an upper bound on |S|. They also presented a data structure with almost optimal space of (1 + o(1))n(log(1/ε) + O(log log n)) bits provided that n > u^0.001, with expected amortized
constant insertion time and worst-case constant lookup time. In this work, we present a filter data structure with improvements in two aspects: it has constant worst-case time for all insertions and
lookups with high probability; it uses space (1 + o(1))n(log(1/ε) + log log n) bits when n > u^0.001, achieving optimal leading constant for all ε = o(1). We also present a dictionary that uses (1 +
o(1))n log(u/n) bits of space, matching the optimal space in terms of the current size, and performs all operations in constant time with high probability.
Original language English (US)
Title of host publication 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020
Editors Artur Czumaj, Anuj Dawar, Emanuela Merelli
Publisher Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic) 9783959771382
State Published - Jun 1 2020
Event 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020 - Virtual, Online, Germany
Duration: Jul 8 2020 → Jul 11 2020
Publication series
Name Leibniz International Proceedings in Informatics, LIPIcs
Volume 168
ISSN (Print) 1868-8969
Conference 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020
Country/Territory Germany
City Virtual, Online
Period 7/8/20 → 7/11/20
All Science Journal Classification (ASJC) codes
• Approximate set membership
• Bloom filters
• Data structures
• Dictionaries
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Gecode::Int::Linear::Lin< Val, P, N, pc > Class Template Reference
#include <linear.hh>
Inherits Gecode::Propagator.
Inherited by Gecode::Int::Linear::ReLin< Val, P, N, PC_INT_BND, BoolView >, and Gecode::Int::Linear::ReLin< Val, P, N, PC_INT_BND, Ctrl >.
Detailed Description
template<class Val, class P, class N, PropCond pc>
class Gecode::Int::Linear::Lin< Val, P, N, pc >
Base-class for n-ary linear propagators.
The type Val can be either double or int, defining the numerical precision during propagation. Positive views are of type P whereas negative views are of type N.
The propagation condition pc refers to all views.
Definition at line 477 of file linear.hh.
Public Member Functions
virtual PropCost cost (void) const
Cost function (defined as dynamic PC_LINEAR_LO).
virtual size_t dispose (Space *home)
Delete propagator and return its size.
Protected Member Functions
Lin (Space *home, bool share, Lin &p)
Constructor for cloning p.
Lin (Space *home, ViewArray< P > &x, ViewArray< N > &y, Val c)
Constructor for creation.
Protected Attributes
ViewArray< P > x
Array of positive views.
ViewArray< N > y
Array of negative views.
Val c
Constant value.
Related Functions
(Note that these are not member functions.)
void bounds_p (const Propagator *, ViewArray< View > &x, Val &c, Val &sl, Val &su)
Compute bounds information for positive views.
void bounds_n (const Propagator *, ViewArray< View > &y, Val &c, Val &sl, Val &su)
Compute bounds information for negative views.
Constructor & Destructor Documentation
template<class Val, class P, class N, PropCond pc>
Gecode::Int::Linear::Lin< Val, P, N, pc >::Lin ( Space * home,
bool share,
Lin< Val, P, N, pc > & p
) [protected]
Constructor for cloning p.
Constructor for creation.
Member Function Documentation
Cost function (defined as dynamic PC_LINEAR_LO).
Implements Gecode::Propagator.
Reimplemented in Gecode::Int::Linear::DomEq< Val, View >.
Delete propagator and return its size.
Reimplemented from Gecode::Actor.
Reimplemented in Gecode::Int::Linear::ReLin< Val, P, N, pc, Ctrl >, Gecode::Int::Linear::ReLin< Val, P, N, PC_INT_BND, Ctrl >, and Gecode::Int::Linear::ReLin< Val, P, N, PC_INT_BND, BoolView >.
Friends And Related Function Documentation
template<class Val, class P, class N, PropCond pc>
void bounds_p ( const Propagator * ,
ViewArray< View > & x,
Val & c,
Val & sl,
Val & su
) [related]
Compute bounds information for positive views.
Definition at line 117 of file nary.icc.
template<class Val, class P, class N, PropCond pc>
void bounds_n ( const Propagator * ,
ViewArray< View > & y,
Val & c,
Val & sl,
Val & su
) [related]
Compute bounds information for negative views.
Definition at line 139 of file nary.icc.
Member Data Documentation
Array of positive views.
Definition at line 480 of file linear.hh.
Array of negative views.
Definition at line 482 of file linear.hh.
Constant value.
Definition at line 484 of file linear.hh.
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Professional development
• The Culturally Responsive Teaching Self-Efficacy Scale (CRTSE) is a survey that assess teachers’ beliefs that engaging in culturally responsive teaching practices will have positive classroom and
student outcomes. The scale is based on Bandura's (1977) research on outcome expectancies (a person's estimate that a given behavior will lead to certain outcomes), and Siwatu's (2006) research
on Culturally Responsive Teaching Competencies. Survey respondents rate from 0 to 100 the probability that a certain culturally responsive teaching behavior will lead to positive classroom and
student outcomes. The scale has primarily been used to support the development of preservice teachers.
• The Misconceptions-Oriented Standards-Based Assessment Resources for Teachers (MOSART) is a set of teacher knowledge assessments linked to the NRC National Science Education Standards.
Assessments are broken down by grade level (K-4, 5-8, and 9-12) and by content area (physical science, earth science, space science, and life science). Assessments are multiple choice with each
question asking for the scientifically correct answer (to measure subject matter knowledge) as well as the most popular incorrect answer given by students (to measure knowledge of student
misconceptions). Student assessments with items linked to teacher assessments are also available.
• The Teachers' Sense of Efficacy Scale (TSES) measures teachers' evaluations of how likely they are to be successful in teaching. TSES conceptualizes teaching as a complex activity and teacher
efficacy as a multi-faceted construct representing at least three distinct factors: Efficacy for Classroom Management, Efficacy to promote Student Engagement, and Efficacy in using Instructional
Strategies. It is designed for and has been used by researchers and school leaders to measure teacher self-efficacy at a particular point in time, as well as before and after participating in
professional development programs. There is a short- and long-form version of the scale.
• K-TEEM is a web-based assessment designed to measure early elementary teachers' mathematical knowledge for teaching.
• The Multicultural Staff Development Teacher Survey analyzes the teachers' perceptions of multicultural education's benefits to their students and who and what is necessary for the students to
receive those perceived benefits; the teachers' interest and motivation for attending a multicultural education workshop given a particular format; and the self-perceived awareness of the basic
elements surrounding multicultural education.
• The Multicultural Efficacy Scale (MES) measures teachers' multicultural efficacy and the multicultural teacher education dimensions of intercultural experiences, minority group knowledge,
attitudes about diversity, and knowledge of teaching skills in multicultural settings.
• Assessing Teacher Learning About Science Teaching (ATLAST) measures teacher knowledge in a specific content area. Assessments exist in the following content areas: Flow of Matter and Energy in
Living Systems; Force and Motion; and Plate Tectonics. All questions are set in instructional contexts, with some items requiring teachers to analyze student thinking or to choose among
instructional responses to student thinking. The measure has been used as a pre- and post-test to assess the effects of professional development programs. There are also corresponding student
written assessments.
• Diagnostic Mathematics Assessments for Elementary School Teachers are written tests that measure mathematics knowledge in four content domains (Whole Number/Computation, Rational Number/
Computation, Geometry/Measurement, Probability/Statistics/Algebra) and across four types of knowledge (Memorized Knowledge; Conceptual Understanding; Problem Solving/ Reasoning; and Pedagogical
Content Knowledge).
• Diagnostic Science Assessments for Middle School Teachers are written tests that measure mathematics knowledge in three content domains (Physical Science; Life Science; Earth/Space Science) and
across four types of knowledge (Declarative Knowledge; Scientific Inquiry and Procedures; Schematic Knowledge; Pedagogical Content Knowledge). Researchers, professional development providers, and
course instructors can administer the assessments as pre- and post-tests before and after workshops, institutes, or courses to determine growth in teachers' content knowledge.
• Diagnostic Mathematics Assessments for Middle School Teachers are written tests that measure mathematics knowledge in four content domains (Number/Computation, Geometry/Measurement, Probability/
Statistics, Algebraic Ideas) and across four types of knowledge (Memorized Knowledge; Conceptual Understanding; Problem Solving/ Reasoning; and Pedagogical Content Knowledge).
• Making Sense of Science (Electric Circuits Assessment) is a written test that measures teachers' pedagogical content knowledge regarding electric circuits. Test items are aligned to the National
Science Education Standards Benchmarks, the Full Option Science System curriculum, and the Science and Technology Concepts curriculum. Items reflect the format and content of the Trends in
International Mathematics and Science Study (TIMSS) and National Assessment of Education Progress (NAEP) assessments.
• The Mathematical Knowledge for Teaching (MKT) instrument measures the specialized knowledge of K-8 teachers of mathematics use in teaching. These measures include items that reflect the real
mathematics tasks teachers face in classrooms, such as assessing student work, representing numbers and operations, and explaining common mathematical rules or procedures. Assessments composed of
these problems can be used to measure the effectiveness of mathematics-focused professional development.
• The Mathematics Teaching Efficacy Beliefs Instrument (MTEBI) is a domain-specific modification of the Science Teaching Efficacy Belief Instrument (STEBI). The MTEBI measures pre-service teachers’
beliefs regarding mathematical instruction, students' ability to learn mathematics and teachers’ sense of efficacy in teaching mathematics.
• The Preservice Teacher Feedback Survey measures beliefs about classroom feedback practices and confidence in giving feedback to students. The measure is designed for use by preservice teachers to
assess the extent to which their beliefs and confidence in giving feedback align with research on quality feedback. The measure has been used as a pre- and post-test to assess the effects of
initial teacher education courses on assessment.
• The Science Teaching Efficacy Belief Instrument (STEBI) can be used to measure science teachers’ self-efficacy and outcome expectancy across all levels of teaching. The STEBI-A is the version
recommended for use with in-service teachers; the STEBI-B is the version recommended for use with pre-service teachers.
• The Technological Pedagogical Content Knowledge (TPACK) instrument is a self-efficacy survey that seeks to monitor and assess teachers' development of TPACK and seven interrelated knowledge
domains. The measure is primarily aimed towards pre-service secondary mathematics teachers and can be used to longitudinally study program effectiveness at TPACK development, and to conduct
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Re: [tlaplus] Missing theorem
Hello Frédéric,
thanks for the suggestion – we always value hints about which lemmas are missing in the standard library. In fact, the next release will see many additions to the SequenceProperties module.
However, I am a little reluctant to add the lemma that you propose, since it is just proved by OBVIOUS. If you have an instance of this lemma in the context of a larger proof where the prover was
unable to prove such a step, I'd be interested in hearing about it.
A theorem that is missing in SequenceProperties. tla (in my opinion).
ASSUME NEW S, NEW J \in Nat, NEW seq \in [ 1 .. J -> S ]
PROVE seq \in Seq(S)
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Spreadsheet Example Formulas and Usage
Related Documentation
• Spreadsheet Example Formulas and Usage
This page provides some common formulas to be used with the Spreadsheet Studies to accomplish specific tasks and explains the particular usage in those cases.
The examples on this page are formatted for entry into the Spreadsheet Study, the Spreadsheet Study for Trading, or the Spreadsheet System/Alert. The concepts in the formulas can be applied to the
Spreadsheet Formula using the Spreadsheet Study Equivalents of Simple Alerts and Main Price Graph Identifiers section. For example, if the entry in the Spreadsheet Study is =B3 - E3 (Open - Last/
Close), then this would be entered in the Spreadsheet Formula as =O - C or alternatively it could be entered as =ID0.SG1 - ID0.SG4 where ID0 is the main price graph.
Common Formulas and Usage
Determining Slope or Direction
Formula that Matches the Coloring of Up and Down Price Bars
This example formula would be entered in any available Formula Column at row 3. Although the formula is coded to rely on it being entered in the T column.
It returns a 1 if price bar is up-colored or a -1 if price bar is down-colored.
=IF(OR(E3>B3,AND(B3>E4,B3=E3)),1, IF(OR(E3<B3,AND(B3<E4,B3=E3)),-1,T4))
Formula that Matches the Slope Coloring of a Study Subgraph Line
This example formula would be entered in any available Formula Column at row 3. Although the formula is coded to rely on it being entered in the Z column.
It returns a 1 if the line is sloped up, or if the current segment is flat and the previous segment was up.
It returns a -1 if the line is sloped down, or if the current segment is flat and the previous segment was down.
ID1 needs to be changed to the particular study ID number it will actually reference. SG1 needs to be changed to the particular Subgraph number within the particular study it will reference. For more
information, refer to References to Study Subgraph Columns.
Moving Averages
Simple Moving Average
Any of the following formulas will return a Simple Moving Average of the Last price, where the Length is in cell H2. A formula would be entered into row 3 of any Formula Column.
Exponential Moving Average
This formula gives an Exponential Moving Average of the Last/Close price. The Length is defined in cell H2. The formula would be entered in cell P3, a Formula Column.
=(E3 - P4) * (2 / (1 + $H$2)) + P4
Simple Moving Average - Skip Zeros
This set of formulas gives a Simple Moving Average skipping over any zero values. The Length is defined in cell H2. The input data is in column P, and the output is in column S.
• Cell Q3: =P3 > 0
• Cell R3: =SUM(OFFSET(Q3, 0, 0, $H$2, 1))
• Cell S3: =SUM(OFFSET(P3, 0, 0, $H$2, 1)) / R3
Using Date Time
The following sections contain information on how to work with Date and Time values in the Spreadsheets.
Internally, Sierra Chart Spreadsheets store the Date and Time in the same manner as Microsoft Excel. As such, the Date is stored as the number of integer days since January 1, 1900. The hours are
then the decimal portion of the number and represent the fractional amount of the day. For example, the Date and Time for August 18, 2018 at 16:32:00 would be 43330.68889. This also means that a
value of 1 represents 1 full day.
Formula Equivalents of Serial Date Time Values
1 millisecond: =1 / 86400000
1 second: =1 / 86400
1 minute: =1 / 1440
1 hour: =1 / 24
1 day: =1
Simple Alert: Bar at Specific Time: BARTIME = FRACTIME(A3)
Simple Alert: Bar at Specific Date: BARDATE = INT(A3)
Simple Alert: Bar at Specific Date and Time: BARDATETIME = A3
Rounding Date Time Values
This section gives examples of how to round the Date and Time to achieve a desired result.
Round to the Nearest Millisecond to remove floating point imprecision to the nearest millisecond: =MROUND(A3, 1 / 86400000)
Round to the Nearest Second to remove floating point imprecision to the nearest second: =MROUND(A3, 1 / 86400)
Round Down to the Nearest Second to remove any millisecond increments: =FLOOR(A3, 1 / 86400)
Round Down to Midnight of the referenced date: =FLOOR(A3, 1)
Trading Formulas
Disable Trading From Spreadsheet Study for Trading
The following sections give examples of formulas to be used to control when trading can occur with the Spreadsheet System for Trading study.
Disable Autotrading Outside of Regular Trading Hours
The following formula disables the Spreadsheet System for Trading study from creating new open positions outside of the specified times. The Spreadsheet System for Trading can still exit from Open
Positions during all hours with this formula.
This formula should be entered into cell J28.
=AND(J8 = 0, OR(FRACTIME(J41) < TIMEVALUE("09:30:00"), FRACTIME(J41) > TIMEVALUE("16:00:00")))
In this example, it will allow the Spreadsheet System for Trading to open new Positions between the hours of 09:30:00 and 16:00:00. Change these values in the formula as needed.
Disable Trading Outside of Regular Trading Hours and for 2 Hours Midday
The following formula disables the Spreadsheet System for Trading study from creating new open positions outside the specified times along with disabling a period in the middle of the day from
11:30:00 to 13:30:00. The Spreadsheet System for Trading can still exit from Open Positions during all hours with this formula.
This formula should be entered into cell J28.
=AND(J8 = 0, OR(OR(FRACTIME(J41) < TIMEVALUE("09:30:00"), FRACTIME(J41) > TIMEVALUE("16:00:00")), AND(FRACTIME(J41) > TIMEVALUE("11:30:00"), FRACTIME(J41) < TIMEVALUE("13:30:00"))))
In this example, it will allow the Spreadsheet System for Trading to open new positions between the hours of 09:30:00 and 11:30:00 and again between 13:30:00 and 16:00:00. Change these values in the
formula as needed.
Cancel Unfilled Working Order n Minutes After Entry
This section lists a set of formulas to be entered in the Spreadsheet System for Trading that will automatically cancel an unfilled order some number of minutes after the close of the entry signal
Adjust the value in Cell H4 as needed to change the number of minutes to wait between the Close of the Entry Bar and Cancelling the Order.
• Cell Y3: =IF(AND (K5 = 0, K4), FRACTIME(A3), Y4)
This formula returns the closing timestamp of a Buy Entry signal in Column K.
• Cell Z3: =IF(AND(M5 = 0, M4), FRACTIME(A3), Z4)
This formula returns the closing timestamp of a Sell Entry signal in Column M.
• Cell H4: 3
This is the number of minutes to wait before canceling an unfilled working order.
• Cell J27: =OR(AND(J4 > 0, FRACTIME(J41) - Y3 >= H4 / 1440), AND(J4 < 0, FRACTIME(J41) - Z3 >= H4 / 1440))
This formula will cancel an unfilled working order 3 minutes after the close of the entry bar. If bars are not time-based, it can occur intrabar.
Calculate Daily Net Closed Profit/Loss After Commissions
The following formulas give ways to calculate the net Profit or Loss and how to display that information when using the Spreadsheet System for Trading.
Calculate the net Profit or Loss including Commissions, not including any Open Positions:
• Cell H2: Enter the round-trip commission cost.
• Cell H45: =J45 - J53 / 2 * H2
Alternatively, the following formulas will calculate the net Profit or Loss including Commissions, and including any Open Positions, once they are closed.
• Cell H2: Enter the round-trip commission cost.
• Cell H45: =J10 + J45 - (ABS(J8) + J53) / 2 * H2
If desired, the results of these formulas can be displayed on the Chart by using the Text Display for Study study using these settings:
• Spreadsheet System for Trading Cell R3: =IF(ROW()=3, $H$45, 0)
• Text Display for Study Study and Subgraph to Display: Spreadsheet System for Trading - R (SG8)
Persistent and Incrementing Variables
The following sections describe ways to create Persistent Variables or Incrementing Variables that can then be used for specific purposes.
Formula to Count Bars Since An Event
The following formula will count the number of bars since a Buy Entry event occurred (cell K3). This will also stop counting if a Sell Entry event occurs (cell M3). This could be used to cancel an
unfilled Order at some number of bars.
The formula below would be entered in cell P3.
=IF(M3, 0,IF(OR(K3, P4 > 0), P4 + 1, P4))
The above formula can also be reversed so that it counts bars since a Sell Entry event (M3) and stops counting if a Buy Entry event occurs (cell K3). The formula below would be entered in cell Q3.
=IF(K3, 0,IF(OR(M3, Q4 > 0), Q4 + 1, Q4))
Cancel an Unfilled Order After n Bars
The following formulas will cancel an unfilled order after some number of bars have passed. This builds on the formulas from Count Bars Since an Event above, which are repeated here.
These formulas only reset the count when the opposing event occurs (a Sell event for a Buy Position, or a Buy even for a Sell Position).
The number of bars (n) is a value stored in cell H4.
These formulas check at the close of a bar because row 4 is referenced in columns P and Q.
• Cell P3: =IF(M3, 0,IF(OR(K3, P4 > 0), P4 + 1, P4))
This counts the bars since a Buy Entry event occurred (see Formula to Count Bars Since An Event above).
• Cell Q3: =IF(K3, 0,IF(OR(M3, Q4 > 0), Q4 + 1, Q4))
This counts the bars since a Sell Entry event occurred (see Formula to Count Bars Since An Event above).
• Cell J27: =OR(AND(P4 >= H4 + 1, J4 > 0), AND(Q4 >= H4 + 1, J4 < 0))
This cancels the unfilled orders after n bars.
Cancel Unfilled Orders After n Bars - Reset Counter on New Order in Same Direction
The following formulas will cancel unfilled orders after some number of bars have passed. This builds on the formulas from Cancel an Unfilled Order After n Bars above, which are repeated here. In
this case, however, the counting of the bars will reset if a new Order is placed.
The number of bars (n) is a value entered in cell H4.
These formulas check at the close of a bar because row 4 is referenced in columns P and Q.
• Cell P3: =IF(BuyZeroingEvent, 0,IF(OR(K3, P4 > 0), P4 + 1, P4))
This counts the bars (in cell P4) since a Buy Entry event occurred (see Formula to Count Bars Since An Event above). The term BuyZeroingEvent should be replaced by an appropriate formula that
will reset the count to 0.
• Cell Q3: =IF(SellZeroingEvent, 0,IF(OR(M3, Q4 > 0), Q4 + 1, Q4))
This counts the bars (in cell Q4) since a Sell Entry event occurred (see Formula to Count Bars Since An Event above). The term SellZeroingEvent should be replaced by an appropriate formula that
will reset the count to 0.
• Cell J27: =OR(AND(P4 >= H4 + 1, J4 > 0), AND(Q4 >= H4 + 1, J4 < 0))
This cancels the unfilled orders after n bars.
Alternatively, the following formulas can be used to simply reset the count to 1 on a new order in the same direction, but does not reset otherwise.
The number of bars (n) is a value entered in cell H4.
• Cell P3: =IF(K3, 1, P4 + 1)
This counts the bars in P4 since a Buy Entry event occurred.
• Cell Q3: =IF (M3, 1, Q4 + 1)
This counts the bars in Q4 since a Sell Entry event occurred.
• Cell J27: =OR(AND(P4 >= H4 + 1, J4 > 0), AND(Q4 >= H4 + 1, J4 < 0))
This cancels the unfilled orders after n bars.
Buy on First Signal and Ignore Multiple Signals
The following set of formulas creates a Buy order per a set of Buy Conditions and then ignores additional Triggers until the corresponding Sell Order is placed.
• Cell P3: =IF(BuyConditions, 1, IF(SellConditions, 0, P4))
In the above formula BuyConditions would be replaced with a desired Buy formula, and the SellConditions would be replaced with a desired Sell formula.
• Cell K3: =AND(P4 = 0, P3)
• Cell M3: =AND(P4, P3 = 0)
Count Events Since First Occurrence
The following set of formulas will count the number of events that occur since a first event triggered the start of the count.
In this example the Spreadsheet System for Trading is used and the number of Buy Entry events is counted since a Sell Entry event occurred. Additionally, the number of Sell Entry events is counted
since a Buy Entry event occurred.
There is also a limit set in cell H4 that will prevent any Buy or Sell trades from occurring if the count is larger than this value. This can be used to limit the number of open positions
• Cell P3: =IF(M3, 0, IF(K3, P4 + 1, P4))
This counts the number of Buy Events from cell K3 and is reset if a Sell Event is detected in cell M3.
• Cell Q3: =IF(K3, 0, IF(M3, Q4 + 1, Q4))
This counts the number of Sell Events in cell K3 and is reset if a Buy Event is detected in cell K3.
• Cell K3: =AND(BuyEntryConditions, P4 < $H$4)
This defines the Buy Entry conditions, but limits the ability to enter the position if the counter in cell P4 is not less than the value in H4. In the above formula BuyEntryConditions would be
replaced with a desired Buy Entry formula.
• Cell M3: =AND(SellEntryConditions, Q4 < $H$4)
This defines the Sell Entry conditions, but limits the ability to enter the position if the counter in cell Q4 is not less than the value in H4. In the above formula SellEntryConditions would be
replaced with a desired Sell Entry formula.
• Cell H4: Enter a value to limit the number of Buy or Sell events.
Data Manipulation
The following sections give formulas and information on how to manipulate the data that is being used in the Spreadsheet Studies.
Removing Floating Point Imprecision
The following formulas can be used to remove floating point imprecision from various items in the Spreadsheet Studies.
1. Remove Floating Point Imprecision of the Tick Size in cell J21. Place this formula in cell H21:
2. Remove Floating Point Imprecision of a Calculated Value to a Multiple of the Tick Size (see above formula):
□ =MROUND(CalculatedValue,$H$21)
In this case the CalculatedValue could be any formula or another cell.
3. Remove Floating Point Imprecision of the OHLC Values to a Multiple of the Tick Size (see above formula):
□ =MROUND(B3, $H$21)
□ =MROUND(C3, $H$21)
□ =MROUND(D3, $H$21)
□ =MROUND(E3, $H$21)
Spreadsheet Study Equivalents of Simple Alerts and Main Price Graph Identifiers
The following table gives different ways to reference data, or to calculate values.
Common Name Main Price Graph Identifier Spreadsheet Equivalent Alternative
Date of the Bar BARDATE =INT(A3)
Time of the Bar BARTIME =FRACTIME(A3)
Date and Time of the Bar BARDATETIME =A3
Bar Open O =B3 or =ID0.SG1@3
Bar High H =C3 or =ID0.SG2@3
Bar Low L =D3 or =ID0.SG3@3
Bar Close/Last C =E3 or ID0.SG4@3
Bar Volume V =F3 or =ID0.SG5@3
Open Interest (Historical Only) OI G3 or =ID0.SG6@3
Number of Trades (Intraday Only) NT =G3 or =ID0.SG6@3
Average of Open, High, Low, and Close OHLC =AVERAGE(B3, C3, D3, E3)
Average of High, Low, and Close HLC =AVERAGE(C3, D3, E3)
Average of High and Low HL =AVERAGE(C3, D3)
Bid Volume BV Add the Bid Volume study and reference SG1
Ask Volume AV Add the Ask Volume study and reference SG1
Open Value for Renko ID0.SG22 Add the Renko Visual Open/Close Values study and reference SG1
Close Value for Renko ID0.SG23 Add the Renko Visual Open/Close Values study and reference SG2
Daily High DAILYHIGH =J17
Daily Low DAILYLOW =J18
Last Trade LASTTRADE
Last Trade Size LASTTRADESIZE
Bid Price BID =J12
Ask Price ASK =J14
Previous Day Close PREVIOUSDAYCLOSE Add the Daily OHLC study and reference SG4 (Set 'Reference Days Back' to 1)
Tick Size TICKSIZE =J21
Volume At Price VAP Set 'Output Volume at Price Data' to Yes on the Spreadsheet Study Inputs
Bid Volume at Price BVAP
Ask Volume at Price AVAP
Other Examples
This section contains examples of formulas to perform tasks that do not fit into the other categories.
Return Last Two Zig Zag Reversal Values
The following set of formulas can be used with the Spreadsheet Study in order to obtain the values of the three most recent Zig Zag reversals. The concept can be continued on to obtain however many
Zig Zag reversal points are desired.
This example assumes that the Zig Zag study is ID1 and that the Zig Zag Input Option for Additional Output for Spreadsheets is set to Yes.
• Cell O3: =IF(ID1.SG3@3 > 0, ID1.SG1@3, O4)
This sets cell O3 to the Price of the most recent Zig Zag reversal point.
• Cell P3: =IF(ID1.SG3@3 > 0, O4, P4)
This sets cell P3 to the Price of the second most recent Zig Zag reversal point.
• Cell Q3: =IF(ID1.SG3@3 > 0, P4, Q4)
This sets cell Q3 to the Price of the third most recent Zig Zag reversal point.
Using the FORECAST Function
The spreadsheet FORECAST function returns the Y-Value for a give X-Value based on a simple linear regression of a set of X and Y values. This can be used to predict (forecast) a value in the future
based on existing information.
A linear regression takes the existing data and does a best fit straight line to the data. This line can then be extrapolated to determine where the data would be expected to be at some point in the
This is only a prediction of the future data based on the existing data. It is not possible to know exactly what will happen as future bars are added to the chart.
The following example uses the Open data for the most recent 60 bars in order to predict the Open value of the 5th bar forward in time.
• Cell K3: =FORECAST(A3 + 5 * (A3 - A4), B3:B62, A3:A62)
The first parameter of the FORECAST() function is the X value for which the predicted value is desired. In this case, we want 5 bars in the future. Therefore, we use the difference in time between
bars and multiply that by 5. This allows for the timeframe of the chart to change without having to update the spreadsheet. For more information on how time is handled in Sierra Chart Spreadsheets,
refer to Serial Date Time Values.
The second parameter of the FORECAST() function is the range of Y values (or outcomes). These are the values for which you want to get the forecasted value. In this case, the Open data (Column B) for
the previous 60 bars is used.
The third parameter of the FORECAST() function is the range of X values. These are inputs to the equation that results in the set of Y values. The time values will often be used for this parameter,
since the charts are time-based, but it is not required. In this case, the time (Column A) for the previous 60 bars is used. It is important that the number of rows being referenced for both the X
and the Y ranges is the same. If there are not the same number of values for both of these parameters then the spreadsheet will return a #NUM! error.
*Last modified Wednesday, 22nd February, 2023. | {"url":"https://www.sierrachart.com/index.php?page=doc/SpreadsheetExampleFormulasAndUsage.php","timestamp":"2024-11-14T01:10:08Z","content_type":"text/html","content_length":"58001","record_id":"<urn:uuid:f1cb5237-5d01-4de7-9249-d796a7850319>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00530.warc.gz"} |
2021-01-23 - DataTypes: Introduction
This series is going to look at how data is stored in a computer. We're going to start with simple things like numbers and references but eventually we will work up to more complicated data
structures like arrays and queues.
My goal with the first part of the series is to get a solid grasp on how computers store simple values and touch a bit on how operations on those values are performed.
The second part will involve some more code and examples of simple versions of more complex combinations of values. These combinations of values form the basis for many of the containers that we use
as programmers so I want to cement the understanding of the basics. | {"url":"https://penguinstew.ca/posts/134","timestamp":"2024-11-02T03:13:03Z","content_type":"text/html","content_length":"5443","record_id":"<urn:uuid:21677cdd-e892-43b9-98f8-a482639fdb9b>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00016.warc.gz"} |
Daniel A. Spielman
I am a Sterling Professor of Computer Science and a Professor Statistics and Data Science and of Mathematics at Yale University.
I am the inagural James A. Attwood Director of the Institute for the Foundations of Data Science (FDS) and a member of TILOS, the NSF Institute for Learning-Enabled Optimization at Scale.
I was a co-Director of the Yale Institute for Network Science (YINS), a Simons Investigator and a MacArthur Fellow.
I am a member of the ACM, AMS, IEEE, SIAM, the National Academy of Sciences, American Academy of Arts and Sciences, and the CT Academy of Science and Engineering. | {"url":"https://www.cs.yale.edu/homes/spielman/","timestamp":"2024-11-02T02:59:37Z","content_type":"text/html","content_length":"17612","record_id":"<urn:uuid:2f8ffad3-7dd6-4d58-9d51-e915056da1ca>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00230.warc.gz"} |
Shafia S.
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Good feedback | {"url":"https://origin-www2.princetonreview.com/academic-tutoring/tutor/shafia%20s--9297300","timestamp":"2024-11-05T21:36:34Z","content_type":"application/xhtml+xml","content_length":"181154","record_id":"<urn:uuid:e5a59fe5-60e1-44b4-81cf-d68c4794e744>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00258.warc.gz"} |
2x table up to 5 - Multiplication by URBrainy.com
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AUTOSYM is an alternative way of specifying symmetry. When AUTOSYM is present, symmetry data is generated. This data is similar to that supplied by the user when SYMMETRY is used. The 'rules' AUTOSYM
uses are these:
• All bond-lengths which are within 0.0001Å are set equal.
• All bond-angles which are within 0.0057
• All dihedral angles which are within 0.0057
• All dihedral angles which are within 0.0057
These 'rules' are a re-wording of internal SYMMETRY functions 1, 2, 3, and 14. The .arc file will include the symmetry data. In order to change the .arc file symmetry data into normal symmetry data,
change AUTOSYM in the .arc file to SYMMETRY.
AUTOSYM only adds symmetry to coordinates that can be optimized. If a coordinate has an optimization flag of "0", it will not be looked at by AUTOSYM.
Consider benzene: If someone built the Z-matrix in internal coordinates, they would likely mark the C-C and C-H distances as optimizable, and lock the angles and dihedrals. If they used AUTOSYM, then
it would symmetry-relate all the C-C distances and all of the C-H distances, but leave the angles and dihedrals untouched. | {"url":"http://openmopac.net/Manual/autosym.html","timestamp":"2024-11-10T09:30:04Z","content_type":"text/html","content_length":"2982","record_id":"<urn:uuid:9fdc46c5-3fba-4657-8516-8a0fb6723a22>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00099.warc.gz"} |
On Imaginary Numbers
When we think about imaginary numbers, with what exactly are we dealing?
If mathematics begins in Euclid’s axioms as a process of finding the simplest elements of an object, then imaginary numbers represent a leap beyond the fundamental premises of normal mathematics.
Since an imaginary number like the standard z=a+bi ,where i represents the imaginary component of z, contain duality. I would propose that imaginary numbers are the mathematics of molecules where
there is no element present, that is no single substance pervades the object rather an indistinguishable duality of substance constitutes the object’s essence.
In conclusion imaginary numbers are the mathematics of molecules and not elements. | {"url":"https://adamsaulkrok.medium.com/on-imaginary-numbers-2475cb11ad6a?source=user_profile_page---------0-------------28a4248e954e---------------","timestamp":"2024-11-05T00:15:56Z","content_type":"text/html","content_length":"79662","record_id":"<urn:uuid:fc7202fb-4069-4e51-9a1b-c0da9b859afb>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00738.warc.gz"} |
ECDIVERSITY procedure • Genstat v21
Calculates measures of diversity with jackknife or bootstrap estimates (D.A. Murray).
PRINT = string Controls printed output (index, estimate); default inde
INDEX = string Controls the type of measurement to be calculated (hshannon, qstatistic, simpsonyule, bergerparker, ibrillouin, ebrillouin, dmcintosh, emcintosh, evar, logseriesalpha,
token lognormallambda, jshannon, margalef, isimpson, richness); default hsha
GROUPS = factor Defines the groups if there is more than one sample
BMETHOD = string Controls whether to use the bootstrap or jackknife method (jackknife, bootstrap); default jack for multiple samples and boot for individual samples
NBOOT = scalar Number of times to resample in bootstrap; default 100
SEED = scalar Seed for random number generator for bootstrap; default 0
CIPROBABILITY = Probability for the confidence interval produced by either jackknife or bootstrap method; default 0.95
INDIVIDUALS = variates Number of individuals per species
SPECIES = variates Number of species
SAVE = variate or pointer Saves the diversity indices
A diversity index is a measure of species diversity within a community that consists of co-occurring populations of several (two or more) different species. There are two components to diversity:
richness and evenness. Richness is the measure of the number of species within a sample where the more species in a community the higher the diversity (or greater richness). Evenness is a measure of
the relative abundance of the different species within a community. The more nearly equal the species relative abundances the higher the diversity.
ECDIVERSITY can be used to calculate several different measures of diversity. Amongst these indices are the log series α and log-Normal λ which are estimated by fitting an underlying species
abundance model, and the Q statistic which is derived from cumulative ranked frequencies. Other available indices include the Margalef and Simpsons 1/D which emphasize the richness component of
diversity. The indices that highlight the evenness component of diversity include Simpsons 1-D, McIntosh D and E, Shannon-Weiner H′ and J′, Brillouin diversity and evenness index, Berger-Parker and
Smith-Wilson evenness measure. Confidence intervals for the measures can be estimated by bootstrapping. For multiple samples, ECDIVERSITY calculates the overall values of the diversity indices, and
provides an option to perform jackknifing to produce less bias estimates with a confidence interval.
The numbers of individuals per species are specified using the INDIVIDUALS parameter. The SPECIES parameter specifies a variate containing the number of species for the associated number of
individuals denoted in the corresponding element of INIDIVIDUALS. SPECIES can be omitted if each of the values in INDIVIDUALS corresponds to one species. The GROUPS option can be used to calculate
measures of diversity for different samples. The SAVE parameter allows the diversity indices to be saved in a variate or in a pointer to a set of variates for each group.
The PRINT option controls printed output, with settings:
index the index of diversity or evenness,
estimate bootstrap or jackknife estimate with confidence limits for the statistic.
The BMETHOD option can be used to select either the bootstrap or jackknife (for multiple samples) method to produce an estimate of the diversity measure with an associated confidence interval. To
produce a bootstrap or jackknife estimate for multiple samples each sample must contain the same number of values where each element corresponds to the same species within each sample. For the
calculation of the bootstrap confidence intervals of the diversity measures, the NBOOT option specifies how many bootstrap samples to take (default 100). The probability level for the confidence
interval can be set by the CIPROBABILITY option; by default 0.95. The SEED option specifies the seed to use in the random number generator used to construct the bootstrap samples. The default value
of zero continues an existing sequence of random numbers or, if the generator has not yet been used in this run of Genstat, it initializes the generator automatically.
Options: PRINT, INDEX, GROUPS, BMETHOD, NBOOT, SEED, CIPROBABILITY.
Parameters: INDIVIDUALS, SPECIES, SAVE.
The log series α index is estimated by fitting a log series model using the ECFIT procedure. The log-Normal λ is the ratio of the S^* and σ parameters estimated by fitting a Poisson-log-Normal
distribution using the ECFIT procedure.
The Q statistic is calculated by:
Q = ( 0.5 × n[R][1] + ∑[r = R1+1 … R2-1] { n[r] } + 0.5 × n[R][2] ) / log( R2 / R1),
where n[r] is the total number of species with abundance r, R1 and R2 are the 25% and 75% quartiles, n[R][1] is the number of species where R1 lies, and n[R][2] is the number of species where R2
The Shannon-Weiner index is evaluated by:
H′ = – ∑[i] (n[i] / N) × log(n[i] / N)
where n[i] are the individuals, N is total number of individuals.
The Shannon-Weiner evenness (Pielou J) is given by
J′ = H′ / log(S)
where H′ is the Shannon index and S is the total number of species.
The Brillouin index is given by
HB = ( log(N!) – ∑[i] {log(n[i]!)} ) / N
where n[i] is the individual in species i and N is total number of individuals.
The Brillouin evenness index is then calculated by
E = HB / HBmax
HBmax = 1 / N × log( N! / ( (N/S)!^S–r × ((N/S)+1)!^r )
where N/S is the integer of N/S and r = N–S(N/S)
Simpsons index D is calculated by
D = ∑[i] {n[i] × (n[i] – 1)} / (N × (N – 1))
and is expressed in the output as both 1-D and 1/D
The Margalef index is:
Dmn = (S – 1) / log(N)
where S is total number of species and N is total number of individuals.
McIntosh’s measure of diversity is expressed as
D = (N – √( ∑[i] {n[i]^2} / (N – √(N))
and the evenness measure is given by
E = (N – √( ∑[i] {n[i]^2} ) / (N – N / √(S))
where n[i] is the individual in species i and N is total number of individuals.
The Berger-Parker index is
d = Nmax / N
where Nmax is the number of individuals in the most abundant species.
The Evar (Smith and Wilson 1996) evenness index is evaluated by
Evar = 1 – 2 / π × arctan( ∑[i] { log(n[i]) – ∑[j] { log(n[j]) } }^2 / S )
where n[i] and n[j] are the number of individuals in species i and j respectively, and S is the total number of species
Species richness is the total number of species.
The jackknife estimate and standard error are generated by the JACKKNIFE procedure where the estimates are calculated from all samples, and then for the situations where one sample is omitted in
turn. The confidence interval is calculated by:
φ +/- t(n-1) × se(φ)
where n is the number of samples.
The bootstrap confidence intervals are generated using the BOOTSTRAP procedure where all individuals are sampled with replacement and the diversity measures are calculated from these samples.
If a parameter is restricted the statistics will be calculated using only those units included in the restriction.
Magurran, A.E. (2003). Measuring Biological Diversity. Blackwell, Oxford.
Smith, B, & Wilson, J.B. (1996). A consumer’s guide to evenness indices. Oikos, 76, 70-82.
See also
Commands for Ecological data.
CAPTION 'ECDIVERSITY example'; STYLE=meta
FACTOR [NVALUES=69; LEVELS=3; VALUES=23(1...3);\
LABELS=!t('Derrycunnihy oakwood','Muckross yew wood',\
'Sitka spruce plot')] Location
VARIATE [VALUES=35,26,25,21,16,11,6,5,3,3,3,3,3,2,2,2,1,1,1,1,0,0,0,\
9,20,10,21, 5,14,0,3,2,6,9,2,0,0,0,6,0,0,0,1,1,1,0,\
14,10, 0,30, 4, 6,0,0,7,3,0,0,0,0,0,0,0,0,0,0,0,0,1]\
ECDIVERSITY [INDEX=hshannon,simpson,berger; GROUPS=Location] Territories | {"url":"https://genstat21.kb.vsni.co.uk/knowledge-base/ecdivers/","timestamp":"2024-11-02T12:47:08Z","content_type":"text/html","content_length":"47356","record_id":"<urn:uuid:795145d4-ff94-4440-a166-d5bab4047b18>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00530.warc.gz"} |
Two-Dimensional Metal in a Parallel Magnetic Field
We have investigated the effect of an in-plane parallel magnetic field [Formula presented] on two high mobility metalliclike dilute two-dimensional hole gas systems in GaAs quantum wells. The
experiments reveal that, while suppressing the magnitude of the low temperature resistance drop, [Formula presented] does not affect [Formula presented], the characteristic energy scale of the
metallic resistance drop. The field [Formula presented] at which the metalliclike resistance drop vanishes is dependent on both the width of the quantum well and the orientation of [Formula
presented]. It is unexpected that [Formula presented] is unaffected by [Formula presented] up to [Formula presented] despite the fact that the Zeeman energy at [Formula presented] is roughly equal to
[Formula presented].
All Science Journal Classification (ASJC) codes
• General Physics and Astronomy
Dive into the research topics of 'Two-Dimensional Metal in a Parallel Magnetic Field'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/two-dimensional-metal-in-a-parallel-magnetic-field-2","timestamp":"2024-11-04T20:58:30Z","content_type":"text/html","content_length":"48824","record_id":"<urn:uuid:e0709e61-1601-4748-a8dc-e5619e39fd98>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00890.warc.gz"} |
Hyperbolic surface code
An extension of the Kitaev surface code construction to hyperbolic manifolds. Given a cellulation of a manifold, qubits are put on \(i\)-dimensional faces, \(X\)-type stabilizers are associated with
\((i-1)\)-faces, while \(Z\)-type stabilizers are associated with \(i+1\)-faces.
Constructions (see code children below) have yielded distances scaling favorably with the number of qubits. The use of hyperbolic surfaces allows one to circumvent bounds on code parameters (such as
BPT bound
) that are valid for lattice geometries.
\((1,D-1)\) surface codes on hyperbolic geometries admit a fault-tolerant implementation of \(C^D Z\) gates [1].
• Holographic tensor-network code — Both holographic tensor-network and hyperbolic surface codes utilize tesselations of hyperbolic surfaces. Encodings for the former are hyperbolically tiled
tensor networks, while the latter is defined on hyperbolically tiled physical-qubit lattices.
• Single-shot code — A 4D hyperbolic surface code can be decoded with the Hastings decoder [2] in time \(O(n\log n)\) and with a logical error scaling inverse polynomially with \(n\).
Y.-F. Wang et al., “Efficient fault-tolerant implementations of non-Clifford gates with reconfigurable atom arrays”, (2024) arXiv:2312.09111
M. B. Hastings, “Decoding in Hyperbolic Spaces: LDPC Codes With Linear Rate and Efficient Error Correction”, (2013) arXiv:1312.2546
Page edit log
Cite as:
“Hyperbolic surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/hyperbolic_surface
Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/topological/surface/hyperbolic/hyperbolic_surface.yml. | {"url":"https://errorcorrectionzoo.org/c/hyperbolic_surface","timestamp":"2024-11-04T10:52:26Z","content_type":"text/html","content_length":"16834","record_id":"<urn:uuid:3d166836-e816-4917-bbf6-c544bed84dac>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00284.warc.gz"} |
American Mathematical Society
Initial Publication Date: January 13, 2014
American Mathematical Society
30,000 members
Education staff lead: AMS Education Committee, comed@ams.org
The mission of AMS is to further the interests of mathematical research and scholarship, serves the national and international community through its publications, meetings, advocacy and other
programs, which
• promote mathematical research, its communication and uses,
• encourage and promote the transmission of mathematical understanding and skills,
• support mathematical education at all levels,
• advance the status of the profession of mathematics, encouraging and facilitating full participation of all individuals,
• foster an awareness and appreciation of mathematics and its connections to other disciplines and everyday life.
The AMS supports programs for both the pre-college and college levels that nurture mathematical talent in the United States:
• AMS has promoted introducing students to research-level mathematics (so called REU programs) by organizing conferences for directors of programs that involve undergraduates in mathematics
research of all types. AMS also keeps track of REU sites and publicizes them online.
• The AMS Epsilon Fund grants provide funding support and recognition for programs that focus on pre-college students and which encourage mathematically talented students to think broadly and
creatively about mathematics and its applications.
• Mathematics Programs That Make a Difference: Since 2006, the AMS has highlighted programs that aim to bring more persons from underrepresented backgrounds into the pipeline beginning at the
undergraduate level and leading to an advanced degree in mathematics, or retain them in the pipeline. This is done through an award.
• AMS website also showcases efforts made by mathematicians in U.S. colleges and universities to improve college-level mathematics education. This is done through the awards to exemplary programs.
The idea is to promote good methods in existing programs and to encourage further innovation, promoting new teaching methods that might scale up to help address the national imperative of
training more well-prepared STEM graduates. Another good example is how AMS collects and organizes information for undergraduate students interested in research.
• The AMS runs several programs specifically for high school students: The Arnold Ross Lecture series for talented high school mathematics students aims to stimulate their interest in mathematics
beyond the traditional classroom and to show them the tremendous opportunities for careers in mathematics. The lectures also intended to illustrate some recent development in mathematical
• Another popular activity is the game "Who wants to be amathematician." Through this national competition AMS hopes to enhance the image of mathematics among students.
• The annual Survey data on the state of Mathematical Sciences is published four times a year in the Notices of the AMS. The survey gives information each year from departments in the mathematical
sciences at four-year colleges and universities in the United States. This provides invaluable information to the mathematical sciences community. The data collected include:
□ New Ph.D. recipients (dissertations, gender, race/ethnicity and citizenship, and employment plans)
□ Faculty (size, recruitment, hiring, and salaries)
□ Graduate students (enrollment status, gender, and citizenship)
□ Degrees awarded (Ph.D., master's, and bachelor's)
□ Course enrollments (graduate and undergraduate)
• Last but not least, AMS is heavily involved in organizing meetings and in public policy that could affect education. AMS is also involved in public awareness campaigns.
Premier Contribution to Faculty Development
AMS organizes Mathematical Research Communities (MRCs) they intend to nurture early-career mathematicians (those who are close to finishing their doctorates or have recently finished) and provides
them with opportunities to build social and collaborative networks to inspire and sustain each other in their work.
The structured program is designed to engage and guide all participants as they start their careers. For each topic, the program includes a one-week summer conference, a Special Session at the
national meeting, a discussion network, ongoing mentoring, and a longitudinal study of early career mathematicians. A wide range of topics have been covered so far.
Additional Undergraduate Education Activities of AMS
• AMS has a number of efforts, it is hard to single one out as the key, but they can be found at:
• The AMS Notices is the flagship newsletter of the society. Over the past few years it has made a effort to discuss educational issues more frequently and now it runs a regular column DOCEAMUS on
• See also
□ Several publications on Mathematics Education have appeared as AMS books. | {"url":"https://serc.carleton.edu/issues/profiles/77068.html","timestamp":"2024-11-01T23:37:50Z","content_type":"text/html","content_length":"47256","record_id":"<urn:uuid:f7a738b2-c4f7-42d4-b97a-cb22eaa43f62>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00013.warc.gz"} |
AP SA2 Maths Question Paper2024 For 9th, 8th, 7th, 6th Class (Pdf)
AP SA2 Maths Question Paper For 9th, 8th, 7th, 6th Class 2024 | AP Summative 2 Annual Exams Maths Question Papers Model Papers
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TS సమ్మేటివ్ 2 వార్షిక పరీక్షల 2024 PDF కోసం AP SA2 మ్యాథ్స్ ప్రశ్నాపత్రం 2024 9వ, 8వ, 7వ మరియు 6వ తరగతి గణితం మోడల్ పేపర్ల కోసం. ఆంధ్రప్రదేశ్ సమ్మేటివ్ 2 గణితం తెలుగు మరియు ఆంగ్ల మాధ్యమంలో మునుపటి ప్రశ్నపత్రం, 6వ,7వ,8వ,9వ తరగతి AP SA2 పరీక్షలకు మోడల్ బిట్ పేపర్. AP/
TS SA2 గణితం 2024లో పేపర్లు 1, 2 మరియు SA2 కోసం 6వ, 7వ, 8వ, మరియు 9వ తరగతి వార్షిక పరీక్షలకు సంబంధించిన EM & TM మునుపటి ప్రశ్న పత్రాలు. AP సమ్మేటివ్ అసెస్మెంట్ 2, TS SA2 మ్యాథమెటిక్స్ పేపర్గా ఇప్పుడు అందుబాటులో ఉంది. డౌన్లోడ్ చేయండి. AP సమ్మేటివ్ మరియు TS SA2 గణిత పరీక్షలు
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SA 2 All Subjects Question Paper For 9th, 8th, 7th, 6th Class Download
SA2 Maths Previous Question Papers Details
Title AP SA 2 Maths Model Question Papers
Name of the State Andra Pradesh (AP) & Telangana (TS)
Name of the Education Board State Council of Educational Research and Training (SCERT) Andhra Pradesh, TS
Exam Classes 6th, 7th, 8th, 9th classes
Name of the Subject Maths SA2
Examination Date Conduct Form April 2024
Official Website https://apscert.gov.in/
Summative 2 Maths Model Question Papers for 6th, 7th, 8th, 9th Classes
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SA2 Maths 6th, 7th, 8th,9th Classes Previous Question Papers
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9 SA 2 Maths Question paper 2- Telugu Medium
1.What are AP SA2 Model papers?
AP SA2 Model papers are sample question papers designed by the Andhra Pradesh Board of Secondary Education (BSEAP) for students to practice and prepare for the Summative Assessment 2 (SA2) exams.
These model papers are based on the latest syllabus and exam pattern and can help students in understanding the type of questions asked in the actual exam.
2.Where can I get AP SA2 Model Papers?
You can get AP SA2 Model papers from various sources such as the official website of BSEAP, online educational portals, coaching centers, and bookstores.
3.How Can AP SA2 Model Papers Help Me?
• AP SA2 Model papers can help you in the following ways:
• They can give you an idea about the exam pattern and marking scheme.
• They can help you in identifying your strengths and weaknesses.
• They can improve your time management skills.
• They can boost your confidence and reduce exam stress.
4.Are AP SA2 Model papers similar to the actual exam papers?
AP SA2 Model papers are designed based on the latest syllabus and exam pattern prescribed by the BSEAP. However, the actual exam papers may have some variations in terms of difficulty level, question
types, and marks allocation.
5.How should I use AP SA2 Model papers for preparation?
You can use AP SA2 Model papers for preparation in the following ways:
• Solve the model papers in a time-bound manner to improve your speed and accuracy.
• Analyze your performance and identify areas that need improvement.
• Take notes of important topics and revise them regularly.
• Seek help from your teachers or mentors to clarify your doubts.
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There are numerous games in many variants using different kinds of dominos. There are also many different dominos sets available, suited for several usages.
The most popular and used set is still the double-six set, having 28 pieces in total, containing:
• A unique piece representing every possible couple of two numbers between 0 and 6 respectively: 0/1, 0/2, 0/3, ..., 0/6, 1/2, 1/3, ... 1/6, 2/3, ... 5/6
• Seven « doubles » for which the two shown numbers are identical: 0/0, 1/1, 2/2, ... 6/6
One can also rather easily find double-nine and double-twelve sets having the same distribution but for numbers between 0 and respectively 9 or 12. These larger sets can accomodate for a greater
number of players, because the classic double-six set can only up to 4 players. The total number of pieces for a defined set of dominos is given by the formula (n+1)!/(n -1)!/2 +(n+1), where n is the
greatest number found in the set. For 6, it gives 7!/5!/2 +7 = 5040/120/2 +7 = 21+7 = 28.
Dominos game rules are numerous and have been invented in many parts of the world across centuries. The following rule, which is the only one applied on the playroom, is only the very basic rule,
smallest common denominator between all existing rules. But many players created their own variants, putting conditions on placing options, changing layouts, or giving special features to particular
Game rules
At the beginning of the game, the dealer deals between 5 and 7 pieces to each player, depending on their number and on the set used. Remaining pieces form the draw pile, which we more commonly call
boneyard when talking about dominos, because they are normally inside a black and thick bag that prevents cheating. When playing with 4 players and a double-six set, everybody gets 7 pieces and there
is no boneyard.
After the distribution is finished, each player plays one after the other in turn, starting with the one having the largest double. In case it's in the boneyard, one can start with doubles smaller in
value. The goal of the game is to be the first one to get rid of all your pieces, so that you get a minimum of points.
The first player must put a double at the center of the table. Then, at your turn, you will have to put, next to already placed dominos chain, a piece on which one of the two shown numbers
corresponds to the one on one of the two ends of the chain. For example if we have the chain 1/2 2/5 5/4 on the table, you can play a domino 4/6 to the right, so that the chain becomes 1/2 2/5 5/4 4/
6, or you can also play a domino 1/3 to the left by turning it, so that the new chain would become 3/1 1/2 2/5 5/4. If you don't have any playable piece, you must pick one in the boneyard. If you
just took a playable domino, you can play it immediately. Otherwise you must keep it in your hand, and it's the next player's turn. If the boneyard is empty, you just skip your turn.
Sometimes, it's possible that a particular piece can be placed at both ends of the chain. In this case, you can remove the ambiguity by pressing D for right or G for left to indicate where you want
to play .
The round finishes when someone has no more dominos in his hand, or when the game is blocked so that nobody can play anymore. One considers the game to be blocked when a complete turn around the
table passed with nobody being able to play and when the boneyard is empty.
The player who ends the round doesn't get any point. Other players count the dominos they still have in their hand. Each piece is worth the sum of the two numbers shown on it, except for the double 0
valued 10 points in case it's the only one you have. If the game ends blocked, all players count their remaining dominos.
The objective of the game is therefore to get as few points as possible. On the playroom, one defines a score limit and all players reaching it are out of the game. The winner is the last player
staying in-game. This limit is proposed to 100 points by default.
Team play
It's possible to play dominos in teams. In this case, players are positioned so that the turn of each team is correctly alternated. The playroom allows to play dominos individually or in teams up to
10 players, forming 2, 3, 4 or 5 teams of 2, 3, 4 or 5 players each.
The player who wins a round makes his team not collect any points, while players in opposing teams sum up their dominos. Dominos that remain in the hands of the winning team are counted as penalty
added to all other teams' scores. This is not standard, but it allows a little more strategy. Just like in individual games, all the teams get points when the round ends blocked.
Keyboard shortcuts summary
• Space: draw a piece from the boneyard
• C: view the pieces at the ends of the chain
• V: view the entire domino chain
• D and G: indicate on which side to play when there is an ambiguity
• S: announce scores
• E: announce game information
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Calculating the Number of Blocks Required for a 5 Room House in South Africa - November 2024
How Many Blocks Do You Need for a 5 Room House in South Africa
When planning to build a house in South Africa, one of the key considerations is determining how many blocks are needed for the construction. The number of blocks required will depend on various
factors such as the size of the house, the number of rooms, and the overall design.
For a 5 room house, the number of blocks needed can vary significantly. The size of each room, the height of the walls, and the thickness of the blocks all play a role in determining the final count.
Additionally, factors such as windows, doors, and other architectural elements need to be taken into account.
One rough estimate suggests that for a 5 room house in South Africa, you would need approximately 5,000 to 7,000 blocks. However, it is important to note that this is just an estimate and the actual
number may vary based on the specifics of your project.
It is always recommended to consult with a professional builder or architect who can accurately assess your specific needs and provide you with a more precise estimate. By considering factors such as
the size of each room, the desired height of the walls, and any additional architectural elements, they will be able to determine the exact number of blocks required for your 5 room house in South
Estimating the Number of Blocks for a 5 Room House in South Africa
When building a house in South Africa, it is important to estimate the number of blocks needed for construction. The number of blocks required can vary depending on the size and layout of the house.
In the case of a 5 room house, there are several factors to consider when estimating the number of blocks needed.
The first step in estimating the number of blocks is to determine the size of the rooms. The average size of a room in South Africa is around 3 meters by 4 meters. Therefore, a 5 room house would
have a total area of approximately 60 square meters (3m x 4m x 5 rooms).
Next, it is important to consider the height of the walls. The standard height for walls in South Africa is 2.4 meters. Therefore, the total volume of the walls can be calculated by multiplying the
area by the height (60 square meters x 2.4 meters = 144 cubic meters).
Once the volume of the walls is known, it is then possible to estimate the number of blocks needed. The average size of a block in South Africa is 390mm x 190mm x 190mm. To calculate the number of
blocks needed, divide the volume of the walls by the volume of a single block (144 cubic meters / 0.007 cubic meters = 20,571 blocks).
However, it is important to consider that additional blocks may be needed for factors such as door and window openings, as well as corners and edges. It is recommended to add an additional 10-15% to
the estimated number of blocks to account for these factors.
In conclusion, estimating the number of blocks for a 5 room house in South Africa involves calculating the area and volume of the walls, and then dividing by the volume of a single block. It is also
important to consider additional blocks for openings and corners. By following these steps, it is possible to get a rough estimate of the number of blocks needed for construction.
Factors to Consider
When determining how many blocks you need for a 5-room house in South Africa, there are several factors that you should consider:
• Size of the rooms: The size of each room will determine how many blocks are needed to construct the walls. Larger rooms will require more blocks, while smaller rooms will require fewer blocks.
• Type of construction: The type of construction method used will also affect the number of blocks needed. For example, if you are building with double walls or using different materials for
certain areas, the block requirements may vary.
• Number of floors: If you are building a multi-story house, you will need to consider the number of blocks needed for each floor. The higher the number of floors, the more blocks will be required.
• Design complexity: The complexity of the house design can also impact the number of blocks needed. If the design includes intricate details or curved walls, additional blocks may be required.
• Building regulations: It’s important to consider any building regulations or codes that may affect the number of blocks needed. Certain regulations may require thicker walls or additional
reinforcement, which can increase the block requirements.
• Quality of blocks: The quality and strength of the blocks you choose will also impact the number needed. Higher quality blocks may require fewer units to achieve the desired structural integrity.
By taking these factors into account, you can estimate the number of blocks required for your 5-room house in South Africa more accurately.
House Size
When building a house in South Africa, it is important to consider the size of the house you want to construct. The number of blocks required will depend on the size of the house.
A 5 room house typically consists of a living room, kitchen, bathroom, and 3 bedrooms. The size of each room can vary, but on average, a room in a South African house is around 15 square meters.
To calculate the total size of the house, you can multiply the size of each room by the number of rooms. In this case, 15 square meters multiplied by 5 rooms would give you a total size of 75 square
Now that you know the size of the house, you can determine how many blocks you will need. The number of blocks required will depend on the size of the blocks you are using, as well as the thickness
of the walls. On average, a block with dimensions of 390mm x 190mm x 90mm can cover an area of approximately 0.07 square meters.
To calculate the number of blocks needed, divide the total size of the house by the area covered by each block. In this case, dividing 75 square meters by 0.07 square meters gives you approximately
1071 blocks.
Keep in mind that this is just an estimate, and the actual number of blocks required may vary depending on other factors such as the design of the house and the presence of additional features such
as a garage or porch.
It is always recommended to consult with a professional builder or architect to get an accurate estimate of the number of blocks needed for your specific house design.
Block Size
When building a house in South Africa, the number of blocks needed depends on the size of the house and the number of rooms. For a 5-room house, it is important to consider the size of the blocks
that will be used.
The size of the blocks can vary, but the most common size used in South Africa is 390mm x 190mm x 140mm. These blocks are commonly referred to as standard blocks.
The number of blocks needed for a 5-room house will depend on the size of the rooms and the layout of the house. It is recommended to consult with a professional builder or architect to determine the
exact number of blocks needed.
However, as a rough estimate, you can calculate the number of blocks needed by multiplying the length, width, and height of the walls that will be built with the blocks. It is important to account
for doors and windows in the calculations as well.
For example, if the average room size is 4 meters by 4 meters and the walls are 3 meters high, the calculation would be as follows:
1. Calculate the total length of all the walls: (4m + 4m + 4m + 4m) = 16 meters.
2. Calculate the total height of the walls: (3m x 5 rooms) = 15 meters.
3. Calculate the total number of blocks needed: (16m x 15m) = 240 blocks.
This is just a basic calculation and the actual number of blocks needed may vary depending on factors such as the thickness of the walls and the design of the house.
It is always recommended to consult with a professional to ensure accuracy and to account for any additional requirements specific to your construction project.
Construction Method
When building a 5-room house in South Africa, the construction method will depend on various factors such as the desired design, available budget, and local building regulations. However, in general,
the construction process can be broken down into several key steps:
1. Planning: Before starting the construction, it is important to carefully plan the layout and design of the house. This includes deciding on the number of rooms, their sizes, and the overall floor
2. Foundation: The construction begins with the excavation of the foundation. This is done by digging trenches and pouring concrete footings to support the weight of the house. The type of
foundation used will depend on the soil conditions and structural requirements.
3. Wall Construction: Once the foundation is in place, the walls of the house can be constructed. The number of blocks needed for the walls will depend on the size of the rooms and the desired
thickness of the walls. Generally, a 5-room house would require a significant number of blocks to build the walls.
4. Roofing: After the walls are erected, the next step is to install the roof. This can be done using various materials such as corrugated iron sheets, tiles, or concrete slabs. The choice of
roofing material will depend on factors such as cost, climate, and personal preference.
5. Finishing: Once the main structure of the house is completed, the finishing touches can be added. This includes installing doors, windows, flooring, plumbing, and electrical systems. The interior
and exterior walls can also be painted or finished with other materials.
6. Inspection and Approval: Before the house can be occupied, it is important to have it inspected by local authorities to ensure that it meets all building codes and regulations. This may involve
inspections of the foundation, walls, roof, and other structural elements.
It is important to note that the construction method may vary depending on the specific requirements and circumstances of each project. Consulting with an experienced contractor or architect is
recommended to ensure that the construction process is carried out efficiently and in compliance with local regulations.
Calculating the Number of Blocks
In order to determine how many blocks you will need to build a 5 room house in South Africa, you need to consider several factors. The size of the rooms, the thickness of the walls, and the overall
design of the house will all impact the number of blocks required.
First, you need to calculate the total area of the house. Measure the length and width of each room and multiply them together to get the area of each room. Add up the areas of all 5 rooms to get the
total area of the house.
Next, you need to determine the thickness of the walls. This will depend on the type of construction and the building regulations in South Africa. Common wall thicknesses range from 100mm to 200mm.
Multiply the total area of the house by the wall thickness to get the volume of the walls.
Once you have the volume of the walls, you can calculate the number of blocks needed. The volume of a block can vary depending on its dimensions, so you will need to consult with the manufacturer or
supplier to get the specific volume. Divide the volume of the walls by the volume of a block to get the number of blocks needed.
Keep in mind that this calculation only gives you an estimate. It’s always a good idea to add some extra blocks to account for waste, breakages, or design changes. Additionally, consider other
materials needed for the construction, such as mortar, cement, and reinforcement.
By accurately calculating the number of blocks required, you can ensure that you have enough materials for the construction of your 5 room house in South Africa.
Determine the Number of Blocks per Square Meter
When building a house in South Africa, it is important to calculate the number of blocks needed per square meter. This will help you determine the overall quantity of blocks required for your 5-room
To calculate the number of blocks, you need to consider the size of the blocks and the area they will cover. The standard size of blocks in South Africa is 390mm x 190mm x 90mm.
First, you need to calculate the area of each block:
Area of block = Length x Width
Area of block = 390mm x 190mm = 74100 mm^2
Next, you need to determine the area of the walls that need to be covered by blocks. This can be calculated by multiplying the height of the walls by the perimeter of the house:
Area of walls = Height x Perimeter
For a 5-room house, the number of walls will depend on the layout and design. Assuming the house has 4 exterior walls and 1 interior wall, you can calculate the perimeter of the house by summing the
lengths of all the walls:
• Exterior wall 1: Length1
• Exterior wall 2: Length2
• Exterior wall 3: Length3
• Exterior wall 4: Length4
• Interior wall: Length5
To calculate the number of blocks required, you can divide the area of the walls by the area of each block:
Number of blocks = Area of walls / Area of block
Once you have determined the number of blocks required, it is always a good idea to add an additional percentage for wastage and breakages.
By following these calculations, you can accurately determine the number of blocks needed per square meter for your 5-room house in South Africa.
FAQ about topic Calculating the Number of Blocks Required for a 5 Room House in South Africa
What is the average size of a room in a 5 room house in South Africa?
The average size of a room in a 5 room house in South Africa is approximately 12 square meters.
How many blocks are needed to build a 5 room house in South Africa?
To build a 5 room house in South Africa, you would typically need around 5,000 blocks.
What is the cost of building a 5 room house in South Africa?
The cost of building a 5 room house in South Africa can vary depending on factors such as location and materials used, but it can range from $50,000 to $100,000.
Video:How Many Blocks Do You Need for a 5 Room House in South Africa
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5 THINGS TO KNOW BEFORE BUYING A HOUSE IN SOUTH AFRICA 2021
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Find the value of , if the mean of the following distribution is
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Question Text Find the value of , if the mean of the following distribution is
Updated On Jun 14, 2023
Topic Statistics
Subject Mathematics
Class Class 9
Answer Type Text solution:1 Video solution: 1
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Calculating Electric Potential - AP Physics C Electricity
All AP Physics C Electricity Resources
Example Questions
Example Question #1 : Calculating Electric Potential
A proton moves in a straight line for a distance of
The charge of a proton is
Correct answer:
Potential difference is given by the change in voltage
Work done by an electric field is equal to the product of the electric force and the distance travelled. Electric force is equal to the product of the charge and the electric field strength.
The charges cancel, and we are able to solve for the potential difference.
Example Question #2 : Calculating Electric Potential
For a ring of charge with radius
Find the expression for electric field produced by the ring.
Correct answer:
We know that
Using the given formula, we can find the electric potential expression for the ring.
Take the derivative and simplify.
Example Question #3 : Calculating Electric Potential
The potential outside of a charged conducting cylinder with radius
What is the electric field at a point located at a distance
Correct answer:
The radial electric field outside the cylinder can be found using the equation
Using the formula given in the question, we can expand this equation.
Now, we can take the derivative and simplify.
Example Question #1 : Calculating Electric Potential
A proton moves in a straight line for a distance of
The charge of a proton is
Correct answer:
Work done by an electric field is given by the product of the charge of the particle, the electric field strength, and the distance travelled.
We are given the charge (
Example Question #5 : Calculating Electric Potential
A negative charge of magnitude
Correct answer:
Relevant equations:
First, find the potential difference between the initial and final positions:
2. Plug this potential difference into the work equation to solve for W:
Example Question #1 : Calculating Electric Potential
Three point charges are arranged around the origin, as shown.
Calculate the total electric potential at the origin due to the three point charges.
Correct answer:
Electric potential is a scalar quantity given by the equation:
To find the total potential at the origin due to the three charges, add the potentials of each charge.
Example Question #21 : Electricity
Three identical point charges with
Correct answer:
The electric potential from point charges is
Knowing that all three charges are identical, and knowing that the center point at which we are calculating the electric potential is equal distance from the charges, we can multiply the electric
potential equation by three.
Plug in the given values and solve for
Example Question #21 : Electricity
Eight point charges of equal magnitude
Correct answer:
By the Pythagorean theorem, each charge is a distance
from the center of the cube, so the potential is
Example Question #1 : Calculating Electric Potential
An infinite plane has a nonuniform charge density given by
You may wish to use the integral:
Correct answer:
Use polar coordinates with the given surface charge density,
Remark: This is exactly the charge distribution that would be induced on an infinite sheet of (grounded) metal if a negative charge
Example Question #2 : Calculating Electric Potential
A nonuniformly charged hemispherical shell of radius
Correct answer:
Use spherical coordinates with the given surface charge density
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Problem E
Is It Even?
You are a leading programmer at $\text {Company}^\text {TM}$, and have been given the following task. Given a list of $N$ integers $x_1,\ldots , x_ N$, is their product $x_1\cdot x_2 \cdot \ldots \
cdot x_ N$ even? You plug away at the problem using every trick in the book, and come up with a very elegant solution. Lo and behold, your supervisor then informs you that the task has changed! It
turns out the higher ups want you to find out if the product $x_1 \cdot x_2 \cdot \ldots \cdot x_ N$ is divisible by $2^ K$ for some integer $K \geq 0$.
You just can’t catch a break!
The input consists of two integers $N, K$ $(1 \leq N \leq 100\, 000$ and $0 \leq K \leq 1000$). These are followed by $N$ lines, each with a single value $x_1,\ldots , x_ N$ respectively ($1 \leq x_
i \leq 10^9$ for each $1 \leq i \leq N$) which form the product.
Display 1 if $2^ K$ divides $x_1\cdot x_2 \cdot \ldots \cdot x_ N$, otherwise display 0.
Sample Input 1 Sample Output 1
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Eva Draws A Line That Includes The Points
Eva Draws A Line That Includes The Points - Which function gives all the points (x,y) on this line? Web 4,539 solutions more related questions algebra write an equation of the line that passes
through the given point and is parallel to the given line. Web y + 2 = 2(x − 1) y + 2 = 2 ( x − 1) 02:00. Maria finds the slope between the points (3,2) and (9,6). Which function gives all the points
(x,y) on this line?
Which function gives all the points ( x , y ) (x,y) ( x , y ) on this line? Web eva finds the slope between the points (0,0) and (3,2). Web y + 2 = 2(x − 1) y + 2 = 2 ( x − 1) 02:00. Which function
gives all the points (x,y) on this line? Y= 2 2−x is the require function. Use the power of algebra to understand and interpret points and lines (something we typically do. We are given the following
in the question:
Question 1 Draw any line segment AB. Mark any point M on it
Y= 2 2−x is the require function. Web 4,539 solutions more related questions algebra write an equation of the line that passes through the given point and is parallel to the given line. Which
HOW TO DRAW A EVA WALLE CUTE, Easy step by step drawing lessons for
X2 + 4y2 = 4 x 2 + 4 y 2 = 4. Determine which of the given points are on the graph of the equation. Explain how you can use mental math to find.
TO DRAW A PERPENDICULAR ON A GIVEN LINE FROM A POINT ON IT & FROM A
Web using the slope formula, find the slope of the line through the points (0,0) and(3,6). X2 + 4y2 = 4 x 2 + 4 y 2 = 4. Explain how you can use mental.
Eva draws a line that includes the points (2,0) and (2,2). Which
Which function gives all the points (x,y) on this line? Web y + 2 = 2(x − 1) y + 2 = 2 ( x − 1) 02:00. Which function gives all the points (x,y).
Lines, Line Segments & Rays Anchor Chart * Jungle Academy Teaching
Determine which of the given points are on the graph of the equation. Y= 2 2−x is the require function. Web y + 2 = 2(x − 1) y + 2 = 2 ( x.
eva draws a line that includes the points (2,0) and (2,2). Which
Web y + 2 = 2(x − 1) y + 2 = 2 ( x − 1) 02:00. Write and solve equations of lines using slope and a point on the line. Level up on.
Eva draws a line that includes the points (2,0) and (2,2). Which
Web eva finds the slope between the points (0,0) and (3,2). Write and solve equations of lines using slope and a point on the line. Web 4,539 solutions more related questions algebra write an
How to Draw Eva Mendes printable step by step drawing sheet
Web 4,539 solutions more related questions algebra write an equation of the line that passes through the given point and is parallel to the given line. Use the power of algebra to understand and
Basics of Drawing Lines Lesson 2 of 3 YouTube
Explain how you can use mental math to find the slope of the line. We are given the following in the question: Maria finds the slope between the points (3,2) and (9,6). Determine which of.
EVA 01. I hope you like this drawing I made a few weeks ago. evangelion
Which function gives all the points (x,y) on this line? Carl finds the slope between the points (3,2) and (6,4). Use the power of algebra to understand and interpret points and lines (something we
Eva Draws A Line That Includes The Points Which function gives all the points (x,y) on this line? Write the equation of a line given the slope and a. Which function gives all the points (x,y) on this
line? Y= 2 2−x is the require function. Web eva finds the slope between the points (0,0) and (3,2).
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Playing with Systems
This 19th article in the series of “Do It Yourself: Electronics”, decodes the working of an RC circuit.
In the previous article, Pugs understood the resistor and capacitor from a high level perspective and used some formulae to design various filters. Are you wondering, how did Pugs get those formulae?
He computed them using the complex notation of the impedance of resistor R & capacitor C. Alongwith, he also used the “A * e^(j * φ)” notation for amplitute A and phase angle φ. Pugs detailed out his
computations in a set of slides. Check them out below:
After the derivation of the formulae, slide 8 above shows their usage. It predicts the change in sine wave amplitude & phase angle at 2 different frequencies of 1KHz & 2KHz compared to the input, for
various pairs of R & C values. And, then Pugs verifies them (within tolerance limits) by comparing the input and output waveforms, on the home-made PC oscilloscope, as created in his previous PC
Oscilloscope article. Note that, while measuring, the GND is kept common for the two waveforms. Also, Pugs used the two stereo lines of the audio jack for the two waveforms, and configured xoscope
Continuing from the previous formulae derivations, the final slide 9 shows the high-pass & low-pass filter computations and the calculation of the cut-off frequency, which has been used in the
previous article.
Basic Filter Design
This 18th article in the series of “Do It Yourself: Electronics”, demonstrates basic resistor-capacitor (RC) based filter design.
Though Pugs got the various waveforms generated, on closer look he realized that there were steps in the waveforms like sine, triangular, …. Expected right? Because of the approximation. Also, the
PWM carrier frequency was present. So, would it produce the results as expected in an resistor-capacitor (RC) experiment with a single frequency sine wave? That was a big question. Or rather, can the
unwanted frequencies be eliminated? That brought Pugs to the exploration of filters. Let’s see what Pugs learnt about them.
High Level understanding
Current i through a capacitor of C farads is given by C dv/dt. And if the voltage is DC that is constant, dv/dt is zero and hence i is zero. So it is said, DC is blocked by a capacitor. If the
voltage is AC that is sine wave of a particular frequency, dv/dt would become a cosine wave, or sine wave with 90° phase shift but with same frequency. Along with, C becomes the multiplication factor
to give C dv/dt as the non-zero current i, hence AC is allowed to pass through a capacitor.
Current i through a resistor of R ohms is given by v / R. So, AC or DC both currents are allowed through it without any phase shift, just with the multiplication factor of 1 / R.
However, combining the two (C & R) in series with a voltage input (as shown in figure below) gives very interesting results as then the current i is being controlled by both the capacitor and the
resistor. Phase shift gets pulled to something between 0° and 90°. And the amplitude turns out to be attenuating depending on the sine wave frequency, with the following relation:
$Amplification = \frac{\omega * R * C}{\sqrt{1 + {(\omega * R * C)}^{2}}}$, where ω = 2 * π * frequency
Filter Design
Now in the amplification formula, observe that the minimum amplification is zero at zero frequency i.e. at DC, and keeps on increasing with frequency, reaching one, at infinite frequency. But
practically, one can’t reach infinity. Hence, for practical purposes, the amplification is considered good enough at a value of 1/√2, which happens when ω * R * C = 1, meaning at frequency = 1/(2 * π
* R * C}, which is often referred as the cut-off frequency. And for practical purposes, the currents of frequencies below this are cut-off, only allowing the ones above it. Note that, by controlling
the values of R & C, one can beautifully control the cut-off frequency.
Now, this current passing through the resistor R gives the voltage v[R] across the resistor as R * i. Thus, this voltage also follows the same cut-off. So, if one considers the voltage v as input and
the voltage v[R] as output, that gives a high pass filter (HPF), which passes the frequencies above the cut-off and (practically) blocks the ones below it.
Interestingly enough, if the positions of R & C are swapped (as shown in figure below), and the voltage v[C] across the capacitor is considered as output, it would become a low pass filter (LPF),
which passes the frequencies below the cut-off and (practically) blocks the ones above it. Why? Because the amplification factor there is as follows:
$Amplification = \frac{1}{\sqrt{1 + {(\omega * R * C)}^{2}}}$, where ω = 2 * π * frequency
With this, Pugs have got the two simplest RC filters.
Filter Application
Now, for proper sine wave, most importantly Pugs wanted to remove the PWM carrier frequency, which is way higher than his various waveform frequencies of 1 & 2 KHz. So, Pugs would need a low pass
filter, which passes 1KHz, 2KHz but filters out the high PWM carrier frequency. A 5KHz cut-off frequency for the filter seems okay. So, let’s compute the values of R & C for it. As per the formula, R
* C = 1/(2* π * frequency) = 1/(2* π * 5000) = 31.831 * 10^-6s. Assuming a value of 1K ohms for R, C could be chosen as the standard 33 nano farads. Using these two selections, Pugs computed back the
cut-off frequency, which turned out to be 4.8KHz – that’s fine for Pugs.
Taking a 1K resistor and a 33 nF capacitor, Pugs connected them in series across his sine wave output.
Pugs then checked the output between RC joint and GND, on the home-made PC oscilloscope, as created in his previous PC Oscilloscope article. It showed up the sine wave as in his previous article but
now with almost no carrier frequency zigzags, though the amplitude was a bit reduced. | {"url":"https://sysplay.in/blog/tag/basic-filter-design/","timestamp":"2024-11-08T18:15:08Z","content_type":"text/html","content_length":"53567","record_id":"<urn:uuid:5ff631ce-cd6f-4a3c-8c90-712238907498>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00760.warc.gz"} |
orthogonal vector
What are orthogonal vectors?
What are orthogonal vectors?
Definition. We say that 2 vectors are orthogonal if they are perpendicular to each other. i.e. the dot product of the two vectors is zero.
What is orthogonal vector formula?
Definition. Two vectors x , y in R n are orthogonal or perpendicular if x · y = 0. Notation: x ⊥ y means x · y = 0. Since 0 · x = 0 for any vector x , the zero vector is orthogonal to every vector in
R n .
What are three orthogonal vectors?
a → , b → , c → are three orthogonal vectors. The dot product of any two vectors should be equal to zero. a → , b → , c → are three orthogonal vectors. The dot product of any two vectors should be
equal to zero.
How do you find orthogonal vectors examples?
Therefore, if the dot product also yields a zero in the components multiplication case, then the 2 vectors are orthogonal. Find whether the vectors a = (5, 4) and b = (8, -10) are orthogonal to one
another or not. Hence, it is proved that the two vectors are orthogonal in nature.
What is orthogonality rule?
Loosely stated, the orthogonality principle says that the error vector of the optimal estimator (in a mean square error sense) is orthogonal to any possible estimator. The orthogonality principle is
most commonly stated for linear estimators, but more general formulations are possible.
What is orthogonality condition?
In Euclidean space, two vectors are orthogonal if and only if their dot product is zero, i.e. they make an angle of 90° (π/2 radians), or one of the vectors is zero. Hence orthogonality of vectors is
an extension of the concept of perpendicular vectors to spaces of any dimension.
Can zero vectors be orthogonal?
The dot product of the zero vector with the given vector is zero, so the zero vector must be orthogonal to the given vector. This is OK. Math books often use the fact that the zero vector is
orthogonal to every vector (of the same type).
How do you determine orthogonality?
To determine if a matrix is orthogonal, we need to multiply the matrix by it’s transpose, and see if we get the identity matrix. Since we get the identity matrix, then we know that is an orthogonal | {"url":"https://thecrucibleonscreen.com/what-are-orthogonal-vectors/","timestamp":"2024-11-10T00:14:41Z","content_type":"text/html","content_length":"52846","record_id":"<urn:uuid:498f4063-2e14-4e54-a6d9-6c351767fed4>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00062.warc.gz"} |
2019 Introduction to AG Codes | homepage
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Introduction to Algberaic-Geometric Codes
Spring 2019
Course Pitch (a.k.a. Syllabus)
Theoretical computer scientists make a regular use of algebraic objects, most prominently polynomials over finite fields, to construct and analyze desired objects such as error correcting codes,
expander graphs, pseudorandom generators, cryptographic protocols, and randomness extractors. Reed-Solomon codes is one such classic example. In 1975, Goppa suggested to construct and analyze codes
using machinery from algebraic geometry or, more precisely, based on algebraic curves. In a long line of research, such codes were eventually constructed. Surprisingly, these codes beat the
Gilbert-Varshamov bound for sufficiently large (constant size) fields.
The mathematics underlying Goppa codes is far deeper than the standard tools generally used by theoretical computer scientists, and the hope is that it can be further exploited in resolving other
open problems, especially problems for which using polynomials one currently obtains the state-of-the-art results. Indeed, many important problems fall into this category. For that to happen, the
underlying mathematics must be grasped by computer scientists. This is the goal of this course (and its follow-up).
In this introductory course we will thoroughly develop the fascinating mathematics underlying algebraic curves to the point where Goppa's framework can be defined and well-understood - a modest yet
challenging goal. The approach we take is mostly algebraic, however, we will not abandon the geometric aspect - we will "think geometrically" and prove theorems algebraically. Students who complete
the course should be able to use the mathematical tools developed beyond the code-specific application.
Who this course is for?
The target audience for the course is computer science graduate and advanced undergraduate students that would like to study fascinating, deep mathematics and employ it in their own research. I will
assume basic knowledge in ring theory and field theory. This was covered in a course I gave last semester as in any standard course on abstract algebra. I will not assume knowledge in topology,
commutative algebra or Galois theory - we'll develop all we need.
So, initially, I was hoping to deliver this course on board like any good old math course. However, this turned out to be difficult in terms of time. So, we are switching to slides now:
Lecture 6 (April 29, 2019)
In this lecture, we studied the following three units + the construction of the field of fractions with respect to a multiplicative set that appears in the next unit.
Lecture 7 (May 6, 2019)
We study the important commutative algebraic tool of localization and use it to prove an important result regarding algebraic curves.
Lecture 8 (May 13, 2019)
Lecture 9 (May 20, 2019)
Lecture 10 (May 27 2019)
Lecture 11 (June 3 2019)
Lecture 12 (June 10 2019)
You can access the old notes here.
Assignments and Final
When and where
Lectures: Monday 12:10-15:00, Orenstein 111
Recitations: Wednesday 10:10-11:00, Orenstein 103
About 8-10 light assignments will be given during the semester. These will account for 50% of the grade. The remaining 50% will be determined by a take-home exam.
Reading material
We will mainly follow the truly marvellous book An Invitation to Arithmetic Geometry by Dino Lorenzini. We may also make use of the standard book in the field of algebraic geometric codes: Algebraic
Function Fields and Codes by Henning Stichtenoth. Other resources include great lecture notes by Morandi, and the books Algebraic Geometry in Coding Theory and Cryptography by Niederreiter and Xing,
and Algebraic-Geometric Codes by Tsfasman and Valdut.
Viewing material
To get some taste of what this course is about, see the introductory talk I gave on the subject at IAS. To go deeper, I recommend Beelen's mini-course on the subject. It assumes prior knowledge but
is clear enough so that much can be gained nonetheless.
A follow-up course
In the following semester, a follow-up course was planned to be given in which we will continue to develop central tools in the study of algebraic curves such as the Riemann-Roch theorem,
factorization of ideals in ring extensions, ramification and discriminants, the zeta function attached to an algebraic curve, and explicit constructions of algebraic curves with desired properties as
well as algebraic-geometric codes. Some of the material may shift from this more advanced course to the introductory course and vice versa.
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SAT Chemistry Chemical Formulas - Laws Of Definite Composition and Multiple Proportions
SAT Chemistry Chemical Formulas - Laws Of Definite Composition and Multiple Proportions
In the problems involving percent composition, we have depended on two things: each unit of an element has the same atomic mass, and every time the particular compound forms, it forms in the same
percent composition. That this latter statement is true no matter the source of the compound is the Law of Definite Composition. There are some compounds formed by the same two elements in which the
mass of one element is constant, but the mass of the other varies. In every case, however, the mass of the other element is present in a small-whole- number ratio to the weight of the first element.
This is called the Law of Multiple Proportions. An example is H[2]O and H[2]O[2].
In H[2]O the proportion of H: O = 2:16 or 1:8
In H[2]O[2] the proportion of H: O = 2:32 or 1:16
The ratio of the mass of oxygen in each is 8:16 or 1:2 (a small-whole-number ratio).
An equation is a simplified way of recording a chemical change. Instead of words, chemical symbols and formulas are used to represent the reactants and the products. Here is an example of how this
can be done. The following is the word equation of the reaction of burning hydrogen with oxygen:
Hydrogen + oxygen yields water.
Replacing the words with the chemical formulas, we have
H[2] + O[2] —> H[2]O
We replaced hydrogen and oxygen with the formulas for their diatomic molecular states and wrote the appropriate formula for water based on the respective oxidation (valence) numbers for hydrogen and
oxygen. Note that the word yields was replaced with the arrow.
Although the chemical statement tells what happened, it is not an equation because the two sides are not equal. While the left side has two atoms of oxygen, the right side has only one. Knowing that
the Law of Conservation of Matter dictates that matter cannot easily be created or destroyed, we must get the number of atoms of each element represented on the left side to equal the number on the
right. To do this, we can only use numbers, called coef¬ficients, in front of the formulas. It is important to note that in attempting to balance equations THE SUBSCRIPTS IN THE FORMULAS MAY NOT BE
Looking again at the skeleton equation, we notice that if 2 is placed in front of H[2]O the numbers of oxygen atoms represented on the two sides of the equation are equal. However, there arc now four
hydrogens on the right side with only two on the left. This can be corrected by using a coefficient of 2 in front of H[2]. Now we have a balanced equation:
2H[2] + O[2] —> 2H[2]O
This equation tells us more than merely that hydrogen reacts with oxygen to form water. It has quantitative meaning as well. It tells us that two molecular masses of hydrogen react with one molecular
mass of oxygen to form two molecular masses of water. Because molecular masses are indirectly related to grams, we may also relate the masses of reactants and products in grams.
This aspect will be important in solving problems related to the masses of substances in a chemical equation.
Here is another, more difficult example: Write the balanced equation for the burning of butane (C[4]H[10]) in oxygen. First, we write the skeleton equation:
C[4]H[10] + O[2] yields CO[2] + H[2]O.
Looking at the oxygens, we see that there are an even number on the left but an odd number on the right. This is a good place to start. If we use a coefficient of 2 for H[2]O, that will even out the
oxygens but introduce four hydrogens on the right while there are ten on the left. A coefficient of 5 will give us the right number of hydrogens but introduces an odd number of oxygens. Therefore, we
have to go to the next even multiple of 5, which is 10. Ten gives us 20 hydrogen atoms on the right. By placing another coefficient of 2 in front of C[4]H[10], we also have 20 hydrogen atoms on the
left. Now the carbons need to be balanced. By placing an 8 in front of CO[2], we have eight carbons on both sides. The remaining step is to balance the oxygens. We have 26 on the right side, so we
need a coefficient of 13 in front of the O[2] on the left to give us 26 oxygens on both sides. Our balanced equation is:
2C[4]H[10] + 13O[2] -> 8CO[2] + 10H[2]O
Once an equation is balanced, you may choose to give additional information in the equation. This can be done by indicating the phases of substances, telling whether each substance is in the liquid
phase (l), the gaseous phase (g), or the solid phase (s). Since many solids will not react to any appreciable extent unless they are dissolved in water, the notation (aq) is used to indicate that the
substance exists in a water (aqueous) solution. Information concerning phase is given in parentheses following the formula for each substance. Several illustrations of this notation are given below:
An example of phase notation in an equation:
2HCl(aq) + Zn(s) → ZnCl[2](aq) + H[2](g)
In words, this says that a water solution of hydrogen chloride (called hydrochloric acid) reacts with solid zinc to produce zinc chloride dissolved in water plus hydrogen gas.
At times, chemists choose to show only the substances that react in the chemical action. These equations are called ionic equations because they stress the reaction and production of ions. If we look
at the preceding equation, we see the complete cast of “actors”:
Notice that nothing happened to the chloride ion. It appears the same on both sides of the equation. It is referred to as a spectator ion. In writing the net ionic equation, spectator ions are
omitted, so the net ionic equation is:
2H^+(aq) + Zn(s) → Zn^2+(aq) + H[2](g)
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: product of elementary matrices
Is there a pg file out there that asks for writing an invertible matrix as a product of elementary matrices? I can't find any in the OPL.
I assume this can be done with MultiAnswer in parserMultiAnswer.pl and Matrix math objects. I tried to find some similar examples in the OPL. I loooked at the diagonalization problems because that
problem is somewhat similar but unfortunately I couldn't find one that uses Matrix math objects.
Are matrix math objects used at all in the OPL? Could someone recommend me a good example.
An ideal simple example would be a math object version of a problem that generates a random matrix A and asks for (using MultiAnswer) two non-identity matrices B and C satisfying A=BC.
Hi Nandor,
I do not have time right now to modify the example below to suit your needs, but you should be able to make the modifications yourself without too much effort. I have been working on updating the OPL
linear algebra questions to use MathObjects, but things are still in development and beta testing.
(Nota: it may be possible to replace $multians->ans_array(2,2,5) with $multians->ans_array(5).)
Best regards,
Paul Pearson
## DESCRIPTION
## Linear Algebra
## KEYWORDS('linear algebra','matrix','multiplication')
## Tagged by cmd6a 5/3/06
## DBsubject(Linear algebra)
## DBchapter('Matrices')
## DBsection(Matrix algebra)
## Date('July 2013')
## Author('Paul Pearson')
## Institution('Hope College')
## TitleText1('Linear Algebra with Applications')
## AuthorText1('Jeffrey Holt')
## EditionText1('1')
## Section1('3.2')
## Problem1('')
$showPartialCorrectAnswers = 0;
$A = Matrix([[0,1],[0,0]]);
$B = Matrix([[0,1],[0,0]]);
$Z = Matrix([[0,0],[0,0]]);
$multians = MultiAnswer($A, $B)->with(
singleResult => 1,
checker => sub {
my ( $correct, $student, $self ) = @_;
my @s = @{$student};
my @c = @{$correct};
$s0 = Matrix($s[0]); $s1 = Matrix($s[1]);
return 0 if $s0 == $Z or $s1 == $Z;
return $s0 * $s1 == $Z;
Give an example of two \( 2 \times 2 \) matrices \( A \) and \( B \),
neither of which is the zero matrix \( \mathbf{0} \),
such that \( AB = \mathbf{0} \).
\( A = \)
\{ $multians->ans_array(2,2,5) \}
\( B = \)
\{ $multians->ans_array(2,2,5) \}
ANS( $multians->cmp() );
COMMENT('MathObject version.');
Thank you, exactly what I needed. The new problems are in the OPL. | {"url":"https://webwork.maa.org/moodle/mod/forum/discuss.php?d=3104","timestamp":"2024-11-14T17:23:45Z","content_type":"text/html","content_length":"90765","record_id":"<urn:uuid:d1edf7b8-6c71-4c57-96fc-b0ab537e9df2>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00261.warc.gz"} |
SPOJ.com - Problem BYTEFOOD
BYTEFOOD - Food Shortage in Byteland
Fanatics from the BBFO blew up all the food factories in the Bytelandian capital! Hurry up! There is still some food left in shops. Some shops are located in the centre, others in the suburbs, so
Johnny has to decide which of them are worth visiting. Some shops can be very big and have plenty of food in them, others may be so small that food dissappears from them at an alarming rate... So?
Help Johnny buy as much food as possible.
There are n open shops, each of them located at position (x[i],y[i]), for i=1,...,n, where 0 <= x[i],y[i] <= 250. The distances between shops are measured using the Manhattan metric (i.e. as sums of
absolute values of differences of x and y coordinates). Besides, every shop is characterized by a linear time function describing how much food is left in the shop at the moment:
f[i] = max{0, a[i] - b[i] * time}
where 0 <= a[i] <= 1000000, 0 <= b[i] <= 1000, while time is the time (in minutes) that has elapsed from the moment Johnny left the house (assume that Johnny does not live in the same place as any
shop). If Johnny decides to stay in a shop, he can buy at most b[i] units of food per minute. Otherwise, he can move along the ortogonal system of streets of the city at a constant speed of unit
distance per minute. Johnny only ever changes the action he is performing at the full minute. Because his family is slowly beginning to starve, he should be back at home not later than m minutes
after he left. Since there are thousands of starving families in the capital, Johnny can't spent more that 1 <= c[i] <= 10 minutes in a shop. Moreover, he will never go into the same shop twice for
fear of being lynched...
The first line of input contains a single positive integer t <= 1000, the number of test cases. Each test case begins with the number of shops in the city 1 <= n <= 1000 and the deadline 1 <= m <=
5000. Then the following n lines consist of four integers x[i] y[i] a[i] b[i] c[i] each, describing the position and the parameters of the function for food availability of the i-th shop. At the end
of every test case comes a line with two integersp q (between 0 and 250), corresponding to the x and y coordinates of the position of Johnny's house.
All the input data are integers.
Process all test cases. The correct output for the i-th test case takes the following form:
i [the number of the test case, in the input order]
s m [s is the number of the target shop and m > 0 is the number of minutes spent in it].
At the end of the series of moves you should always write a line conisting of two zeros ('0 0').
All the output data should be integers.
The score of your program is the total amount of food that Johnny bought (summed over all the testcases in which he managed to come back home before the deadline).
Score = 1261
hide comments
ankur_gupta54: 2015-06-24 13:44:56
Can anyone please explain the output... because for the given sample input I m getting more score... but on submission its showing wrong answer
Virat Goyal: 2013-12-28 12:26:19
please elaborate the format of output for different possible cases as I am unable to predict why my code is told to be giving wrong answer
Aditya Pande: 2012-06-06 09:30:26
why am i getting WA in submission id 7099121
plz check
Added by: mima
Date: 2004-06-09
Time limit: 20s
Source limit: 50000B
Memory limit: 1536MB
Cluster: Cube (Intel G860)
Languages: All except: NODEJS PERL6 VB.NET
Resource: - | {"url":"https://www.spoj.com/problems/BYTEFOOD/","timestamp":"2024-11-04T21:37:15Z","content_type":"text/html","content_length":"26198","record_id":"<urn:uuid:d300a1f7-708b-41ad-b44b-c6a33f2fe094>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00231.warc.gz"} |
Question ID - 151701 | SaraNextGen Top Answer
Question ID - 151701 | SaraNextGen Top Answer
Arrange the order of power dissipated in the given circuits, if the same current is passing through all the circuits. The resistance of each resistor is
1 Answer
127 votes
Answer Key / Explanation : (b)
So it is obvious that
127 votes
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John von Neumann: Selected Letterssearch
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John von Neumann: Selected Letters
A co-publication of the AMS and London Mathematical Society
Softcover ISBN: 978-1-4704-6863-7
Product Code: HMATH/27.S
List Price: $125.00
MAA Member Price: $112.50
AMS Member Price: $100.00
eBook ISBN: 978-1-4704-3895-1
Product Code: HMATH/27.E
List Price: $120.00
MAA Member Price: $108.00
AMS Member Price: $96.00
Softcover ISBN: 978-1-4704-6863-7
eBook: ISBN: 978-1-4704-3895-1
Product Code: HMATH/27.S.B
List Price: $245.00 $185.00
MAA Member Price: $220.50 $166.50
AMS Member Price: $196.00 $148.00
Click above image for expanded view
John von Neumann: Selected Letters
A co-publication of the AMS and London Mathematical Society
Softcover ISBN: 978-1-4704-6863-7
Product Code: HMATH/27.S
List Price: $125.00
MAA Member Price: $112.50
AMS Member Price: $100.00
eBook ISBN: 978-1-4704-3895-1
Product Code: HMATH/27.E
List Price: $120.00
MAA Member Price: $108.00
AMS Member Price: $96.00
Softcover ISBN: 978-1-4704-6863-7
eBook ISBN: 978-1-4704-3895-1
Product Code: HMATH/27.S.B
List Price: $245.00 $185.00
MAA Member Price: $220.50 $166.50
AMS Member Price: $196.00 $148.00
• History of Mathematics
Volume: 27; 2005; 301 pp
MSC: Primary 00; 01
John von Neumann was perhaps the most influential mathematician of the twentieth century. Not only did he contribute to almost all branches of mathematics, he created new fields and was a
pioneering influence in the development of computer science.
During and after World War II, he was a much sought-after technical advisor. He served as a member of the Scientific Advisory Committee at the Ballistic Research Laboratories, the Navy Bureau of
Ordinance, and the Armed Forces Special Weapons Project. He was a consultant to the Los Alamos Scientific Laboratory and was appointed by U.S. President Dwight D. Eisenhower to the Atomic Energy
Commission. He received the Albert Einstein Commemorative Award, the Enrico Fermi Award, and the Medal of Freedom.
This collection of about 150 of von Neumann's letters to colleagues, friends, government officials, and others illustrates both his brilliance and his strong sense of responsibility. It is the
first substantial collection of his letters, giving a rare inside glimpse of his thinking on mathematics, physics, computer science, science management, education, consulting, politics, and war.
With an introductory chapter describing the many aspects of von Neumann's scientific, political, and social activities, this book makes great reading. Readers of quite diverse backgrounds will be
fascinated by this first-hand look at one of the towering figures of twentieth century science.
To read a review published in the Notices of the American Mathematical Society, click here.
Also of interest and available from the AMS is John von Neumann: The Scientific Genius Who Pioneered the Modern Computer, Game Theory, Nuclear Deterrence, and Much More and Invariant Measures.
General audience; readers of diverse backgrounds.
□ Chapters
□ Introductory comments
□ Letter to N. Aronszajn
□ Letters to F. Aydelotte
□ Letter to E.F. Beckenbach
□ Letter to H. Bethe
□ Letters to G. Birkhoff
□ Letter to W.J.E. Blaschke
□ Letter to R.S. Burington
□ Letters to V. Bush
□ Letter to R. Carnap
□ Letter to W. Cattell
□ Letter to T.M. Cherry
□ Letter to H. Cirker
□ Letter to H. Crocker
□ Letter to M.R. Davie
□ Letter to W.E. Deming
□ Letter to J.L. Destouches
□ Letter to P.A.M. Dirac
□ Letters to J. Dixmier
□ Letter to P.A. Dodd
□ Letter to W.M. DuMond
□ Letter to R.E. Duncan
□ Letter to editor of Evening Star
□ Letter to R. Farquharson
□ Letter to A. Flexner
□ Letter to R.O. Fornaguera
□ Letter to N.H. Goldsmith
□ Letter to W.H. Gottschalk (and Hans Rademacher)
□ Letters to K. Gödel
□ Letter to G. Haberler
□ Letters to I. Halperin
□ Letter to G.B. Harrison
□ Letter to M. de Horvath
□ Letter to A.S. Householder
□ Letters to C.C. Hurd
□ Letter to K. Husimi
□ Letters to P. Jordan
□ Letters to I. Kaplansky
□ Letter to C.E. Kemble
□ Letter to J.R. Killian
□ Letters to H.D. Kloosterman
□ Letter to H. Kuhn
□ Letter to J. Lederberg
□ Letter to W.E. Lingelbach
□ Letter to S. MacLane
□ Letter to J.C.C. McKinsey
□ Letter to M.M. Mitchell
□ Letter to T.V. Moore
□ Letter to O. Morgenstern
□ Letters to M. Morse
□ Letter to E. Nagel
□ Letter to J.R. Oppenheimer
□ Letters to R. Ortvay
□ Letter to W. Overbeck
□ Letter to H.H. Rankin
□ Letter to H.P. Robertson
□ Letter to E. Schrödinger
□ Letter to E. Segre
□ Letters to F.B. Silsbee
□ Letter to L. Spitzer
□ Letters to M. Stone
□ Letters to L.L. Strauss
□ Letter to J. Stroux
□ Letter to T. Tannaka
□ Letter to E. Teller
□ Letters to L.B. Tuckerman
□ Letters to S. Ulam
□ Letter to E.R. van Kampen
□ Letters to O. Veblen
□ Letters to N. Wiener
□ Letter to H. Wold
□ This selection of letters by the great mathematician John von Neumann (1903-1957) has been edited with remarkable care and expertise. Nearly all of his letters are published here for the
first time. ....The extensive introductory comments by the editor (pp. 1-40) furnish important background information that will prove itself useful, especially for those readers who cannot
rely on a thorough knowledge of von Neumann's seminal contributions to the foundations of mathematics, mathematical logic, quantum mechanics, quantum logic, computer science, and game theory.
Mathematics and Mechanics
□ ...this fine volume of selected letters will be of great interest to mathematicians, scientists, educators, and, in particular, those interested in the history of the fields to which John von
Neumann contributed so much.
Historia Mathematica
□ ...this will be an enduring reference book ...delivers an impressive sampling of the breadth and influence of von Neumann's work.
Notices of the American Mathematical Society
□ I recommend that you check out a copy of the book yourself. These letters are put together in a very nice volume, and give quite a bit of insight into one of the great mathematical minds of
all time ...
MAA Reviews
□ ...the volume under review will certainly contribute to our knowledge about John von Neumann and his personality. It will be interesting for historians of science, especially of mathematics
and mathematical physics.
Zentrablatt MATH
• Book Details
• Table of Contents
• Additional Material
• Reviews
• Requests
Volume: 27; 2005; 301 pp
MSC: Primary 00; 01
John von Neumann was perhaps the most influential mathematician of the twentieth century. Not only did he contribute to almost all branches of mathematics, he created new fields and was a pioneering
influence in the development of computer science.
During and after World War II, he was a much sought-after technical advisor. He served as a member of the Scientific Advisory Committee at the Ballistic Research Laboratories, the Navy Bureau of
Ordinance, and the Armed Forces Special Weapons Project. He was a consultant to the Los Alamos Scientific Laboratory and was appointed by U.S. President Dwight D. Eisenhower to the Atomic Energy
Commission. He received the Albert Einstein Commemorative Award, the Enrico Fermi Award, and the Medal of Freedom.
This collection of about 150 of von Neumann's letters to colleagues, friends, government officials, and others illustrates both his brilliance and his strong sense of responsibility. It is the first
substantial collection of his letters, giving a rare inside glimpse of his thinking on mathematics, physics, computer science, science management, education, consulting, politics, and war. With an
introductory chapter describing the many aspects of von Neumann's scientific, political, and social activities, this book makes great reading. Readers of quite diverse backgrounds will be fascinated
by this first-hand look at one of the towering figures of twentieth century science.
To read a review published in the Notices of the American Mathematical Society, click here.
Also of interest and available from the AMS is John von Neumann: The Scientific Genius Who Pioneered the Modern Computer, Game Theory, Nuclear Deterrence, and Much More and Invariant Measures.
General audience; readers of diverse backgrounds.
• Chapters
• Introductory comments
• Letter to N. Aronszajn
• Letters to F. Aydelotte
• Letter to E.F. Beckenbach
• Letter to H. Bethe
• Letters to G. Birkhoff
• Letter to W.J.E. Blaschke
• Letter to R.S. Burington
• Letters to V. Bush
• Letter to R. Carnap
• Letter to W. Cattell
• Letter to T.M. Cherry
• Letter to H. Cirker
• Letter to H. Crocker
• Letter to M.R. Davie
• Letter to W.E. Deming
• Letter to J.L. Destouches
• Letter to P.A.M. Dirac
• Letters to J. Dixmier
• Letter to P.A. Dodd
• Letter to W.M. DuMond
• Letter to R.E. Duncan
• Letter to editor of Evening Star
• Letter to R. Farquharson
• Letter to A. Flexner
• Letter to R.O. Fornaguera
• Letter to N.H. Goldsmith
• Letter to W.H. Gottschalk (and Hans Rademacher)
• Letters to K. Gödel
• Letter to G. Haberler
• Letters to I. Halperin
• Letter to G.B. Harrison
• Letter to M. de Horvath
• Letter to A.S. Householder
• Letters to C.C. Hurd
• Letter to K. Husimi
• Letters to P. Jordan
• Letters to I. Kaplansky
• Letter to C.E. Kemble
• Letter to J.R. Killian
• Letters to H.D. Kloosterman
• Letter to H. Kuhn
• Letter to J. Lederberg
• Letter to W.E. Lingelbach
• Letter to S. MacLane
• Letter to J.C.C. McKinsey
• Letter to M.M. Mitchell
• Letter to T.V. Moore
• Letter to O. Morgenstern
• Letters to M. Morse
• Letter to E. Nagel
• Letter to J.R. Oppenheimer
• Letters to R. Ortvay
• Letter to W. Overbeck
• Letter to H.H. Rankin
• Letter to H.P. Robertson
• Letter to E. Schrödinger
• Letter to E. Segre
• Letters to F.B. Silsbee
• Letter to L. Spitzer
• Letters to M. Stone
• Letters to L.L. Strauss
• Letter to J. Stroux
• Letter to T. Tannaka
• Letter to E. Teller
• Letters to L.B. Tuckerman
• Letters to S. Ulam
• Letter to E.R. van Kampen
• Letters to O. Veblen
• Letters to N. Wiener
• Letter to H. Wold
• This selection of letters by the great mathematician John von Neumann (1903-1957) has been edited with remarkable care and expertise. Nearly all of his letters are published here for the first
time. ....The extensive introductory comments by the editor (pp. 1-40) furnish important background information that will prove itself useful, especially for those readers who cannot rely on a
thorough knowledge of von Neumann's seminal contributions to the foundations of mathematics, mathematical logic, quantum mechanics, quantum logic, computer science, and game theory.
Mathematics and Mechanics
• ...this fine volume of selected letters will be of great interest to mathematicians, scientists, educators, and, in particular, those interested in the history of the fields to which John von
Neumann contributed so much.
Historia Mathematica
• ...this will be an enduring reference book ...delivers an impressive sampling of the breadth and influence of von Neumann's work.
Notices of the American Mathematical Society
• I recommend that you check out a copy of the book yourself. These letters are put together in a very nice volume, and give quite a bit of insight into one of the great mathematical minds of all
time ...
MAA Reviews
• ...the volume under review will certainly contribute to our knowledge about John von Neumann and his personality. It will be interesting for historians of science, especially of mathematics and
mathematical physics.
Zentrablatt MATH
Please select which format for which you are requesting permissions. | {"url":"https://bookstore.ams.org/view?ProductCode=HMATH/27","timestamp":"2024-11-05T16:07:24Z","content_type":"text/html","content_length":"167730","record_id":"<urn:uuid:40b7d5a0-2283-4ecc-8796-82259357f22e>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00867.warc.gz"} |
How do you derive y = (1 + sin(2x)]/[1 - sin(2x)) using the quotient rule? | HIX Tutor
How do you derive #y = (1 + sin(2x)]/[1 - sin(2x))# using the quotient rule?
Answer 1
$f ' \left(x\right) = \frac{4 \cos \left(2 x\right)}{1 - \sin \left(2 x\right)} ^ 2 = \frac{\mathrm{dy}}{\mathrm{dx}}$.
By the quotient rule, the derivative of #(u(x))/(v(x))# is #(u'(x)v(x) - u(x)v'(x))/(v(x)^2)#. Let's say #f(x) = (1+sin(2x))/(1-sin(2x))#.
Here, #u(x) = 1 + sin(2x)# and #v(x) = 1 - sin(2x)#. The derivative of #sin(2x)# is #2cos(2x)# by the chain rule, so #u'(x) = 2cos(2x)# and #v'(x) = -u'(x)#.
So, by applying the quotient rule : #f'(x) = (2cos(2x)(1-sin(2x)) - (1+sin(2x))(-2cos(2x)))/(1-sin(2x))^2#
#f'(x) = (2cos(2x)(1-sin(2x) + 1 + sin(2x)))/(1-sin(2x))^2 = (4cos(2x))/(1-sin(2x))^2#
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Answer 2
To derive ( y = \frac{1 + \sin(2x)}{1 - \sin(2x)} ) using the quotient rule, follow these steps:
The quotient rule is given by (\frac{d}{dx}\left(\frac{f}{g}\right) = \frac{f'g - fg'}{g^2}), where (f = 1 + \sin(2x)) and (g = 1 - \sin(2x)).
First, find the derivatives of (f) and (g):
• (f' = \frac{d}{dx}(1 + \sin(2x)) = 2\cos(2x))
• (g' = \frac{d}{dx}(1 - \sin(2x)) = -2\cos(2x))
Now, apply the quotient rule:
[y' = \frac{f'g - fg'}{g^2} = \frac{(2\cos(2x))(1 - \sin(2x)) - (1 + \sin(2x))(-2\cos(2x))}{(1 - \sin(2x))^2}]
[= \frac{2\cos(2x) - 2\sin(2x)\cos(2x) + 2\cos(2x) + 2\sin(2x)\cos(2x)}{(1 - \sin(2x))^2}]
[= \frac{4\cos(2x)}{(1 - \sin(2x))^2}]
That is the derivative of (y) with respect to (x).
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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A Super Simple Introduction to Options (Calls, puts) - F.S. Comeau
Okay, so in our latest “Super Simple” article, we covered what short selling is. I highly recommend you study that post and understand it well before trying to understand options. I’ll be honest:
options are a bit more complicated and a bit harder to understand, but don’t let that stop you, I made this post ridiculously easy to understand. I’ll be honest: everyone can understand the basic
part of options if he truly wants to.
As always, we will cover only a very small subset of what options are. Understand that options go extremely far, well past the Ph.D. level. There are, for instance, options on options, but that’s out
of the scope of this article.
As I said, don’t let the name or any financial blockage stop you. I am going to make options so simple you’ll understand the basics 100% fifteen minutes from now.
OK so let’s say you buy a phone. The salesman says to you, “At any time during the next 30 days, you can buy another phone for 50% off.”
That’s it?
Yes, that’s it. Next question.
Come on, go a little bit more in depth than that.
An option is the OPTION, but NOT THE OBLIGATION, to do something. I’m going to repeat it because I taught “Introduction to derivatives” in college for two years and half the class didn’t understand
it even at the end of the year. I’m serious: I think I repeated over a hundred times and over half the class still didn’t get it. Some people would have graduated and would diplomas by now if they
had understood the simple fact:
An option is an OPTION, you do NOT EVER EVER have the obligation to USE IT. You can just drop it in your personal safe and forget all about it forever
Please don’t yell…
So in the example above, in addition to buying a phone, you also got an option. The option states: “You buy a phone at 50% off at any time from now until 30 days have lapsed.” This means that on the
28th day, you could go there and tell the guy, “Hey, I want my 50% off phone.”
YOU COULD ALSO CHOOSE NOT TO DO IT.
Hey, I said no yelling
Sorry but hopefully this time it sticks. You are under no obligation whatsoever to go there and buy that phone at 50% off. Also, note that you cannot go there on the 31th day. If you did, the
salesman would say, “sorry, this was only valid for 30 days following your purchase, it’s too late.”
Now, what’s the point of that phone option? Well, let’s say the phone you bought was the iPhone 7 and you plan to give that iPhone to your mother on her birthday. You also want a phone for yourself,
but you’re thinking to yourself, “hey, maybe there’ll be another sale in the next 30 days and the iPhone 7 will be 75% off!”. Your reasoning is as follows:
• If the iPhone 7 is off by more than 50% in the next 30 days, you will buy that phone.
• Otherwise, you will go to the salesman at the end of the 30th day (just in case there’s a surprise sale at the last minute) and say, “Hey, I got an option to buy an iPhone 7 at 50% off. I want my
iPhone 7 and I want it now!”
That’s basically it. You can see right away that an option has a value. Let’s try to simulate the value of our option to buy an iPhone 7 at 50% off.
It would be: [Probability that the iPhone 7 is not more than 50% off in the next 30 days] * [Money you save with that 50% off coupon]
So if an iPhone is $1,000 and there’s a 90% chance the iPhone 7 will not be more than 50% off in the next 30 days, your coupon is worth:
90% * [$1,000 - $500] = $450
That’s it! The salesman just “gave you” $450.
Wait, I thought options were supposed to be complicated!?
Another way to see it is my local cinema: if you go there on Saturday, they give you a coupon for a movie at 50% off at any time, expiring in 30 days. But, you won’t go to the cinema just for the
hell of it: you’ll only go back if there’s a new movie that comes out that you want to see (assume that demand doesn’t change with price, for the purpose of this article). If the normal ticket is
$10, how much is that coupon worth? Well:
[Probability there’ll be a movie you want to see and that you’ll go back to the cinema in the next 30 days] * [$10-$5]
Get it?
Okay, time to introduce the last element of options (already!): interest.
This is a bit complicated and not really relevant (for now), but you should know that a dollar one month from now is worth less than a dollar today.
What would you prefer? Having a dollar today or having a dollar a month from now?
Yeah, thought so, but let’s push it a little bit:
What would you prefer? Having a dollar today or having $1,000,000 a month from now?
This is the basic idea of interest: that money can be carried through time. If you invest $1,000 at a bank and the money grows to $1,010 a year from now, the paid has paid you $10 in interest, or 1%.
Yes, I know it’s simple, but it’s meant to be. One day, I’ll write an entire article on interest, but for now, let’s stick with that.
Options are affected by interest too. Let’s take the iPhone 7 example in the beginning. I said the coupon was worth $450 because I assumed you would not invest it. But remember: you only have to pay
your 50% off iPhone when you buy it. So for the next 30 days, you can place your money in a bank account, earning interest!
Your thought process would be this:
• You buy an iPhone 7 for your mom. You get a 50% off coupon (option). You can buy an iPhone at 50% at any time during the next 30 days.
• The cost to buy the iPhone 7 at 50% off would be $500. $500 30 days from now is worth $495 today (i.e. if you place $495 in a bank account right now, you’ll have $500 in it in 30 days).
• If at any time during the next 30 days you see the iPhone 7 at more than 50% off, you will buy that phone and tear that coupon.
• Otherwise, at the end of those 30 days, you will withdraw the money from your bank account and buy that phone.
Thus, it only cost you $495 today to get that iPhone at most 30 days from now. It’s a gross simplification, but understand the general idea.
But I want my iPhone now!
We all do.
What about financial options?
OK, the main topic. Let’s go for a non-depreciating, intangible product. Instead of an iPhone, let’s talk about Apple shares. Apple shares are intangible and, unlike material object, no one is going
to offer a sale on shares. There is one and only one price for a stock: on the stock market.
Let’s say Apple is at $100 per share. You want to buy an option on Apple. How would it worth?
Well, first, let’s understand that there are two kinds of options: calls and puts.
Calls give you the right to BUY apple shares. Puts give you a right to SELL apple shares. Options typically worth in lots of 100, so one call on Apple would give you the right to buy 100 apple
shares. Pretty!
Calls = buy, Puts = sell. Okay!
When you exercise a call or a put, you use the option. It’s gone afterwards. If you exercise a call, you buy 100 shares of Apple. If you exercise a put, you sold 100 shares of AAPL. Note that you do
not need to own shares of Apple to exercise a put, i.e. sell them. If you had 0 shares of apple and exercised a put, you’d end up with -100 shares, a negative amount of shares. See: short selling.
So let’s say you buy a call on Apple, expiring in 30 days, at a price of $120. Options have a STRIKE, meaning the price at which they can be exercised. Here, the strike is $120, meaning you’ll have
to pay $120 to buy your AAPL shares. The strike can be anything you want; $5`0,000 if you feel like it.
Let’s analyse this call for a minute:
• You have the right, but not the obligation, to buy 100 shares of Apple at any time in the next 30 days
• You can buy those shares at $120 per share. Since there are 100 shares per lot, it would take you $120*100=$12,000.
That’s it. That’s a call.
But if AAPL currently trades at $100, I don’t want to buy it for $120!
Not now, that’s for sure, but let’s say that 15 days from now, AAPL jumped to $150 a share. Wouldn’t you like to be able to buy AAPL for $120 then?
An option is said to be in-the-money when it can be exerciced profitably. In our case, with AAPL at $100 and the strike at $120, it would certainly make no sense to exercise our call and buy AAPL at
$120 when we can just buy it at $100.
An option is said to be out-the-money when it cannot be exercised profitably, as is the case in the paragraph above.
An option is said to be at-the-money when the price equals the strike, i.e. if AAPL climbed to $120 and we had calls with a strike of $120, the call would be at-the-money.
Too complicated
It does get complex, but let’s go back to the question: how much is our call worth? Well, like our 50% off iPhone coupon earlier:
[Probability we will exercise our call in 30 days] * [Money we save when we exercise our call].
First, understand that it is never optimal to exercise our call before expiration, i.e. the last day. Like in our phone example above, you’ll wait as long as you can to use it. Why? Imagine AAPL goes
up to $125. You exercise your option and one second after after you buy it at $120, AAPL crashes to $115. If you had waited just one more second, you wouldn’t have exercised your calls and could have
bought it for $115.
It is never optimal to exercise an optimal before the last day when it’s due to expire
An astute reader might note that there are exceptions to the rules above, but they are very rare in practice. In 20 years of trading, I’ve never seen a single situation where it was optimal to
exercise an option before expiration (for those curious, with an option, you are not entitled to receive dividends. Say AAPL decides to pay a $50 dividend a week for now. You could choose to exercise
your option to buy the shares so you’d get the $50 dividend. It could also be optimal to exercise a put if interest rates are really, really high, but both of those scenarios haven’t happened in
decades, if ever. Theory stuff.).
But but but what if I buy a call, the stock goes up and I think the stock will go down next! Couldn’t I exercise my calls? I want to lock my profits!!!
Let me reformulate your question.
Let’s say AAPL is at $120. You buy a call with a $120 strike price for $1. AAPL jumps to $130 and your call is now valued at $15. You have gotten a 1,400% profit.
You think AAPL is going to drop and you want to take your profits on that call. Your question is, “Why not exercice the calls to buy 100 AAPL shares and then resell them?”
Well, you could do that, but you’d lose money: the price of a call depends on two elements: the exercise value, which is how much you’d get if you exercised it, and the time value. In the example
above, the exercise value is $130-$120=$10 (since our call is in-the-money by $10). But the call itself is valued at $15, not $10, because there are chances the stock could go even higher. The time
value of that call would be the price of the call minus the $10 above, thus $15-$10.
Thus, if you’re really convinced AAPL is going back down, you’d be reselling your calls for $15 and not exercising them to earn $10. Yes, you could exercise your calls then, but you’d be a moron.
Back to pricing…
Now, let’s go back to our formula. The value of a call is:
[Probability we will exercise our call in 30 days] * [Money we save when we exercise our call]
We will exercice our call only if AAPL > $120 in 30 days. If it’s at $119.9, we aren’t going to buy it at $120 when we could buy it at $119.9. Thus, the value of a call is:
[Probability AAPL > $120 in 30 days] * [Money we save when we exercise our call]
For the second bracket, let us consider that: how much money will we save? Well, we will buy it for $120, so we will save whatever difference there is to Apple’s stock price in 30 days. If it’s at
$121, we’ll save a dollar. If it’S at $200, we’ll say $80. Thus:
[Probability AAPL > $120 in 30 days] * max[0,Value of AAPL in 30 days - $120]
Note the max element because, again, we will not exercise our option if AAPL is under $120 in 30 days. Say AAPL is at $110 in 30 days, we won’t exercise our calls and we’ll just throw it away. Max
(a,b) means the maximum of a and b, so max(1,5)=5.
One last time: remember interest from earlier? Well, again, we are (possibly) buying AAPL 30 days from now. $120 a month from now is worth less today. Let’s say you could place $115 in a bank account
and have $120 a month from now. Thus, if you are exercising your call, you are buying AAPL for $115 since you only need $115 today to potentially buy it later.
Price of a call= Value today of: [[Probability AAPL > $120 in 30 days] * max[0,Value of AAPL in 30 days - $120]]
Woah, calls are amazing! But wait a minute… How can I know the value of AAPL in 30 days?
And that’s the true kicker, the punchline to the story: you can’t! Nobody can! And nobody can guess the probability AAPL will be over $120 in 30 days neither!
That’s what makes options fun! If I think the probability of AAPL being above $120 is high and if I think the value of AAPL in 30 days will be super high, I would be ready to pay a lot for that call!
But if I thought there were no chances AAPL will be above $120, then I wouldn’t pay much for that option.
Wait, where’s my pretty formula to calculate how much a call is price?
There are none.
But I heard something about Black-Scholes and… and…
Black-Scholes is one model used to price options. It’s simply a model that uses a set of assumptions to project potential stock prices. Truth be told, you could use any model you want to price your
options. Black-Scholes is popular due to its theoretical backing and ease of use. It’s an extremely effective model in 95% of cases, but a terrible model in the remaining 5% of cases.
Like all models, Black-Scholes has its strengths and flaws. But we still remain with the fact that nobody really knows how much an option is worth. The price of an option is set by the market, i.e.
buyers and sellers.
What other models are used to price options?
A common approach to pricing options is the binomial approach, which is, quite ironically, one of the most effective in statistical terms. Here’s the super simple:
AAPL is at $100. You want to buy a call at $120 expiring in a year. During the next year, AAPL can only do two things: go up to $150 or go down to $50, each with a 50% probability. The interest
rate is 0%. What is the value of that call?
Well, the call won’t be exerciced if AAPL goes down to $50 since the price would be under our strike price ($50<$120). It will only be exerced if AAPL goes up to $150. Since there is no interest
rate, the price of the call should be:
50% * ($150-$120)=$15
And our call is priced at $15!
This example is borderline an undergrad-level financial problem. I’m not joking. Of course, our models assumes that AAPL can only go to $150 or $50, but all models make assumptions.
It can be shown that the binomial approach eventually becomes the Black-Scholes model under some circumstances. Personally, I like the mesh approach, which is well beyond the scope of this guide.
Lastly, note that there are financial laws and theorems that determine options prices and that sound bound can be establish to limit their value. For instance, for the same expiration T, the same
strike K and the same continually compounded, risk-free interest rate r:
Price of the call - Price of the put = Price of the Underlying Stock - K*exp(-rt)
This is the put-call parity, it is based on the risk-neutral approach (also, as some readers might note, on arbitrage constraint). Don’t fret if you didn’t get all that, this is grad school financial
stuff, but you can see there is a relationship between the price of a put and the price of a call.
What’s the catch?
Let’s say you buy a call. It means someone, somewhere sold it to you. In other words, someone gave you the right to buy AAPL at $120.
That person won’t do that for free.
I mean, nobody is going to give you something for free, in finance the last place of them all. That person will expect to get compensated. You will have to pay that person. All options carry a
premium that the buyer pays and the seller receives (minus a commission).
The more likely the option is to be “in the money”, the higher the premium will be. The higher it can go, the higher the premium will be. Stocks that tend to move a lot tend to have higher option
premiums because people expect it to move a lot and thus potentially be worth a lot more if the stock moves in the right direction.
Compare AAPL with Boring Inc. Both companies trade at $100 and you buy options with a strike of $120. If AAPL beats earnings, it can jump to $140 per share. Thus, your option would be worth $140-$120
($120 is the strike) * 100 shares per lot = $2,000. Compare that to Boring Inc. If Boring Inc jumps, it will only jump to $125. Thus, your option would be worth ($125-120) * 100 = $5,000, a lot less.
So the call will be priced at a lower price.
Of course, nobody knows for sure what’s going to happen and Boring Inc. might jump to $200 for all we know, but people use models to predict potential moves. If Boring Inc. has never moved bigly in
the past, they assume it’s very unlikely it will move bigly in the future. Again, all models have assumptions. Those models sometimes fail and funds go bankrupt all the time.
One last question: what if I buy options, they’re in-the-money and I forget all about them?
Options that are in the money by $0.01 or more are automatically exercised at the date of expiration. Typically, that’s Friday 4:30PM, after market closes. Yes, magic! So if you have a call at a $120
strike and AAPL is at $120.01 and the market closes, congratulations, you just bought 100 shares of AAPL for $120 per share. You can also tell your broker, “hey man, I know my calls are in the
money, but you know what, I don’t really want AAPL shares in my portfolio and $0.01 is not enough. Plus, it might fall over the week-end. Please don’t exercise those calls, m’kay?” Fun fact: a major
source of conflict happens at options expiration when there’s a misunderstanding and major lawsuits and complaints happen against brokers all the time. “I thought you said exercise half! I thought
there was a week left on them! I thought I rolled them to next week!” etc. Like casinos, brokers almost always win those complaints.
About the seller of calls…
Yes, you can sell options. You can buy and sell options just like you buy and sell shares (see: Short Selling). I buy and sell options all the time.
Say you sell AAPL calls at $120 strike expiring a year from now. I receive $10 for those calls. If a year from now AAPL is at $119.99, congratulations, I get to keep the $10 for ABSOLUTELY FREE!
FREE MONEY!!!!!
Remember: once a call is expired, it’s GONE GONE GONE.
That’s it?
That’s it. People buy options because they provide more leverage. For instance, let’s say AAPL was worth $100. If you wanted to buy 100 shares, you would need $10,000. If AAPL went up to $150, you
would make 100*($150-$120)=$3,000. You have earned $3,000 from an investment of $10,000, a 30% return.
Let’s say instead you paid $2 for some $120 calls on apple. Options are on a lot of 100, so you would only have to pay $200 today. If AAPL jumped to $130, you would earn 100*($130-$120)=$1,000, a
669% return on your money.
Woah! Why don’t everyone buy options?
Because once they expire, they are GONE GONE GONE! You can’t go back and say, “You know what guys, I think I changed my mind. I don’t want this call after all, please give me my money back.” NOPE! I
do sell calls all the time and by the time the call expires, usually, I’ll have the money already spent somewhere.
In the example above, if you buy shares instead and hold them and they go to $119.99, you just earned 100 * ($119.99-$100= $1,999, a 19.99% return
If you bought options instead, you just earned 100 * max (0, $119.99-$120)=0, a -100% return.
So options are more risky?
You have no idea how painful it can be when the clock is ticking and your options are barely out-the-money. Similarly, you have NO IDEA how painful it is when you sell calls and the price goes up day
after day after day after day… Say you sold a call at a $100 strike for $5 expiring in 30 days. The stock is at $90 on day 1, then $95 on day 2, $98 on day 3, $101 on day 4, $108 on day 5… you have
no idea how painful it is emotionally. You can literally bankrupt yourself overnight by selling options and in fact, several people did precisely that. I still remember the day I sold a massive
amount of Facebook puts at $80 and every ten minutes, I had lost another thousand of so.
But options can also be used to mitigate risk. Say you bought AAPL for $100 and it goes up to $120. You are afraid it’s going back to $100. You could buy puts with a $120 exercice price. If AAPL went
down to $100, you would lose your gain on your AAPL shares (you paid $100 and they are now at $100), but you would earn 100 * ($120-$100)= $2,000 on your puts. Less, of course, the premium you paid
for your calls. Say those calls are $5. Since there are 100 shares in a lot, you paid $5*100 and your total gain is now $2,000-$500 = $1,500.
Why not sell the shares directly if you think they are going to fall?
Because you would be forced to pay taxes on your capital gains.
Welcome to the wonderful world of finance.
There are other reasons, of course, but… That’s enough for one day!
2 Responses to A Super Simple Introduction to Options (Calls, puts)
1. MB January 31, 2017 at 3:05 pm #
I think you mean “$500”?
Thus, your option would be worth ($125-120) * 100 = $5,000, a lot less. So the call will be priced at a lower price.
2. wt January 31, 2017 at 3:46 pm #
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Risk Assessment
During the Risk Assessment phase of this project, you are asked to evaluate:
• What are the risks and rewards if the solution is implemented and results in the desired outcome?
Consider the potential amount of loss and the probability of the loss occurring for each of the solutions to your problem(s) you identified during Week Three if each of the solutions are implemented
and the outcome is the desired result.
• What are the risks and rewards if the solution is implemented but does not result in the desired outcome?
Consider the potential amount of loss and the probability of the loss occurring for each of the solutions to your problem(s) you identified during Week Three if the solution does not result in the
desired outcome. This is done for each risk associated for each option.
• What are the risks and rewards if nothing is done (continuing with the status quo)?
Risk Assessment based on last paper of identifying solutions. See attached. | {"url":"https://therealwriters.com/risk-assessment-7/","timestamp":"2024-11-06T23:42:38Z","content_type":"text/html","content_length":"26998","record_id":"<urn:uuid:539635ff-abfc-4ecb-8b4b-e886f748fc68>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00324.warc.gz"} |
7 Top Operators in C programming Language - ZareenAcademy
Operators in C programming Language
Operators in C programming language are fundamental components that enable developers to perform various operations on operands. These operations can range from simple arithmetic calculations to
complex logical manipulations. Understanding the different types of operators and how they work is essential for writing efficient and concise C code. In this comprehensive guide, we’ll explore the
different categories of operators in C, their usage, and examples to illustrate their functionality.
Arithmetic Operators
Basic mathematical operations, including addition, subtraction, multiplication, and division, are carried out using arithmetic operators. These operators work on numeric operands and produce numeric
results. The following are the arithmetic operators in C:
Addition (+)
Subtraction (-)
Multiplication (*)
Division (/)
Modulus (%)
For example:
int a = 10;
int b = 5;
int sum = a + b; // sum = 15
int difference = a – b; // difference = 5
int prod = a * b; // prod = 50
int quotient = a / b; // quotient = 2
int remainder = a % b; // remainder = 0
Relational Operators
Relational operators are used to compare the values of two operands. They return a boolean value (true or false) based on the comparison result. Relational operators are often used in conditional
statements and loops to control program flow. The following are the relational operators in C:
Equal to (==)
Not equal to (!=)
Greater than (>)
Less than (<)
Greater than or equal to (>=)
Less than or equal to (<=)
For example:
int x = 10;
int y = 5;
if (x > y) {
printf(“x is greater than y\n”);
Logical Operators
Logical operators are used to perform logical operations on boolean operands. They are typically used to combine multiple conditions and determine the overall truth value of a compound expression.The
logical operators in C are as follows:
Logical AND (&&)
Logical OR (||)
Logical NOT (!)
For example:
int age = 25;
if (age >= 18 && age <= 30) {
printf(“You are a young adult\n”);
Bitwise Operators
Bit-level operations on integer operands are carried out using bitwise operators. They manipulate individual bits within the binary representation of numbers. Bitwise operators are commonly used in
low-level programming and device driver development. The following are the bitwise operators in C:
Bitwise AND (&)
Bitwise OR (|)
Bitwise XOR (^)
Bitwise NOT (~)
Left shift (<<)
Right shift (>>)
For example:
unsigned int x = 5; // Binary representation: 0000 0101
unsigned int result = x << 2; // Left shift by 2 bits: 0001 0100 (20 in decimal)
Assignment Operators
Variable values are assigned using assignment operators. To carry out an operation and assign the outcome to a variable, they mix the assignment (=) operator with bitwise, arithmetic, and other
operators. The assignment operators in C are as follows:
Simple assignment (=)
Addition assignment (+=)
Subtraction assignment (-=)
Multiplication assignment (*=)
Division assignment (/=)
Modulus assignment (%=)
Bitwise AND assignment (&=)
Bitwise OR assignment (|=)
Bitwise XOR assignment (^=)
Left shift assignment (<<=)
Right shift assignment (>>=)
For example:
int x = 10;
x += 5; // Equivalent to x = x + 5
Miscellaneous Operators
C programming language also provides several miscellaneous operators for special purposes:
sizeof operator: Provides a variable’s or data type’s size in bytes.
Conditional operator (ternary operator): Provides a compact syntax for conditional expressions.
Comma operator: Evaluates multiple expressions sequentially and returns the value of the last expression.
For example:
int size = sizeof(int); // size = 4 (on a typical 32-bit system)
int max = (x > y) ? x : y; // max is assigned the larger of x and y
Operators are essential building blocks of C programming language, enabling developers to perform a wide range of operations on operands. By understanding the different categories of operators and
their usage, developers can write efficient and concise C code for various applications. Mastery of operators is crucial for becoming proficient in C programming and developing robust and scalable
software solutions.
This comprehensive guide has covered arithmetic, relational, logical, bitwise, assignment, and miscellaneous operators in C, providing examples to illustrate their functionality and usage. Armed with
this knowledge, developers can confidently tackle programming challenges and create high-quality C programs.
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EViews Help: @val
Number from a string.
Syntax: @val(arg[, fmt])
arg: string
fmt: (optional) numeric format string
Return: scalar
Returns the number that string arg represents. You may provide an optional numeric format string fmt.
In most cases, EViews will be able to convert your string into the corresponding numeric value without additional input. If EViews is unable to perform a conversion, it will return a missing (NA)
There are a few important conventions used in the conversion process:
• A leading “$” in the string will be ignored.
• Strings enclosed in “( )” or with a leading “–” will be treated as negative numbers. All other numeric strings, including those with a leading “+” will be treated as positive numbers. You may not
have a leading “+” or “–” inside of the parentheses.
• A trailing “%” sign instructs EViews to treat the input string as a percentage—the resulting value will be divided by 100.
There are some situations where you must provide a numeric format string so that EViews can properly interpret your input. The syntax for the format string depends on the type of number the string
Real-Value Formats
EViews will properly interpret non-delimited decimal and scientific notation numeric input strings as numbers.
If your string uses “,” to separate thousands, you should specify the “t” format string to remove “,” delimiters prior to conversion. If the string uses “.” to separate thousands, you should use
“t..” to instruct EViews to remove “.” delimiters.
If your input string represents a number with suppressed decimal format, you should include a format string beginning with the letter “s”:
s.precision suppressed decimal point format (precision determines the number of digits to the right of the decimal)
EViews will divide the resulting number by 10 raised to the power of the specified precision. The “s” format specification may be followed by a “t.” or a “t..” specification if necessary.
Integer Formats
r ratio (e.g., “30 1/5”).
i integer
h hexadecimal
o octal
b binary
You should use the “r”, “h”, “o”, or “b” formats to indicate that your input is in the specified format. The “i” format is generally not necessary unless you wish to produce a missing value for a
non-integer input string.
scalar num = @val("$1.23")
assigns the scalar NUM the numeric value 1.23.
series ser1 = @val("-$123.88")
returns the value -123.88.
scalar sperct = @val("478%")
divides the value by 100, setting the scalar SPERCT to 4.78.
scalar sratio = @val("(321 1/5)", "r")
sets the scalar SRATIO equal to -321.2
scalar shexa = @val("f01a", "h")
treats the string “f01a” as a hexadecimal number, converts it into the decimal equivalent, 61466, and assigns it to the scalar object SHEXA.
scalar sbin = @val("11110101", "b")
interprets the string “11110101” as a binary number, converts it into the decimal equivalent, 245, and assigns it to the scalar SBIN.
To ensure that a value is an integer, you may use the “i” option.
scalar sintna = @val("98.32", "i")
scalar sint = @val("96", "i")
SINTNA will contain a missing value NA since the input represents a non-integer value, while SINT is set to 96.
You may use @val to convert values in an svector into a vector. The matrix command,
vector v = @val(sv1)
converts the string values of svector SV1 to numeric values and returns the values in the svector V. If the vector V exists it will be sized to match the rows of SV1 and non-numeric strings will be
converted to NA.
The series command
series x = @val(alpha1)
converts the string values in the alpha series ALPHA1 to numeric values and returns the values in the series X. Non-numeric strings will be converted to NA.
Format strings may be used to govern the conversion,
vector vbin = @val(svbin, "b")
interprets the strings in the svector SVBIN as binary numbers, converts it into their decimal equivalents and assigns it to the vector VBIN. If for example, SVBIN contained “110” “001” and “010”, the
resultant VBIN will contain 6, 1, and 2.
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Fraction interactives seminar
Steve O'Connor
Dec. 3, 2011, 3:50 p.m.
As a fifth grade teacher in New York State, I am interested in aligning my fraction instruction with the Common Core State Standards. With 45 states having adopted the standards, I feel that it is
important to keep the new standards in mind. I have been working my way through several relevant readings on the topic.
I look forward to working on this project in general.
Linda FS Dec. 4, 2011, 2:30 p.m.
In Reply To: Steve O'Connor Dec. 3, 2011, 3:50 p.m.
I agree that we must keep in mind the common core standards. And in fact the standards themselves about fractions are reasonable. But I notice that the Draft Progressions is brand new (August 2011).
When the first drafts of the Common Core came out 4-5 years ago, there were pages with examples and then these pages were abruptly removed from the internet. (I had written several responses with
details about some of the inadequacies of the examples for the algebra: expressions and equations so I still have that set.)
Now I am looking at this Draft Progressions and I see several similar problems. It seems to me that the people who write these things have never taught kiddies in the classroom (I myself have never
taught young ones - I get the older kids who are the result of this teaching and I simply start over.)
My complaints about the Draft Progressions after a brief look at it.
(a) There is no verbalization. Teaching is done verbally - particularly fractions. Any explanation should include how to actually talk about fractions. Two years ago, I made a video about my pet
peeve about people talking fractions: http://youtu.be/xOQWmLGPmWQ , i.e. how do you say "1/4". Many people say "one over four". What does that mean? "One-fourth" is a picture in my head; "one over
four" is 3 words.
(b) I also looked at the pictures in this draft progressions. They take fractions out of context. We teach kids at a very early age that we cannot compare an apple to an orange. They understand this
and they understand that if we have 2 apples and 2 oranges that we DO have the same number of each type of fruit but we DON'T have the same fruit. But apparently a child will compare 1/4 of a giant
pizza to 1/2 of a small pizza and conclude that 1/4>1/2 (see page 4). I simply cannot believe this. (I might believe they think 1/4>1/2 since 4>2, but not from a silly pizza example - please correct
me if I am wrong.)
From this - in the very next paragraph - it goes on to explain how important it is to compare 2/5 > 2/7 (which indeed is a much, much harder concept) and use this example in SPITE of the fact that
the standards specifically state: Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.
Then this draft continues onto 4th grade and immediately they go on to the stuff that (I think) Sue was talking about: complicated expressions "over" complicated expressions.
There is no way to understand this reasoning in terms of defining fractions as parts of a whole and it is easy to see exactly where we lose them. First we lose the teachers, who then lose the
(The draft progression actually mentions presenting to 4th graders
$\frac{28}{36}=\frac{28\div 4}{36\div 4}=\frac{7}{9}$
which (I think) is ridiculous at that age and again in SPITE of the fact that the standards specifically state: Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4,
5, 6, 8, 10, 12, and 100.)
So while keeping the standards in mind, we need to focus on teaching methods that really communicate what fractions are and how they help us function; teaching methods that work in the classroom and
have been used by real teachers teaching grades 3 and 4.
Steve O'Connor Dec. 4, 2011, 5:31 p.m.
In Reply To: Linda FS Dec. 4, 2011, 2:30 p.m.
I understand your concerns with the draft progressions. That being said, they are drafts (In fact the blog in which it was posted referred to the fractions progression as being a drafty draft) . Let
me address some of your points.
(a) There is no verbalization. Teaching is done verbally - particularly fractions. Any explanation should include how to actually talk about fractions. Two years ago, I made a video about my pet
peeve about people talking fractions: http://youtu.be/xOQWmLGPmWQ , i.e. how do you say "1/4". Many people say "one over four". What does that mean? "One-fourth" is a picture in my head; "one over
four" is 3 words.
I agree correct verbalization is critical in math. It may be that the writers of these progressions make the mistaken assumption that practitioners are disciplined in their use of proper mathematical
language. This is a hazard of the standards having been largely written by those removed from the classroom. Imprecise and inaccurate use of terminology is rampant in elementary math. Teachers must
be conscious of the dangers of slipping into the vernacular.
Hung-hsi Wu's (One of the contributors to the progressions) Understanding Numbers in Elementary School Mathematics gives a great deal of attention to precision in speaking in mathematics instruction.
(b) I also looked at the pictures in this draft progressions. They take fractions out of context. We teach kids at a very early age that we cannot compare an apple to an orange. They understand this
and they understand that if we have 2 apples and 2 oranges that we DO have the same number of each type of fruit but we DON'T have the same fruit. But apparently a child will compare 1/4 of a giant
pizza to 1/2 of a small pizza and conclude that 1/4>1/2 (see page 4). I simply cannot believe this. (I might believe they think 1/4>1/2 since 4>2, but not from a silly pizza example - please correct
me if I am wrong.)
I agree that the pizza illustration may not have been the best illustration of the importance comparing the same whole. I think most students in grade 3 could tell me why this illustration is a
fallacy. (I can test this by asking some third graders). This illustration appears to be an over simplification of the pitfall discussed by Wu in pp. 193-195 of Understanding Numbers.
The use of examples that are outside of the domain of denominators reflects a certain inattention to the standards (this is a draft); nonetheless, I still believe there is value to what is being
presented. I think that examples may have been pulled from other works without revising them to comply with the common core.
I cannot speak to fourth grade, but I can certainly say that examples such as 28 ÷ 4/36 ÷ 4 are within the realm of what is done currently in (my) fifth grade, and I am positive that that similar
examples using multiplication (and perhaps division too) to create equivalent fractions are routinely used in grade four.
I don't know exactly who wrote this progression. It is part of a project from the University of Arizona: http://ime.math.arizona.edu/progressions/ , and Wu is a writer. There is interesting active
discussion of this document where it was posted: http://commoncoretools.wordpress.com/2011/08/12/drafty-draft-of-fractions-progression/. You might consider adding your legitimate points.
In any case, I now view these materials more critically, and I am spending more time with Wu's Understanding Numbers in Elementary School. If anyone is interested, the book is currently discounted at
the American Mathematical Society's website: http://www.ams.org/bookstore-getitem/item=MBK-79
suehellman Dec. 5, 2011, 8:55 a.m.
In Reply To: Linda FS Dec. 4, 2011, 2:30 p.m.
Linda, after reading your post and viewing the video, several points come to mind:
1) I don't think there is enough verbalisation by students in math classes PERIOD. They get used to sort of distancing themselves verbally from what's on a math page and float on or swim in or drown
in a sea of numbers bereft of language. I think that problemtic way of referring to a fraction comes from trying to make up a language that works inside weird kind of 'math think' that they get into.
The concepts never get anchored in real language.
I used to do something called 'words first' in which students had to be able to say a process for solving a problem without using any specific numbers. They had to be able to generalise processes and
mix math vocabulary and real language to express themselves. So given a question like the one in your video, they would have had to explain they why's and how's without numberical what's. At first
they felt and acted like mathematical 'Helen Kellers' -- struggling to get out even the simplest of real sentences. Eventually they started to turn their ability to verbalise back on and the language
flowed more easily. New neuro-research shows that math learning and language are closely associated in the brain --so I think I was on the right track.
I think that to get people to be fluid and flexible in math, they have to be able to talk themselves through experiementing with different problem-solving approaches. This can't happen easily if they
are not used to conceptualising math in real language. People talk about 'number sense' as being intuitive, but I believe that a lot of our intuitions come in bundled in language. Unless one is so
steeped in deep understanding of math that one lives in a math landscape, thoughts, flashes of ideas, the abilty to switch streams are partly a function of the ability to give language to insights.
We keep dream journals. We write down ideas to come back to at another time. How can kids get close to being comfortable inside math if they leave their language sklls at the door?
2) Your unique perspective (you see the end result of years of math learning) is very helpful. Math mislearning it seems has a coninuum as does math learning.
3) Trying to create a math course for students I will not be teaching is a challenge that is bringing a lot of these kinds of insights to light.
4) Pushing children into doing overly complex questions too early in a way forces the teacher and the kids into memorizing processes before the concepts are fully grasped. We hope that as they get
older their understanding of why they are doing the processes will magically mature, but the early 'why' learning is just let go by brains concentrating on what is the apparent higher priority --
storage of rote learning.
The brain prunes neurons that contain learning which is not used and reused and added to, so if we think that ensuring understanding of size and number of pieces (denominator and numerator) in an
early grade ensures continued knowledge of that relationship, we're fooling ourselves. This is where 'following the standards' can defeat math understanding -- because early learning that is not
deemed as useful by a brain is pruned. Recylcling through the fundamentals is the only thing that will keep them in kids' heads (literally).
Also if a concept is understood at one time but then incorrectly or only partially reactivated at a later time (eg.during review), the neural architecture that housed the original correct learning
will be physically destroyed and new structures that embody the wrong or incomplete learning will grow instead. Unless the fundamental understandings are revisited SUCCESSFULLY (for every kid; not
for the teacher) several times in a year and then many years in a row, they will be lost or replaced and show up in your adults as so much mathematical swiss cheese.
Linda FS Nov. 30, 2011, 3:37 p.m.
1. Warm regards to all!
2. Neither deep nor profound, but I try desparately to get my kids to use decimals whenever possible. However, I have yet to find a way to solve word problems involving "work" without fractions. But
I don't call them fractions. I call them "the part of the work done per unit time" and I never require LCM just ACM (any common multiple). Does anyone know an easier way to explain work problems?
e.g. YouTube demo: http://youtu.be/tOD9rpn_7fY
It should be visible on a smartphone and for my amusement I will try to insert a QR code (Unfortunately in the video I forgot to change my 1's for US l's - They all are "flagged" like this type :) )
suehellman Nov. 30, 2011, 11:28 a.m.
Some of the reading I've been doing lately has to do with
(1) learners being able to transition from relying in meaning and metaphor to being able to accept fractions as numbers in their own right an manipulate them with ease. Little kids learn that 1 + 4
= 5 by counting concrete objects, but eventually the quantities turn into number objects that can be maniputated without the kids having to constantly count or reconnect with the meaning.
(2) teachers confusing dressing up traditional problems in real world clothes (along the line of the Emperor's New Clothes) with creating authentic problems that are real to the students.
I want to create metaphors and learning experiences to help students connect wiht the concepts and skills on a more intuitive level, but I also want them to make that transition to being able to
handle the procedures flexibly. I believe that the first can lead to the second, but that for all kids this is not an automatic leap. They have to be guided across that chasm.
I'm interested in exploring the metaphors and authentic experiences that can give meaning to concepts many of of take for granted.
Interesting article -- http://math.berkeley.edu/~wu/NCTM2010.pdf
Maria Droujkova Nov. 30, 2011, 7:27 a.m.
I would like to focus on deep, profound and modern definition of "fractions." I see it as closely connected to ratios and proportions. Fraction operations need to be de-emphasized, and proportional
reasoning (with fraction notation) expanded.
karen Nov. 30, 2011, 10:45 a.m.
In Reply To: Maria Droujkova Nov. 30, 2011, 7:27 a.m.
I'll be interested to hear how this fits in with the open math textbook materials that are there now. Is this an added section or does it call into question the whole structure of the unit?
MrSteve Nov. 30, 2011, 12:18 a.m.
My Initial Thoughts:
1. Would like to focus not just on multimedia and interactive materials, but concrete physical things kids can do/construct and play with as well
2. Also ideas for students and teachers on how to use the "interactive materials" like a lesson plan and questions to get kids engaged and thinking, so it is more about kids discussing and
exploring, rather than us delivering our vast fraction knowledge from on high :) | {"url":"https://courses.p2pu.org/en/groups/fraction-interactives-seminar/content/getting-started/","timestamp":"2024-11-14T07:32:08Z","content_type":"text/html","content_length":"60129","record_id":"<urn:uuid:7c4f237b-86d0-45fa-850f-ce62d18ff429>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00409.warc.gz"} |
Publications Search
This report describes research and development of the large eddy simulation (LES) turbulence modeling approach conducted as part of Sandia's laboratory directed research and development (LDRD)
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A ball moving with velocity 2 ms^(-1) collides head on with another st
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How to Do Break-Even Analysis in Excel
Last Modified: October 17, 2024 - 6 min read
Nikesh Vora
Break-even analysis is an important tool when evaluating the sales volume or level of business activity needed to become profitable. Break-even analysis can be done in multiple software solutions,
including Microsoft Excel, a popular platform for business owners and financial analysts.
Let’s explore how it works!
Step-by-Step Guide: Break-Even Analysis in Microsoft Excel
In order to perform a break-even analysis in Excel, you must first compile your product’s sales price, variable costs, and fixed costs. These components are the inputs that will determine your
break-even point.
• Sales price is the per unit price you will charge customers for your product.
• Variable costs are expenses that change in proportion to how many units are produced and then subsequently sold. These typically include the materials and direct labor involved in producing the
• Fixed costs are expenses that don’t change regardless of how many units are produced and subsequently sold. An example of this would be rent expense. No matter how many units are produced,
your monthly rent expense for the building will remain the same.
Step 1: Set Up Your Environment
Set up your Excel sheet with your sales price, variable costs, and fixed costs that you can reference to build out your break-even analysis.
Step 2: Create a Data Table for Units Sold
Set up your worksheet to include intervals of units that you would anticipate selling as part of your business. This will vary depending on the gross margin (sales price minus variable costs) and
how significant your fixed costs are.
Typically, items that sell for a lower price per unit will require more units to break even.
Step 3: Build the Break-Even Formula
After you’ve outlined the number of units you anticipate selling, build a formula in Excel to calculate the net income at the volumes that you outlined.
Use this break-even point formula: =((Sales Price – Variable Costs) x Units Sold) – Fixed Costs
When setting up this formula, use a $ between the column letter and the row number (see the screenshot below). This locks in the row referenced for the sales price, variable costs, and fixed
costs so the formula can be copied down for each units sold quantity without manual changes.
If you have significant variable expenses and/or fixed costs, use a SUM function to total all expenses in each category.
Step 4: Calculate Net Income at Different Sales Volumes
After you’ve populated your formula you can copy it down by dragging the bottom right corner of the cell down to the other units sold amounts. This will allow you to approximate your break-even
point and assess profitability at different sales volumes.
In this example, the break-even point is between 3,000 and 3,500 units produced.
Step 5: Prepare for Goal Seek
To get the exact break-even point in units, you’ll use the Goal Seek function. The Goal Seek function allows you to get the correct input value, in this case units sold, when the desired output
value is known, in this case the break-even point.
Copy the formula from Step 3 down to a row without any units sold input.
Step 6: Access the Data tab
Once this has been done navigate to the Data tab in Excel.
Step 7: Use the What-If Analysis Tool
Within the Data tab, in the Forecast section, find the drop-down labeled ‘What-If Analysis’. Click this dropdown and select ‘Goal Seek’.
Step 8: Set Up Goal Seek
Once Goal Seek has been selected, three prompts will pop up (set cell, to value, and by changing cell), as outlined below.
“Set cell”
For this prompt, you will need to select the profit cell that you copied down in step 5 (in this example E11).
We are trying to calculate how many units we need to produce break-even, which means producing a net income/profit of $0.
“To value”
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For this prompt you will want to input 0. When performing a break-even analysis, you are calculating how many units you would need to produce and sell to break-even or have $0 in net income.
“By changing cell”
For this prompt you will need to select the units sold cell (D11). Â The profit in cell E11 is calculating based off the units sold in this cell.
Once you’ve filled out the three prompts in the Goal Seek function select the OK button.
Step 9: Interpret the results
After you select’ OK’ Excel will calculate the break-even point. In this example, the number of units needed to break-even would be 3,400.
Next Steps: Performing Sensitivity Analysis in Your Break-Even Excel Model
After calculating the break-even point, perform sensitivity analysis within your Excel financial model to see how changes in inputs affect your break-even point in units.
Examples include:
• Adjusting your sales price: If you increase or decrease your price per unit by $5, recalculate how many units you need to sell to break even.
• Modifying variable costs: Determine how negotiating lower raw materials costs or increasing direct labor costs by $1 per unit affects your break-even point.
• Changing fixed costs: Calculate how reducing rent by $1,000 or adding a new facility impacts the number of units needed to break even.
You can analyze these scenarios individually or combine multiple changes for comprehensive what-if analysis.
Use Cases for Break-Even Analysis:
A break-even analysis determines the break-even point – where total revenue equals total costs, resulting in zero profit or loss. It’s valuable for:
• Calculating the minimum sales volume needed for profitability when starting a new business or expanding an existing one.
• Evaluating your selling price, variable expenses, and total fixed costs to identify products requiring price increases or areas for cost reduction.
• Assessing the viability of adding a new product to your current portfolio.
Break-even analysis is a crucial tool for business planning, financial modeling, and performing cost-volume-profit analysis (CVP) to make informed decisions about pricing, costs, and sales targets.
Improve Your Break-Even Analysis with Live Data
Break-even analysis in Excel is a powerful tool for business planning and financial modeling. By following this guide, you can make informed decisions about pricing, costs, and sales targets.
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PG Calculator is an excellent replacement for a standard calculator.
PG Calculator works in Algebraic and RPN modes.
PG Calculator is available in three different versions: Free, Standard and Pro.
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Bonus: Efficient ways of determining whether a number is a power of two - James Kitchen Games
Bonus: Efficient ways of determining whether a number is a power of two
My solution to BinaryGap provides a straight forward way to determine if an integer is a power of two: by looping through its binary representation there should be only one ‘1’. But is there a way to
determine this without looping?
Solution 1: Decrement and compare
If you are familiar with subtracting binary numbers from each other, then you will know that the lone ‘1’ bit in a power of two is replaced by a ‘0’ and all lower bits become ‘1’. Therefore if a
number is not zero and is a power of two, by taking the bitwise AND of x and (x-1), we should get zero, as illustrated below:
if( ( 0 != x ) && 0 == ( x & ( x-1 ) ) ){ return true }
Solution 2: Complement and compare
Two’s complement means to take the complement of a number and then add one. It is used to represent integers and their negative equivalents without incorrectly representing zero as a negative that
can happen with one’s complement.
Therefore if a non-zero number is a power of two, then the bitwise AND of a number x and it’s twos complement (~x+1) representation will equal the original number x.
This is because, as shown in the above table, the arithmetic of adding 1 to a power of two’s complement shifts all numbers below the lone ‘1’ bit which will be ‘0’ after taking the complement to zero
and putting the bit back to ‘1’ again.
if( ( 0 != x ) && x == ( x & (~x + 1 ) ) ){ return true; }
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Merge Sort | Data Structure | Geekboots
Fri Apr 23 2021
Merge Sort
A natural approach to problem solving is divide and conquer. In terms of sorting, a simple way to do this would be to split the list in half, sort the halves, and then merge the sorted halves
together. This is the idea behind Merge sort.
Merge sort is one of the simplest sorting algorithms conceptually, and has good performance both in the asymptotic sense and in empirical running time. Surprisingly, even though it is based on a
simple concept, it is relatively difficult to implement in practice.
In sorting n objects, merge sort has an average and worst-case performance of O(n log n). If the running time of merge sort for a list of length n is T(n), then the recurrence T(n) = 2T(n/2) + n.
File Name: merge-sort-algorithm.c
function merge(left,right)
var list result
while length(left) > 0 or length(right) > 0
if length(left) > 0 and length(right) > 0
if first(left) <= first(right)
append first(left) to result
left = rest(left)
append first(right) to result
right = rest(right)
else if length(left) > 0
append first(left) to result
left = rest(left)
else if length(right) > 0
append first(right) to result
right = rest(right)
end while
return result | {"url":"https://www.geekboots.com/datastructure/merge-sort","timestamp":"2024-11-04T14:19:48Z","content_type":"text/html","content_length":"903539","record_id":"<urn:uuid:cfdc9b7f-8376-40f4-bb0b-59930a3a073f>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00643.warc.gz"} |
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the
ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle (the hypotenuse), and the cosine is the ratio of the length of the adjacent leg to that
of the hypotenuse. For an angle ${\displaystyle \theta }$, the sine and cosine functions are denoted as ${\displaystyle \sin(\theta )}$ and ${\displaystyle \cos(\theta )}$.
Sine and cosine
General definition {\displaystyle {\begin{aligned}&\sin(\theta )={\frac {\textrm {opposite}}{\textrm {hypotenuse}}}\\[8pt]&\cos(\theta )={\frac {\textrm {adjacent}}{\textrm {hypotenuse}}}\\[8pt]\
Fields of Trigonometry, Fourier series, etc.
The definitions of sine and cosine have been extended to any real value in terms of the lengths of certain line segments in a unit circle. More modern definitions express the sine and cosine as
infinite series, or as the solutions of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers.
The sine and cosine functions are commonly used to model periodic phenomena such as sound and light waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, and
average temperature variations throughout the year. They can be traced to the jyā and koṭi-jyā functions used in Indian astronomy during the Gupta period.
Elementary descriptions
Right-angled triangle definition
For the angle α, the sine function gives the ratio of the length of the opposite side to the length of the hypotenuse.
To define the sine and cosine of an acute angle ${\displaystyle \alpha }$ , start with a right triangle that contains an angle of measure ${\displaystyle \alpha }$ ; in the accompanying figure, angle
${\displaystyle \alpha }$ in a right triangle ${\displaystyle ABC}$ is the angle of interest. The three sides of the triangle are named as follows:
• The opposite side is the side opposite to the angle of interest; in this case, it is ${\displaystyle a}$ .
• The hypotenuse is the side opposite the right angle; in this case, it is ${\displaystyle h}$ . The hypotenuse is always the longest side of a right-angled triangle.
• The adjacent side is the remaining side; in this case, it is ${\displaystyle b}$ . It forms a side of (and is adjacent to) both the angle of interest and the right angle.
Once such a triangle is chosen, the sine of the angle is equal to the length of the opposite side divided by the length of the hypotenuse, and the cosine of the angle is equal to the length of the
adjacent side divided by the length of the hypotenuse: ${\displaystyle \sin(\alpha )={\frac {\text{opposite}}{\text{hypotenuse}}},\qquad \cos(\alpha )={\frac {\text{adjacent}}{\text{hypotenuse}}}.}$
The other trigonometric functions of the angle can be defined similarly; for example, the tangent is the ratio between the opposite and adjacent sides or equivalently the ratio between the sine and
cosine functions. The reciprocal of sine is cosecant, which gives the ratio of the hypotenuse length to the length of the opposite side. Similarly, the reciprocal of cosine is secant, which gives the
ratio of the hypotenuse length to that of the adjacent side. The cotangent function is the ratio between the adjacent and opposite sides, a reciprocal of a tangent function. These functions can be
formulated as: {\displaystyle {\begin{aligned}\tan(\theta )&={\frac {\sin(\theta )}{\cos(\theta )}}={\frac {\text{opposite}}{\text{adjacent}}},\\\cot(\theta )&={\frac {1}{\tan(\theta )}}={\frac {\
text{adjacent}}{\text{opposite}}},\\\csc(\theta )&={\frac {1}{\sin(\theta )}}={\frac {\text{hypotenuse}}{\text{opposite}}},\\\sec(\theta )&={\frac {1}{\cos(\theta )}}={\frac {\textrm {hypotenuse}}{\
textrm {adjacent}}}.\end{aligned}}}
Special angle measures
As stated, the values ${\displaystyle \sin(\alpha )}$ and ${\displaystyle \cos(\alpha )}$ appear to depend on the choice of a right triangle containing an angle of measure ${\displaystyle \alpha }$ .
However, this is not the case as all such triangles are similar, and so the ratios are the same for each of them. For example, each leg of the 45-45-90 right triangle is 1 unit, and its hypotenuse is
${\displaystyle {\sqrt {2}}}$ ; therefore, ${\textstyle \sin 45^{\circ }=\cos 45^{\circ }={\frac {\sqrt {2}}{2}}}$ . The following table shows the special value of each input for both sine and cosine
with the domain between ${\textstyle 0<\alpha <{\frac {\pi }{2}}}$ . The input in this table provides various unit systems such as degree, radian, and so on. The angles other than those five can be
obtained by using a calculator.
Angle, x sin(x) cos(x)
Degrees Radians Gradians Turns Exact Decimal Exact Decimal
0° 0 0^g 0 0 0 1 1
30° 1/6π 33+1/3^g 1/12 1/2 0.5 ${\displaystyle {\frac {\sqrt {3}}{2}}}$ 0.8660
45° 1/4π 50^g 1/8 ${\displaystyle {\frac {\sqrt {2}}{2}}}$ 0.7071 ${\displaystyle {\frac {\sqrt {2}}{2}}}$ 0.7071
60° 1/3π 66+2/3^g 1/6 ${\displaystyle {\frac {\sqrt {3}}{2}}}$ 0.8660 1/2 0.5
90° 1/2π 100^g 1/4 1 1 0 0
Law of sines and cosines' illustration
The law of sines is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known. Given that a triangle ${\displaystyle ABC}$ with sides ${\displaystyle a}
$ , ${\displaystyle b}$ , and ${\displaystyle c}$ , and angles opposite those sides ${\displaystyle \alpha }$ , ${\displaystyle \beta }$ , and ${\displaystyle \gamma }$ . The law states, ${\
displaystyle {\frac {\sin \alpha }{a}}={\frac {\sin \beta }{b}}={\frac {\sin \gamma }{c}}.}$ This is equivalent to the equality of the first three expressions below: ${\displaystyle {\frac {a}{\sin \
alpha }}={\frac {b}{\sin \beta }}={\frac {c}{\sin \gamma }}=2R,}$ where ${\displaystyle R}$ is the triangle's circumradius.
The law of cosines is useful for computing the length of an unknown side if two other sides and an angle are known. The law states, ${\displaystyle a^{2}+b^{2}-2ab\cos(\gamma )=c^{2}}$ In the case
where ${\displaystyle \gamma =\pi /2}$ from which ${\displaystyle \cos(\gamma )=0}$ , the resulting equation becomes the Pythagorean theorem.
Vector definition
The cross product and dot product are operations on two vectors in Euclidean vector space. The sine and cosine functions can be defined in terms of the cross product and dot product. If ${\
displaystyle \mathbb {a} }$ and ${\displaystyle \mathbb {b} }$ are vectors, and ${\displaystyle \theta }$ is the angle between ${\displaystyle \mathbb {a} }$ and ${\displaystyle \mathbb {b} }$ , then
sine and cosine can be defined as: {\displaystyle {\begin{aligned}\sin(\theta )&={\frac {|\mathbb {a} \times \mathbb {b} |}{|a||b|}},\\\cos(\theta )&={\frac {\mathbb {a} \cdot \mathbb {b} }{|a||b|}}.
Analytic descriptions
Unit circle definition
The sine and cosine functions may also be defined in a more general way by using unit circle, a circle of radius one centered at the origin ${\displaystyle (0,0)}$ , formulated as the equation of ${\
displaystyle x^{2}+y^{2}=1}$ in the Cartesian coordinate system. Let a line through the origin intersect the unit circle, making an angle of ${\displaystyle \theta }$ with the positive half of the $
{\displaystyle x}$ -axis. The ${\displaystyle x}$ - and ${\displaystyle y}$ -coordinates of this point of intersection are equal to ${\displaystyle \cos(\theta )}$ and ${\displaystyle \sin(\theta )}$
, respectively; that is, ${\displaystyle \sin(\theta )=y,\qquad \cos(\theta )=x.}$
This definition is consistent with the right-angled triangle definition of sine and cosine when ${\textstyle 0<\theta <{\frac {\pi }{2}}}$ because the length of the hypotenuse of the unit circle is
always 1; mathematically speaking, the sine of an angle equals the opposite side of the triangle, which is simply the ${\displaystyle y}$ -coordinate. A similar argument can be made for the cosine
function to show that the cosine of an angle when ${\textstyle 0<\theta <{\frac {\pi }{2}}}$ , even under the new definition using the unit circle.
Graph of a function and its elementary properties
Animation demonstrating how the sine function (in red) is graphed from the y-coordinate (red dot) of a point on the unit circle (in green), at an angle of θ. The cosine (in blue) is the x-coordinate.
Using the unit circle definition has the advantage of drawing a graph of sine and cosine functions. This can be done by rotating counterclockwise a point along the circumference of a circle,
depending on the input ${\displaystyle \theta >0}$ . In a sine function, if the input is ${\textstyle \theta ={\frac {\pi }{2}}}$ , the point is rotated counterclockwise and stopped exactly on the $
{\displaystyle y}$ -axis. If ${\displaystyle \theta =\pi }$ , the point is at the circle's halfway. If ${\displaystyle \theta =2\pi }$ , the point returned to its origin. This results that both sine
and cosine functions have the range between ${\displaystyle -1\leq y\leq 1}$ .
Extending the angle to any real domain, the point rotated counterclockwise continuously. This can be done similarly for the cosine function as well, although the point is rotated initially from the $
{\displaystyle y}$ -coordinate. In other words, both sine and cosine functions are periodic, meaning any angle added by the circumference's circle is the angle itself. Mathematically, ${\displaystyle
\sin(\theta +2\pi )=\sin(\theta ),\qquad \cos(\theta +2\pi )=\cos(\theta ).}$
A function ${\displaystyle f}$ is said to be odd if ${\displaystyle f(-x)=-f(x)}$ , and is said to be even if ${\displaystyle f(-x)=f(x)}$ . The sine function is odd, whereas the cosine function is
even. Both sine and cosine functions are similar, with their difference being shifted by ${\textstyle {\frac {\pi }{2}}}$ . This means, {\displaystyle {\begin{aligned}\sin(\theta )&=\cos \left({\frac
{\pi }{2}}-\theta \right),\\\cos(\theta )&=\sin \left({\frac {\pi }{2}}-\theta \right).\end{aligned}}}
The fixed point iteration x[n+1] = cos(x[n]) with initial value x[0] = −1 converges to the Dottie number.
Zero is the only real fixed point of the sine function; in other words the only intersection of the sine function and the identity function is ${\displaystyle \sin(0)=0}$ . The only real fixed point
of the cosine function is called the Dottie number. The Dottie number is the unique real root of the equation ${\displaystyle \cos(x)=x}$ . The decimal expansion of the Dottie number is approximately
Continuity and differentiation
The quadrants of the unit circle and of sin(x), using the Cartesian coordinate system
The sine and cosine functions are infinitely differentiable. The derivative of sine is cosine, and the derivative of cosine is negative sine: ${\displaystyle {\frac {d}{dx}}\sin(x)=\cos(x),\qquad {\
frac {d}{dx}}\cos(x)=-\sin(x).}$ Continuing the process in higher-order derivative results in the repeated same functions; the fourth derivative of a sine is the sine itself. These derivatives can be
applied to the first derivative test, according to which the monotonicity of a function can be defined as the inequality of function's first derivative greater or less than equal to zero. It can also
be applied to second derivative test, according to which the concavity of a function can be defined by applying the inequality of the function's second derivative greater or less than equal to zero.
The following table shows that both sine and cosine functions have concavity and monotonicity—the positive sign (${\displaystyle +}$ ) denotes a graph is increasing (going upward) and the negative
sign (${\displaystyle -}$ ) is decreasing (going downward)—in certain intervals. This information can be represented as a Cartesian coordinates system divided into four quadrants.
Quadrant Angle Sine Cosine
Degrees Radians Sign Monotony Convexity Sign Monotony Convexity
1st quadrant, I ${\displaystyle 0^{\circ }<x<90^{\circ }}$ ${\displaystyle 0<x<{\frac {\pi }{2}}}$ ${\displaystyle +}$ Increasing Concave ${\displaystyle +}$ Decreasing Concave
2nd quadrant, II ${\displaystyle 90^{\circ }<x<180^{\circ }}$ ${\displaystyle {\frac {\pi }{2}}<x<\pi }$ ${\displaystyle +}$ Decreasing Concave ${\displaystyle -}$ Decreasing Convex
3rd quadrant, III ${\displaystyle 180^{\circ }<x<270^{\circ }}$ ${\displaystyle \pi <x<{\frac {3\pi }{2}}}$ ${\displaystyle -}$ Decreasing Convex ${\displaystyle -}$ Increasing Convex
4th quadrant, IV ${\displaystyle 270^{\circ }<x<360^{\circ }}$ ${\displaystyle {\frac {3\pi }{2}}<x<2\pi }$ ${\displaystyle -}$ Increasing Convex ${\displaystyle +}$ Increasing Concave
Both sine and cosine functions can be defined by using differential equations. The pair of ${\displaystyle (\cos \theta ,\sin \theta )}$ is the solution ${\displaystyle (x(\theta ),y(\theta ))}$ to
the two-dimensional system of differential equations ${\displaystyle y'(\theta )=x(\theta )}$ and ${\displaystyle x'(\theta )=-y(\theta )}$ with the initial conditions ${\displaystyle y(0)=0}$ and $
{\displaystyle x(0)=1}$ . One could interpret the unit circle in the above definitions as defining the phase space trajectory of the differential equation with the given initial conditions. It can be
interpreted as a phase space trajectory of the system of differential equations ${\displaystyle y'(\theta )=x(\theta )}$ and ${\displaystyle x'(\theta )=-y(\theta )}$ starting from the initial
conditions ${\displaystyle y(0)=0}$ and ${\displaystyle x(0)=1}$ .
Integral and the usage in mensuration
Their area under a curve can be obtained by using the integral with a certain bounded interval. Their antiderivatives are: ${\displaystyle \int \sin(x)\,dx=-\cos(x)+C\qquad \int \cos(x)\,dx=\sin(x)
+C,}$ where ${\displaystyle C}$ denotes the constant of integration. These antiderivatives may be applied to compute the mensuration properties of both sine and cosine functions' curves with a given
interval. For example, the arc length of the sine curve between ${\displaystyle 0}$ and ${\displaystyle t}$ is ${\displaystyle \int _{0}^{t}\!{\sqrt {1+\cos ^{2}(x)}}\,dx={\sqrt {2}}\operatorname {E}
\left(t,{\frac {1}{\sqrt {2}}}\right),}$ where ${\displaystyle \operatorname {E} (\varphi ,k)}$ is the incomplete elliptic integral of the second kind with modulus ${\displaystyle k}$ . It cannot be
expressed using elementary functions. In the case of a full period, its arc length is ${\displaystyle L={\frac {4{\sqrt {2\pi ^{3}}}}{\Gamma (1/4)^{2}}}+{\frac {\Gamma (1/4)^{2}}{\sqrt {2\pi }}}={\
frac {2\pi }{\varpi }}+2\varpi \approx 7.6404\ldots }$ where ${\displaystyle \Gamma }$ is the gamma function and ${\displaystyle \varpi }$ is the lemniscate constant.
Inverse functions
The usual principal values of the arcsin(x) and arccos(x) functions graphed on the Cartesian plane
The inverse function of sine is arcsine or inverse sine, denoted as "arcsin", "asin", or ${\displaystyle \sin ^{-1}}$ . The inverse function of cosine is arccosine, denoted as "arccos", "acos", or $
{\displaystyle \cos ^{-1}}$ .^[a] As sine and cosine are not injective, their inverses are not exact inverse functions, but partial inverse functions. For example, ${\displaystyle \sin(0)=0}$ , but
also ${\displaystyle \sin(\pi )=0}$ , ${\displaystyle \sin(2\pi )=0}$ , and so on. It follows that the arcsine function is multivalued: ${\displaystyle \arcsin(0)=0}$ , but also ${\displaystyle \
arcsin(0)=\pi }$ , ${\displaystyle \arcsin(0)=2\pi }$ , and so on. When only one value is desired, the function may be restricted to its principal branch. With this restriction, for each ${\
displaystyle x}$ in the domain, the expression ${\displaystyle \arcsin(x)}$ will evaluate only to a single value, called its principal value. The standard range of principal values for arcsin is from
${\textstyle -{\frac {\pi }{2}}}$ to ${\textstyle {\frac {\pi }{2}}}$ , and the standard range for arccos is from ${\displaystyle 0}$ to ${\displaystyle \pi }$ .
The inverse function of both sine and cosine are defined as:${\displaystyle \theta =\arcsin \left({\frac {\text{opposite}}{\text{hypotenuse}}}\right)=\arccos \left({\frac {\text{adjacent}}{\text
{hypotenuse}}}\right),}$ where for some integer ${\displaystyle k}$ , {\displaystyle {\begin{aligned}\sin(y)=x\iff &y=\arcsin(x)+2\pi k,{\text{ or }}\\&y=\pi -\arcsin(x)+2\pi k\\\cos(y)=x\iff &y=\
arccos(x)+2\pi k,{\text{ or }}\\&y=-\arccos(x)+2\pi k\end{aligned}}} By definition, both functions satisfy the equations:${\displaystyle \sin(\arcsin(x))=x\qquad \cos(\arccos(x))=x}$ and {\
displaystyle {\begin{aligned}\arcsin(\sin(\theta ))=\theta \quad &{\text{for}}\quad -{\frac {\pi }{2}}\leq \theta \leq {\frac {\pi }{2}}\\\arccos(\cos(\theta ))=\theta \quad &{\text{for}}\quad 0\leq
\theta \leq \pi \end{aligned}}}
Other identities
According to Pythagorean theorem, the squared hypotenuse is the sum of two squared legs of a right triangle. Dividing the formula on both sides with squared hypotenuse resulting in the Pythagorean
trigonometric identity, the sum of a squared sine and a squared cosine equals 1:^[b] ${\displaystyle \sin ^{2}(\theta )+\cos ^{2}(\theta )=1.}$
Sine and cosine satisfy the following double-angle formulas:{\displaystyle {\begin{aligned}\sin(2\theta )&=2\sin(\theta )\cos(\theta ),\\\cos(2\theta )&=\cos ^{2}(\theta )-\sin ^{2}(\theta )\\&=2\cos
^{2}(\theta )-1\\&=1-2\sin ^{2}(\theta )\end{aligned}}}
Sine function in blue and sine squared function in red. The x-axis is in radians.
The cosine double angle formula implies that sin^2 and cos^2 are, themselves, shifted and scaled sine waves. Specifically,^[26] ${\displaystyle \sin ^{2}(\theta )={\frac {1-\cos(2\theta )}{2}}\qquad
\cos ^{2}(\theta )={\frac {1+\cos(2\theta )}{2}}}$ The graph shows both sine and sine squared functions, with the sine in blue and the sine squared in red. Both graphs have the same shape but with
different ranges of values and different periods. Sine squared has only positive values, but twice the number of periods.
Series and polynomials
This animation shows how including more and more terms in the partial sum of its Taylor series approaches a sine curve.
Both sine and cosine functions can be defined by using a Taylor series, a power series involving the higher-order derivatives. As mentioned in § Continuity and differentiation, the derivative of sine
is cosine and that the derivative of cosine is the negative of sine. This means the successive derivatives of ${\displaystyle \sin(x)}$ are ${\displaystyle \cos(x)}$ , ${\displaystyle -\sin(x)}$ , $
{\displaystyle -\cos(x)}$ , ${\displaystyle \sin(x)}$ , continuing to repeat those four functions. The ${\displaystyle (4n+k)}$ -th derivative, evaluated at the point 0: ${\displaystyle \sin ^
{(4n+k)}(0)={\begin{cases}0&{\text{when }}k=0\\1&{\text{when }}k=1\\0&{\text{when }}k=2\\-1&{\text{when }}k=3\end{cases}}}$ where the superscript represents repeated differentiation. This implies the
following Taylor series expansion at ${\displaystyle x=0}$ . One can then use the theory of Taylor series to show that the following identities hold for all real numbers ${\displaystyle x}$ —where $
{\displaystyle x}$ is the angle in radians. More generally, for all complex numbers: {\displaystyle {\begin{aligned}\sin(x)&=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots \\&=\
sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1}\end{aligned}}} Taking the derivative of each term gives the Taylor series for cosine: {\displaystyle {\begin{aligned}\cos(x)&=1-{\frac {x^{2}}
{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots \\&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}x^{2n}\end{aligned}}}
Both sine and cosine functions with multiple angles may appear as their linear combination, resulting in a polynomial. Such a polynomial is known as the trigonometric polynomial. The trigonometric
polynomial's ample applications may be acquired in its interpolation, and its extension of a periodic function known as the Fourier series. Let ${\displaystyle a_{n}}$ and ${\displaystyle b_{n}}$ be
any coefficients, then the trigonometric polynomial of a degree ${\displaystyle N}$ —denoted as ${\displaystyle T(x)}$ —is defined as: ${\displaystyle T(x)=a_{0}+\sum _{n=1}^{N}a_{n}\cos(nx)+\sum _{n
The trigonometric series can be defined similarly analogous to the trigonometric polynomial, its infinite inversion. Let ${\displaystyle A_{n}}$ and ${\displaystyle B_{n}}$ be any coefficients, then
the trigonometric series can be defined as: ${\displaystyle {\frac {1}{2}}A_{0}+\sum _{n=1}^{\infty }A_{n}\cos(nx)+B_{n}\sin(nx).}$ In the case of a Fourier series with a given integrable function $
{\displaystyle f}$ , the coefficients of a trigonometric series are: {\displaystyle {\begin{aligned}A_{n}&={\frac {1}{\pi }}\int _{0}^{2\pi }f(x)\cos(nx)\,dx,\\B_{n}&={\frac {1}{\pi }}\int _{0}^{2\pi
Complex numbers relationship
Complex exponential function definitions
Both sine and cosine can be extended further via complex number, a set of numbers composed of both real and imaginary numbers. For real number ${\displaystyle \theta }$ , the definition of both sine
and cosine functions can be extended in a complex plane in terms of an exponential function as follows: {\displaystyle {\begin{aligned}\sin(\theta )&={\frac {e^{i\theta }-e^{-i\theta }}{2i}},\\\cos(\
theta )&={\frac {e^{i\theta }+e^{-i\theta }}{2}},\end{aligned}}}
Alternatively, both functions can be defined in terms of Euler's formula: {\displaystyle {\begin{aligned}e^{i\theta }&=\cos(\theta )+i\sin(\theta ),\\e^{-i\theta }&=\cos(\theta )-i\sin(\theta ).\end
When plotted on the complex plane, the function ${\displaystyle e^{ix}}$ for real values of ${\displaystyle x}$ traces out the unit circle in the complex plane. Both sine and cosine functions may be
simplified to the imaginary and real parts of ${\displaystyle e^{i\theta }}$ as: {\displaystyle {\begin{aligned}\sin \theta &=\operatorname {Im} (e^{i\theta }),\\\cos \theta &=\operatorname {Re} (e^
{i\theta }).\end{aligned}}}
When ${\displaystyle z=x+iy}$ for real values ${\displaystyle x}$ and ${\displaystyle y}$ , where ${\displaystyle i={\sqrt {-1}}}$ , both sine and cosine functions can be expressed in terms of real
sines, cosines, and hyperbolic functions as:{\displaystyle {\begin{aligned}\sin z&=\sin x\cosh y+i\cos x\sinh y,\\\cos z&=\cos x\cosh y-i\sin x\sinh y.\end{aligned}}}
Polar coordinates
${\displaystyle \cos(\theta )}$ and ${\displaystyle \sin(\theta )}$ are the real and imaginary parts of ${\displaystyle e^{i\theta }}$ .
Sine and cosine are used to connect the real and imaginary parts of a complex number with its polar coordinates ${\displaystyle (r,\theta )}$ : ${\displaystyle z=r(\cos(\theta )+i\sin(\theta )),}$
and the real and imaginary parts are {\displaystyle {\begin{aligned}\operatorname {Re} (z)&=r\cos(\theta ),\\\operatorname {Im} (z)&=r\sin(\theta ),\end{aligned}}} where ${\displaystyle r}$ and ${\
displaystyle \theta }$ represent the magnitude and angle of the complex number ${\displaystyle z}$ .
For any real number ${\displaystyle \theta }$ , Euler's formula in terms of polar coordinates is stated as ${\textstyle z=re^{i\theta }}$ .
Complex arguments
Domain coloring of sin(z) in the complex plane. Brightness indicates absolute magnitude, hue represents complex argument.
Vector field rendering of sin(z)
Applying the series definition of the sine and cosine to a complex argument, z, gives:
{\displaystyle {\begin{aligned}\sin(z)&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}z^{2n+1}\\&={\frac {e^{iz}-e^{-iz}}{2i}}\\&={\frac {\sinh \left(iz\right)}{i}}\\&=-i\sinh \left(iz\right)\\
\cos(z)&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}z^{2n}\\&={\frac {e^{iz}+e^{-iz}}{2}}\\&=\cosh(iz)\\\end{aligned}}}
where sinh and cosh are the hyperbolic sine and cosine. These are entire functions.
It is also sometimes useful to express the complex sine and cosine functions in terms of the real and imaginary parts of its argument:
{\displaystyle {\begin{aligned}\sin(x+iy)&=\sin(x)\cos(iy)+\cos(x)\sin(iy)\\&=\sin(x)\cosh(y)+i\cos(x)\sinh(y)\\\cos(x+iy)&=\cos(x)\cos(iy)-\sin(x)\sin(iy)\\&=\cos(x)\cosh(y)-i\sin(x)\sinh(y)\\\
Partial fraction and product expansions of complex sine
Using the partial fraction expansion technique in complex analysis, one can find that the infinite series ${\displaystyle \sum _{n=-\infty }^{\infty }{\frac {(-1)^{n}}{z-n}}={\frac {1}{z}}-2z\sum _{n
=1}^{\infty }{\frac {(-1)^{n}}{n^{2}-z^{2}}}}$ both converge and are equal to ${\textstyle {\frac {\pi }{\sin(\pi z)}}}$ . Similarly, one can show that ${\displaystyle {\frac {\pi ^{2}}{\sin ^{2}(\pi
z)}}=\sum _{n=-\infty }^{\infty }{\frac {1}{(z-n)^{2}}}.}$
Using product expansion technique, one can derive ${\displaystyle \sin(\pi z)=\pi z\prod _{n=1}^{\infty }\left(1-{\frac {z^{2}}{n^{2}}}\right).}$
Usage of complex sine
sin(z) is found in the functional equation for the Gamma function,
${\displaystyle \Gamma (s)\Gamma (1-s)={\pi \over \sin(\pi s)},}$
which in turn is found in the functional equation for the Riemann zeta-function,
${\displaystyle \zeta (s)=2(2\pi )^{s-1}\Gamma (1-s)\sin \left({\frac {\pi }{2}}s\right)\zeta (1-s).}$
As a holomorphic function, sin z is a 2D solution of Laplace's equation:
${\displaystyle \Delta u(x_{1},x_{2})=0.}$
The complex sine function is also related to the level curves of pendulums.^[35]
Complex graphs
Sine function in the complex plane
Real component Imaginary component Magnitude
Arcsine function in the complex plane
Real component Imaginary component Magnitude
The word sine is derived, indirectly, from the Sanskrit word jyā 'bow-string' or more specifically its synonym jīvá (both adopted from Ancient Greek χορδή 'string'), due to visual similarity between
the arc of a circle with its corresponding chord and a bow with its string (see jyā, koti-jyā and utkrama-jyā). This was transliterated in Arabic as jība, which is meaningless in that language and
written as jb (جب). Since Arabic is written without short vowels, jb was interpreted as the homograph jayb (جيب), which means 'bosom', 'pocket', or 'fold'. When the Arabic texts of Al-Battani and
al-Khwārizmī were translated into Medieval Latin in the 12th century by Gerard of Cremona, he used the Latin equivalent sinus (which also means 'bay' or 'fold', and more specifically 'the hanging
fold of a toga over the breast'). Gerard was probably not the first scholar to use this translation; Robert of Chester appears to have preceded him and there is evidence of even earlier usage.^[42]
The English form sine was introduced in the 1590s.^[c]
The word cosine derives from an abbreviation of the Latin complementi sinus 'sine of the complementary angle' as cosinus in Edmund Gunter's Canon triangulorum (1620), which also includes a similar
definition of cotangens.
Quadrant from 1840s Ottoman Turkey with axes for looking up the sine and versine of angles
While the early study of trigonometry can be traced to antiquity, the trigonometric functions as they are in use today were developed in the medieval period. The chord function was discovered by
Hipparchus of Nicaea (180–125 BCE) and Ptolemy of Roman Egypt (90–165 CE).^[44]
The sine and cosine functions can be traced to the jyā and koṭi-jyā functions used in Indian astronomy during the Gupta period (Aryabhatiya and Surya Siddhanta), via translation from Sanskrit to
Arabic and then from Arabic to Latin.
All six trigonometric functions in current use were known in Islamic mathematics by the 9th century, as was the law of sines, used in solving triangles.^[45] With the exception of the sine (which was
adopted from Indian mathematics), the other five modern trigonometric functions were discovered by Arabic mathematicians, including the cosine, tangent, cotangent, secant and cosecant.^[45]
Al-Khwārizmī (c.780–850) produced tables of sines, cosines and tangents.^[46]^[47] Muhammad ibn Jābir al-Harrānī al-Battānī (853–929) discovered the reciprocal functions of secant and cosecant, and
produced the first table of cosecants for each degree from 1° to 90°.^[47]
The first published use of the abbreviations sin, cos, and tan is by the 16th-century French mathematician Albert Girard; these were further promulgated by Euler (see below). The Opus palatinum de
triangulis of Georg Joachim Rheticus, a student of Copernicus, was probably the first in Europe to define trigonometric functions directly in terms of right triangles instead of circles, with tables
for all six trigonometric functions; this work was finished by Rheticus' student Valentin Otho in 1596.
In a paper published in 1682, Leibniz proved that sin x is not an algebraic function of x.^[48] Roger Cotes computed the derivative of sine in his Harmonia Mensurarum (1722).^[49] Leonhard Euler's
Introductio in analysin infinitorum (1748) was mostly responsible for establishing the analytic treatment of trigonometric functions in Europe, also defining them as infinite series and presenting "
Euler's formula", as well as the near-modern abbreviations sin., cos., tang., cot., sec., and cosec.
Software implementations
There is no standard algorithm for calculating sine and cosine. IEEE 754, the most widely used standard for the specification of reliable floating-point computation, does not address calculating
trigonometric functions such as sine. The reason is that no efficient algorithm is known for computing sine and cosine with a specified accuracy, especially for large inputs.
Algorithms for calculating sine may be balanced for such constraints as speed, accuracy, portability, or range of input values accepted. This can lead to different results for different algorithms,
especially for special circumstances such as very large inputs, e.g. sin(10^22).
A common programming optimization, used especially in 3D graphics, is to pre-calculate a table of sine values, for example one value per degree, then for values in-between pick the closest
pre-calculated value, or linearly interpolate between the 2 closest values to approximate it. This allows results to be looked up from a table rather than being calculated in real time. With modern
CPU architectures this method may offer no advantage.
The CORDIC algorithm is commonly used in scientific calculators.
The sine and cosine functions, along with other trigonometric functions, are widely available across programming languages and platforms. In computing, they are typically abbreviated to sin and cos.
Some CPU architectures have a built-in instruction for sine, including the Intel x87 FPUs since the 80387.
In programming languages, sin and cos are typically either a built-in function or found within the language's standard math library. For example, the C standard library defines sine functions within
math.h: sin(double), sinf(float), and sinl(long double). The parameter of each is a floating point value, specifying the angle in radians. Each function returns the same data type as it accepts. Many
other trigonometric functions are also defined in math.h, such as for cosine, arc sine, and hyperbolic sine (sinh). Similarly, Python defines math.sin(x) and math.cos(x) within the built-in math
module. Complex sine and cosine functions are also available within the cmath module, e.g. cmath.sin(z). CPython's math functions call the C math library, and use a double-precision floating-point
Turns based implementations
Some software libraries provide implementations of sine and cosine using the input angle in half-turns, a half-turn being an angle of 180 degrees or ${\displaystyle \pi }$ radians. Representing
angles in turns or half-turns has accuracy advantages and efficiency advantages in some cases.^[51]^[52] In MATLAB, OpenCL, R, Julia, CUDA, and ARM, these functions are called sinpi and cospi.^[51]^[
53]^[52]^[54]^[55]^[56] For example, sinpi(x) would evaluate to ${\displaystyle \sin(\pi x),}$ where x is expressed in half-turns, and consequently the final input to the function, πx can be
interpreted in radians by sin.
The accuracy advantage stems from the ability to perfectly represent key angles like full-turn, half-turn, and quarter-turn losslessly in binary floating-point or fixed-point. In contrast,
representing ${\displaystyle 2\pi }$ , ${\displaystyle \pi }$ , and ${\textstyle {\frac {\pi }{2}}}$ in binary floating-point or binary scaled fixed-point always involves a loss of accuracy since
irrational numbers cannot be represented with finitely many binary digits.
Turns also have an accuracy advantage and efficiency advantage for computing modulo to one period. Computing modulo 1 turn or modulo 2 half-turns can be losslessly and efficiently computed in both
floating-point and fixed-point. For example, computing modulo 1 or modulo 2 for a binary point scaled fixed-point value requires only a bit shift or bitwise AND operation. In contrast, computing
modulo ${\textstyle {\frac {\pi }{2}}}$ involves inaccuracies in representing ${\textstyle {\frac {\pi }{2}}}$ .
For applications involving angle sensors, the sensor typically provides angle measurements in a form directly compatible with turns or half-turns. For example, an angle sensor may count from 0 to
4096 over one complete revolution.^[57] If half-turns are used as the unit for angle, then the value provided by the sensor directly and losslessly maps to a fixed-point data type with 11 bits to the
right of the binary point. In contrast, if radians are used as the unit for storing the angle, then the inaccuracies and cost of multiplying the raw sensor integer by an approximation to ${\textstyle
{\frac {\pi }{2048}}}$ would be incurred.
See also
1. ^ The superscript of −1 in ${\displaystyle \sin ^{-1}}$ and ${\displaystyle \cos ^{-1}}$ denotes the inverse of a function, instead of exponentiation.
2. ^ Here, ${\displaystyle \sin ^{2}(x)}$ means the squared sine function ${\displaystyle \sin(x)\cdot \sin(x)}$ .
3. ^ The anglicized form is first recorded in 1593 in Thomas Fale's Horologiographia, the Art of Dialling.
1. ^ "OEIS A003957". oeis.org. Retrieved 2019-05-26.
2. ^ "Sine-squared function". Retrieved August 9, 2019.
3. ^ "Why are the phase portrait of the simple plane pendulum and a domain coloring of sin(z) so similar?". math.stackexchange.com. Retrieved 2019-08-12.
4. ^ Various sources credit the first use of sinus to either See Merlet (2004). See Maor (1998), Chapter 3, for an earlier etymology crediting Gerard. See Katz (2008), p. 210.
5. ^ Brendan, T. (February 1965). "How Ptolemy constructed trigonometry tables". The Mathematics Teacher. 58 (2): 141–149. doi:10.5951/MT.58.2.0141. JSTOR 27967990.
6. ^ ^a ^b Gingerich, Owen (1986). "Islamic Astronomy". Scientific American. Vol. 254. p. 74. Archived from the original on 2013-10-19. Retrieved 2010-07-13.
7. ^ Jacques Sesiano, "Islamic mathematics", p. 157, in Selin, Helaine; D'Ambrosio, Ubiratan, eds. (2000). Mathematics Across Cultures: The History of Non-western Mathematics. Springer
Science+Business Media. ISBN 978-1-4020-0260-1.
8. ^ ^a ^b "trigonometry". Encyclopedia Britannica. 17 June 2024.
9. ^ Nicolás Bourbaki (1994). Elements of the History of Mathematics. Springer. ISBN 9783540647676.
10. ^ "Why the sine has a simple derivative Archived 2011-07-20 at the Wayback Machine", in Historical Notes for Calculus Teachers Archived 2011-07-20 at the Wayback Machine by V. Frederick Rickey
Archived 2011-07-20 at the Wayback Machine
11. ^ ^a ^b "MATLAB Documentation sinpi
12. ^ ^a ^b "R Documentation sinpi
13. ^ "OpenCL Documentation sinpi
14. ^ "Julia Documentation sinpi
15. ^ "CUDA Documentation sinpi
16. ^ "ARM Documentation sinpi
17. ^ "ALLEGRO Angle Sensor Datasheet Archived 2019-04-17 at the Wayback Machine
Works cited
• Abramowitz, Milton; Stegun, Irene A. (1970), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover Publications, Ninth printing
• Adlaj, Semjon (2012), "An Eloquent Formula for the Perimeter of an Ellipse" (PDF), American Mathematical Society, 59 (8): 1097
• Axler, Sheldon (2012), Algebra and Trigonometry, John Wiley & Sons, ISBN 978-0470-58579-5
• Bourchtein, Ludmila; Bourchtein, Andrei (2022), Theory of Infinite Sequences and Series, Springer, doi:10.1007/978-3-030-79431-6, ISBN 978-3-030-79431-6
• Gunter, Edmund (1620), Canon triangulorum
• Howie, John M. (2003), Complex Analysis, Springer Undergraduate Mathematics Series, Springer, doi:10.1007/978-1-4471-0027-0, ISBN 978-1-4471-0027-0
• Traupman, Ph.D., John C. (1966), The New College Latin & English Dictionary, Toronto: Bantam, ISBN 0-553-27619-0
• Katz, Victor J. (2008), A History of Mathematics (PDF) (3rd ed.), Boston: Addison-Wesley, “The English word "sine" comes from a series of mistranslations of the Sanskrit jyā-ardha (chord-half).
Āryabhaṭa frequently abbreviated this term to jyā or its synonym jīvá. When some of the Hindu works were later translated into Arabic, the word was simply transcribed phonetically into an
otherwise meaningless Arabic word jiba. But since Arabic is written without vowels, later writers interpreted the consonants jb as jaib, which means bosom or breast. In the twelfth century, when
an Arabic trigonometry work was translated into Latin, the translator used the equivalent Latin word sinus, which also meant bosom, and by extension, fold (as in a toga over a breast), or a bay
or gulf.”
• Maor, Eli (1998), Trigonometric Delights, Princeton University Press, ISBN 1-4008-4282-4
• Merlet, Jean-Pierre (2004), "A Note on the History of the Trigonometric Functions", in Ceccarelli, Marco (ed.), International Symposium on History of Machines and Mechanisms, Springer, doi
:10.1007/1-4020-2204-2, ISBN 978-1-4020-2203-6
• Merzbach, Uta C.; Boyer, Carl B. (2011), A History of Mathematics (3rd ed.), John Wiley & Sons, “It was Robert of Chester's translation from Arabic that resulted in our word "sine". The Hindus
had given the name jiva to the half-chord in trigonometry, and the Arabs had taken this over as jiba. In the Arabic language, there is also the word jaib meaning "bay" or "inlet". When Robert of
Chester came to translate the technical word jiba, he seems to have confused this with the word jaib (perhaps because vowels were omitted); hence, he used the word sinus, the Latin word for "bay"
or "inlet".”
• Plofker (2009), Mathematics in India, Princeton University Press
• Powell, Michael J. D. (1981), Approximation Theory and Methods, Cambridge University Press, ISBN 978-0-521-29514-7
• Rudin, Walter (1987), Real and complex analysis (3rd ed.), New York: McGraw-Hill, ISBN 978-0-07-054234-1, MR 0924157
• Smith, D. E. (1958) [1925], History of Mathematics, vol. I, Dover Publications, ISBN 0-486-20429-4
• Varberg, Dale E.; Purcell, Edwin J.; Rigdon, Steven E. (2007), Calculus (9th ed.), Pearson Prentice Hall, ISBN 978-0131469686
• Vince, John (2023), Calculus for Computer Graphics, Springer, doi:10.1007/978-3-031-28117-4, ISBN 978-3-031-28117-4
• Young, Cynthia (2012), Trigonometry (3rd ed.), John Wiley & Sons, ISBN 978-1-119-32113-2
• ——— (2017), Trigonometry (4th ed.), John Wiley & Sons, ISBN 978-1-119-32113-2
• Zimmermann, Paul (2006), "Can we trust floating-point numbers?", Grand Challenges of Informatics (PDF), p. 14/31
• Zygmund, Antoni (1968), Trigonometric Series (2nd, reprinted ed.), Cambridge University Press, MR 0236587
External links
Look up sine in Wiktionary, the free dictionary.
• Media related to Sine function at Wikimedia Commons
Look up sine and cosine in Wiktionary, the free dictionary. | {"url":"https://www.knowpia.com/knowpedia/Sine_and_cosine","timestamp":"2024-11-08T05:00:49Z","content_type":"text/html","content_length":"535810","record_id":"<urn:uuid:cb8b9421-3424-42c5-96a1-c39f8902a4b0>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00330.warc.gz"} |
Introduction to Integer Numbers | Curious Toons
Table of Contents
Understanding Integers
Definition of Integers
Integers are a fundamental set of numbers in mathematics that include all whole numbers, both positive and negative, as well as zero. In simpler terms, if we think about the number line, integers
consist of the numbers you can find without any fractions or decimals. They include numbers like -3, -2, -1, 0, 1, 2, 3, and so on, extending infinitely in both directions. The symbol commonly used
to represent the set of integers is ( \mathbb{Z} ), which comes from the German word “Zahlen,” meaning numbers. Integers are important because they help us understand the concept of opposites and
additions and subtractions without limitations from fractional parts. The inclusion of zero is also significant, as it serves as a neutral point between positive and negative integers. Understanding
integers sets the stage for exploring other mathematical concepts, such as equations and inequalities. As we learn about integers, keep in mind their applications in real-world situations, from
temperature changes to financial transactions, where we often need to deal with both gains and losses.
Properties of Integer Numbers
Integers possess several important properties that make them unique and useful in mathematics. First, let’s talk about closure. This property states that when you add or multiply two integers, the
result will always be another integer. For example, if you add -3 and 5, you get 2, which is an integer. The second property is associativity, which means that when you add or multiply integers, the
grouping of the numbers does not affect the outcome. For instance, ( (2 + 3) + 4 ) is equal to ( 2 + (3 + 4) ). The third property is commutativity, which tells us that the order in which we add or
multiply integers does not matter, so ( 4 + 5 ) is the same as ( 5 + 4 ). Additionally, integers have an identity element: zero is the additive identity (since adding zero doesn’t change a number),
while one is the multiplicative identity (since multiplying by one doesn’t change a number). Lastly, we also have the property of negatives, meaning every positive integer has a corresponding
negative integer. Understanding these properties not only helps us work with integers effectively but also lays the groundwork for more advanced mathematical concepts.
Representation of Integers
Number Line Representation
Understanding integers begins with the concept of the number line. Imagine a straight line stretching infinitely in both directions. At the center of this line is the number zero. To the right of
zero, we plot positive integers (1, 2, 3, and so on), while to the left, we find negative integers (-1, -2, -3, etc.). Each point on the number line represents a specific integer, spaced evenly
apart, which helps us visualize their relationships. For instance, the distance between -3 and 0 is the same as between 0 and 3—both have a distance of three units.
The number line is not just a tool for location; it also helps us understand operations with integers. For addition, we can think about moving to the right for positive numbers and to the left for
negative numbers. For instance, if you are at -2 and add 3, you move three steps to the right, landing at +1. This visual representation makes it easier to grasp concepts like greater than or less
than, as the distance from zero indicates the value of an integer, helping you develop a strong foundational understanding.
Visualizing Integers with Graphs
Graphs expand our understanding of integers beyond a simple number line. When we use coordinate planes, we can represent integers as points in a two-dimensional space. In this representation, we have
an x-axis (horizontal) and a y-axis (vertical). Each point on the graph is defined by a pair of coordinates, (x, y), where both x and y can be positive or negative integers.
Visualizing integers with graphs has numerous practical applications. For example, if we plotted the temperature in degrees Celsius over a week, we could see patterns, trends, or changes, even if
some temperatures were below zero. The graphical representation not only highlights relationships but also makes it easy to interpret data, facilitating comparisons between different sets of
Graphs also assist in more complex concepts like functions and transformations, emphasizing how integers can interact with each other in varied contexts. By connecting abstract numerical ideas to
visual elements, we enhance our comprehension, making math not just a series of calculations, but a dynamic and vibrant subject.
Operations with Integers
Addition and Subtraction of Integers
When we talk about the addition and subtraction of integers, we’re handling positive and negative whole numbers. To add integers, we need to consider their signs. If you add two positive numbers, the
result is positive. If you add two negative numbers, the result is negative, and the absolute values are combined. However, when you add one positive and one negative integer, you need to subtract
the smaller absolute value from the larger one, and the result takes the sign of the integer with the larger absolute value.
For example, ( 5 + (-3) ) equals ( 2 ) because 5 is larger than 3. Subtraction can be understood as adding the opposite: for example, subtracting ( -3 ) is the same as adding ( 3 ). Hence, ( 5 – (-3)
= 5 + 3 = 8 ).
Understanding these rules helps us navigate real-life applications, such as temperature changes or debt. Remember, keeping track of the signs is crucial as they guide you to the correct answer.
Practice using a number line can also help visualize these operations!
Multiplication and Division of Integers
When we move to multiplication and division of integers, the rules are a bit different but equally logical. For multiplication, if both integers have the same sign—either both positive or both
negative—the product is positive. For example, ( 4 \times 5 = 20 ) and ( (-4) \times (-5) = 20 ). Conversely, if the integers have different signs (one positive and one negative), the product is
negative—for instance, ( 4 \times (-5) = -20 ).
Dividing integers follows similar rules to multiplication. If two integers with the same sign are divided, the result is positive, while a division of integers with different signs results in a
negative outcome. For example, ( 20 ÷ 4 = 5 ) and ( (-20) ÷ (-4) = 5 ); however, ( (-20) ÷ 4 = -5 ).
These operations are essential in various contexts, like calculating areas and understanding finances. Practice helps strengthen your ability to switch signs correctly as you work with these
Integers in Real Life
Applications in Daily Life
Integers are all around us in our daily lives, and understanding them helps us make sense of various situations. For example, consider sports scores. If a team scores 3 points and then loses 2 points
due to a foul, we can represent this as an integer addition problem: (3 – 2 = 1). This illustrates how we can track changes, such as maintaining a lead in a game.
Another real-life application of integers is in measuring temperatures. When it’s cold outside, temperatures can drop below zero degrees, which we represent as negative integers. If it is 3 degrees
at noon and drops to -2 degrees in the evening, we can see how temperatures can fluctuate, helping us prepare for weather changes.
In general, integers help us quantify and represent real-world situations, whether we are talking about gains and losses, distances, or even time management. They allow us to communicate and
calculate in a precise manner, making them fundamental in our decision-making processes and daily activities.
Integers in Money and Temperature
Integers play a crucial role in our understanding of money and temperature, two concepts that impact our everyday decisions. When it comes to money, we often deal with positive and negative integers.
For example, if you have $50, that value is a positive integer. If you spend $70, your financial state can be expressed as -$20, indicating that you’re in debt. This shows how we can use integers to
track our balance, highlight our financial situation, and make informed budgeting decisions.
Similarly, temperature can be expressed using integers on a number scale. Positive integers represent temperatures above freezing, while negative integers represent temperatures below freezing. For
instance, if the temperature is -5 degrees on a winter morning but rises to 10 degrees by noon, we can understand the drastic change in weather. Knowledge of integers helps us interpret forecasts,
dress appropriately, and plan outdoor activities.
In both instances, integers serve as fundamental tools for measuring and understanding concepts that affect our lives, whether we are managing finances or adapting to changing weather conditions.
Common Mistakes with Integers
Common Errors in Operations
When working with integer numbers, students often encounter a variety of common errors that can lead to incorrect answers. One frequent mistake is confusing the rules for addition and subtraction of
negative and positive numbers. For example, when adding a negative number, some students mistakenly treat it as a positive, leading to incorrect sums. Similarly, when subtracting a negative, many
forget that it is equivalent to adding the absolute value of that number, causing confusion in calculations.
Another error arises during multiplication and division. Students sometimes overlook the rule where the product (or quotient) of two negative integers results in a positive integer, while a positive
and a negative number yield a negative product (or quotient). Misapplying these rules can result in significant errors.
Additionally, some students fail to use parentheses when necessary, particularly in expressions involving multiple operators. This can lead to misunderstandings in the order of operations. Overall,
being aware of these common operational errors is crucial for mastering integer calculations and building a strong foundation for future math concepts.
Tips to Avoid Mistakes
To minimize errors when dealing with integer numbers, it’s essential to adopt a systematic approach. First and foremost, clearly remember the rules governing positive and negative integers. Creating
a cheat sheet can be incredibly helpful; this could include key rules for addition, subtraction, multiplication, and division of integers.
Another effective strategy is to practice visualizing problems on a number line. This can provide a clear and concrete understanding of where integers fall in relation to one another, making it
easier to grasp operations like adding and subtracting negatives.
Additionally, always double-check your work. After performing calculations, take a moment to verify your answers. You can do this by estimating whether the result seems reasonable based on the
numbers involved.
Finally, work collaboratively with classmates. Explaining your thought process to someone else can illuminate any areas of misunderstanding and reinforce your learning. By following these tips and
maintaining a positive attitude toward practice and review, you’ll find that working with integers becomes much more manageable and error-free!
As we wrap up our exploration of integer numbers, let’s take a moment to reflect on the profound role they play in our daily lives and the world around us. Integers, whether positive, negative, or
zero, serve as the backbone of mathematics, representing everything from temperature fluctuations to financial transactions. They are not just abstract symbols; they reflect real-world concepts,
allowing us to quantify situations and solve problems.
Consider the beauty of balancing equations: just as in life, we must strive for equilibrium, whether it’s in our personal relationships or mathematical expressions. The interplay between positive and
negative integers teaches us resilience and perspective. Sometimes, we face setbacks (negative numbers), but they often lead to growth and understanding (positive outcomes).
As we move forward, remember that math isn’t merely about numbers; it’s a language that helps us articulate our thoughts and experiences. Whether you’re measuring progress or evaluating challenges,
the integers you’ve learned will guide you. Keep questioning, stay curious, and leverage your growing mathematical toolkit. Each chapter, like each integer, builds on the last, preparing you for the
complexities of the future. So embrace this foundation and let your mathematical journey continue to flourish! | {"url":"https://curioustoons.in/introduction-to-integer-numbers/","timestamp":"2024-11-09T19:13:49Z","content_type":"text/html","content_length":"107746","record_id":"<urn:uuid:523f0fc1-7b73-482b-8934-bcdc3aaafe4f>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00720.warc.gz"} |
Nautical Leagues (International) to Megaparsecs Converter
Enter Nautical Leagues (International)
β Switch toMegaparsecs to Nautical Leagues (International) Converter
How to use this Nautical Leagues (International) to Megaparsecs Converter π €
Follow these steps to convert given length from the units of Nautical Leagues (International) to the units of Megaparsecs.
1. Enter the input Nautical Leagues (International) value in the text field.
2. The calculator converts the given Nautical Leagues (International) into Megaparsecs in realtime β using the conversion formula, and displays under the Megaparsecs label. You do not need to
click any button. If the input changes, Megaparsecs value is re-calculated, just like that.
3. You may copy the resulting Megaparsecs value using the Copy button.
4. To view a detailed step by step calculation of the conversion, click on the View Calculation button.
5. You can also reset the input by clicking on button present below the input field.
What is the Formula to convert Nautical Leagues (International) to Megaparsecs?
The formula to convert given length from Nautical Leagues (International) to Megaparsecs is:
Length[(Megaparsecs)] = Length[(Nautical Leagues (International))] / 5553775309874243000
Substitute the given value of length in nautical leagues (international), i.e., Length[(Nautical Leagues (International))] in the above formula and simplify the right-hand side value. The resulting
value is the length in megaparsecs, i.e., Length[(Megaparsecs)].
Calculation will be done after you enter a valid input.
Consider that a luxury cruise ship sails 20 nautical leagues during a day at sea.
Convert this distance from nautical leagues to Megaparsecs.
The length in nautical leagues (international) is:
Length[(Nautical Leagues (International))] = 20
The formula to convert length from nautical leagues (international) to megaparsecs is:
Length[(Megaparsecs)] = Length[(Nautical Leagues (International))] / 5553775309874243000
Substitute given weight Length[(Nautical Leagues (International))] = 20 in the above formula.
Length[(Megaparsecs)] = 20 / 5553775309874243000
Length[(Megaparsecs)] = 0
Final Answer:
Therefore, 20 nautical league is equal to 0 Mpc.
The length is 0 Mpc, in megaparsecs.
Consider that an exploration vessel navigates through 10 nautical leagues of ocean.
Convert this distance from nautical leagues to Megaparsecs.
The length in nautical leagues (international) is:
Length[(Nautical Leagues (International))] = 10
The formula to convert length from nautical leagues (international) to megaparsecs is:
Length[(Megaparsecs)] = Length[(Nautical Leagues (International))] / 5553775309874243000
Substitute given weight Length[(Nautical Leagues (International))] = 10 in the above formula.
Length[(Megaparsecs)] = 10 / 5553775309874243000
Length[(Megaparsecs)] = 0
Final Answer:
Therefore, 10 nautical league is equal to 0 Mpc.
The length is 0 Mpc, in megaparsecs.
Nautical Leagues (International) to Megaparsecs Conversion Table
The following table gives some of the most used conversions from Nautical Leagues (International) to Megaparsecs.
Nautical Leagues (International) (nautical league) Megaparsecs (Mpc)
0 nautical league 0 Mpc
1 nautical league 0 Mpc
2 nautical league 0 Mpc
3 nautical league 0 Mpc
4 nautical league 0 Mpc
5 nautical league 0 Mpc
6 nautical league 0 Mpc
7 nautical league 0 Mpc
8 nautical league 0 Mpc
9 nautical league 0 Mpc
10 nautical league 0 Mpc
20 nautical league 0 Mpc
50 nautical league 0 Mpc
100 nautical league 0 Mpc
1000 nautical league 0 Mpc
10000 nautical league 0 Mpc
100000 nautical league 0 Mpc
Nautical Leagues (International)
A nautical league (international) is a unit of length used in maritime contexts. One nautical league is equivalent to 3 nautical miles, which is approximately 5,556 meters or 3.452 miles.
The nautical league is defined as three times the length of a nautical mile, based on the Earth's circumference and one minute of latitude.
Nautical leagues are used historically for measuring distances at sea. While not commonly used in modern navigation, they remain a part of maritime history and are occasionally referenced in
literature and older navigational texts.
A megaparsec (Mpc) is a unit of length used in astronomy to measure extremely large astronomical distances. One megaparsec is equivalent to one million parsecs, or approximately 3.262 million
light-years or about 3.086 Γ 10^22 meters.
The megaparsec is defined as one million times the length of a parsec, where one parsec is the distance at which one astronomical unit subtends an angle of one arcsecond.
Megaparsecs are used to measure vast distances between galaxies, clusters of galaxies, and other large-scale structures in the universe. They provide a convenient scale for expressing the immense
distances encountered in cosmology and extragalactic astronomy.
Frequently Asked Questions (FAQs)
1. What is the formula for converting Nautical Leagues (International) to Megaparsecs in Length?
The formula to convert Nautical Leagues (International) to Megaparsecs in Length is:
Nautical Leagues (International) / 5553775309874243000
2. Is this tool free or paid?
This Length conversion tool, which converts Nautical Leagues (International) to Megaparsecs, is completely free to use.
3. How do I convert Length from Nautical Leagues (International) to Megaparsecs?
To convert Length from Nautical Leagues (International) to Megaparsecs, you can use the following formula:
Nautical Leagues (International) / 5553775309874243000
For example, if you have a value in Nautical Leagues (International), you substitute that value in place of Nautical Leagues (International) in the above formula, and solve the mathematical
expression to get the equivalent value in Megaparsecs. | {"url":"https://convertonline.org/unit/?convert=nautical_leagues-megaparsecs","timestamp":"2024-11-12T04:06:31Z","content_type":"text/html","content_length":"92302","record_id":"<urn:uuid:078c5c3f-e087-41a3-8003-c79f42c7eb6b>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00045.warc.gz"} |
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Solving Quadratic Equations By Factoring Worksheet Answer Key
Solving Quadratic Equations By Factoring Worksheet Answer Key
Then after learning the quadratic formula the student can apply it. If a student is new to the quadratic formula he or she can refer to the answer key first before using the application.
Maze Freebie Solve Quadratic Equation By Factoring Level 1 Solving Quadratic Equations Quadratics Quadratic Equation
In the given quadratic equation the coefficient of x 2 is 1.
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3 different C++ programs to find permutation and combination nPr and nCr - CodeVsColor
C++ program to find permutation and combination nPr and nCr :
This C++ program will show you how to find the permutation and combination of two user-provided values in three different ways. We need two values to find the permutation and combination. The total
number of items and the items to pick on each selection.
A permutation is the number of ways we can arrange a given set of items and the order matters. Similarly, a combination is the number of ways we can select a given set of items without considering
the order. If n is the total number of items of a set and r is the number of items to select on each subset, the permutation and combination are denoted by nPr and nCr respectively.
We will write one program that takes the values of n and r as inputs from the user and print out the calculated values.
Formula to find permutation and combination :
Before starting the program, let’s learn the formula to calculate the permutation and combinations.
As I explained before, permutation and combination are calculated by considering n distinct elements r at a time. So, we will take the values of n and r from the user. Below is the formula to find
out a permutation:
Similarly, we can calculate the value of a combination with the below formula:
The symbol ! is used to define factorial.
As you can see, if we know how to find the factorial of a number, we can easily find out the nPr and nCr values of any two numbers.
How to calculate the factorial of a number:
We can calculate the factorial of a number in different ways. The factorial of a number is equal to the multiplication of all the positive numbers equal to or smaller than the number starting from 1.
We have different ways to find the factorial of a number:
Iterative approach by using a loop:
• Initialize one variable as 1 to hold the final factorial result value.
• Run one loop from 1 to the number, by incrementing the value by 1 on each step.
• Multiply the value with the result on each step.
• At the end of the loop, the result will hold the factorial.
Recursive method:
We can also use a recursive method to find the factorial. It will call the same method again and again to calculate the factorial of a number.
Let’s learn how to use these methods to find the permutation and combination in C++.
Method 1: C++ program to find permutation and combination by using a for loop:
The following program uses a for loop to find the factorial of a number.
#include <iostream>
using namespace std;
int findFact(int n)
int factorial = 1;
for(int i = 1; i <= n; i++)
factorial *= i;
return factorial;
int findNpR(int n, int r)
return findFact(n) / findFact(n - r);
int findNcR(int n, int r)
return findFact(n) / (findFact(n - r) * findFact(r));
int main()
int n, r, nPr, nCr;
cout << "Enter the value of n:" << endl;
cin >> n;
cout << "Enter the value of r:" << endl;
cin >> r;
nPr = findNpR(n, r);
nCr = findNcR(n, r);
cout << "Permutation,nPr : "<< nPr << endl;
cout << "Combination,nCr : "<< nCr << endl;
Download this example on Github
• The findFact method finds the factorial of a number. It takes a number as its parameter and returns the factorial value.
□ It initializes a variable factorial as 1 to hold the factorial value of the number n.
□ The for loop runs from i = 1 to i = n. On each step, it increases the value of i by 1.
□ It returns the value of factorial.
• The findNpR method is used to calculate the permutation for the provided n and r values. Similarly, the findNcR method is used to calculate the combination.
• It takes the values of n and r as inputs from the user.
• It uses the findNpR and findNcR methods to calculate the permutation and combination of the user-provided numbers. These values are assigned to the nPr and nCr variables.
• The last two lines are used to print the calculated values.
This program will print outputs as below:
Enter the value of n:
Enter the value of r:
Permutation,nPr : 132
Combination,nCr : 66
Method 2: C++ program to find permutation and combination by using a while loop:
We can use a while loop to find the factorial of a number. Let me change the above example to use a while loop to calculate the factorial:
#include <iostream>
using namespace std;
int findFact(int n)
int factorial = 1;
int i = 1;
while (i <= n)
factorial *= i;
return factorial;
int findNpR(int n, int r)
return findFact(n) / findFact(n - r);
int findNcR(int n, int r)
return findFact(n) / (findFact(n - r) * findFact(r));
int main()
int n, r, nPr, nCr;
cout << "Enter the value of n:" << endl;
cin >> n;
cout << "Enter the value of r:" << endl;
cin >> r;
nPr = findNpR(n, r);
nCr = findNcR(n, r);
cout << "Permutation,nPr : " << nPr << endl;
cout << "Combination,nCr : " << nCr << endl;
Download this example on Github
• The value of i is initialized before the loop starts. It will keep running until the value of i is smaller than or equal to i. On each step, we are multiplying the value of i with the variable
factorial to calculate the required factorial.
It will print similar output.
Method 3: C++ program to find permutation and combination recursively:
This method will use a recursive method to find the permutation. This is more concise than the previous two methods:
#include <iostream>
using namespace std;
int findFact(int n)
return n == 1 ? 1 : n * findFact(n - 1);
int findNpR(int n, int r)
return findFact(n) / findFact(n - r);
int findNcR(int n, int r)
return findFact(n) / (findFact(n - r) * findFact(r));
int main()
int n, r, nPr, nCr;
cout << "Enter the value of n:" << endl;
cin >> n;
cout << "Enter the value of r:" << endl;
cin >> r;
nPr = findNpR(n, r);
nCr = findNcR(n, r);
cout << "Permutation,nPr : "<< nPr << endl;
cout << "Combination,nCr : "<< nCr << endl;
Download this example on Github
• The findFact method is a recursive method in this example.
• It returns n == 1 ? 1 : n * findFact(n - 1) i.e. if the current value of n is 1, it returns 1. Else, it returns n * findFact(n - 1) i.e. it recursively calls the same method with n - 1 as the
parameter and multiplies the value with n. Similarly, findFact(n - 1) will return (n - 1) * findFact(n - 2) etc. and the recursive loop will stop when the value of the parameter will be 1.
You will get similar output with this example as well.
Enter the value of n:
Enter the value of r:
Permutation,nPr : 720
Combination,nCr : 120
You can try to run this application and find out permutations and combinations for different values.
You might also like: | {"url":"https://www.codevscolor.com/c-plus-plus-find-permutation-combination","timestamp":"2024-11-11T16:44:20Z","content_type":"text/html","content_length":"102900","record_id":"<urn:uuid:e1847170-99a5-489c-9a00-c1faed682711>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00496.warc.gz"} |
Suspension Design [Archive] - Page 2 - FSAE.com Forums
WHY A SOFT TWIST-MODE???
It has been suggested (on another thread) that while a soft Twist-mode might be advantageous on bumpy roads, it might offer NO such advantages on smooth, sealed roads, such as those typical in
circuit racing.
Those of you who prefer to "follow the numbers", rather than the unquantified opinions of experts, please read on...
Milliken's RCVD, Chapter 18 "Wheel Loads" starts with,
"The [vertical] loads at each wheel are extremely important in determining a car's maximum steady-state cornering capability." (my emphasis).
The chapter goes on to give examples of how to calculate these vertical wheel loads. Quite reasonably, these calculations are simplified by assumptions such as,
"...steady-state operating conditions - that is, smooth roadway, constant speed cornering, constant longitudinal acceleration, constant grade, etc.
... roll rates, spring rates ... are linear,
... chassis of the car ... is [torsionally] rigid.", and so on.
In fact, there are about 5 pages discussing the importance of a torsionally stiff chassis, because,
"... if the chassis torsional spring is weak, attempts to control the lateral load transfer distribution (and "balance" the car's handling by resisting more of the rolling moment on one track than
the other) will be confusing at best and impossible at worst." (my emphasis again).
Equations for calculating the variations to wheel loads from a large number of different factors are then given, including,
* CG position,
* lateral and longitudinal load transfer from horizontal Gs,
* banking,
* crests and dips in the road (albeit in a 2-D vertical-longitudinal plane only),
* aero loads,
* engine torque reaction (for front-engine -> live-rear-axle drivetrain).
It is quite clear, however, from the seven-plus pages devoted to it, that the Millikens believe that Lateral Load Transfer Distribution is the most important factor to be considered when "adjusting
handling balance" (which, in this particular area, I agree with). I repeat this for emphasis, if the wheel loads do NOT change as per your intended LLTD (or Claude's "Magic Number"), then the car
will not handle the way you expect it to.
At the end of the chapter is "18.11 Summary Example". This works through some of the above calculations for what might be a "sportscar", or perhaps a fairly softly sprung racecar (the corner-spring
and ARB rates are a lot less than the tyre rates, so the car is not a very stiffly-sprung aero-car). Right at the very end of the chapter, on page 708 (my older version), is Table 18.1 summarising
the changes in wheel loads due to the various factors. For this particular example the "Banking" effect is quite large (ie. oval track racing), the "Aero" effect quite small (ie. no big wings), and,
quite clearly, the LLTD is by far the most important effect.
Please go through the RCVD example in more detail, but for now take it that the car is slightly front heavy, but roughly with about 900 lbs weight per wheel. There is a Total LLT of about 800 lbs
(from the two inside wheels, to the two outside wheels). This is distributed by the "springs, bars, and RC heights" as +/-430 lbs front, and +/-370 lbs rear, giving LLTD = 54%F, 46%R.
Now the twist in the story. :) Nowhere in this 40 page chapter is any mention made of any TWIST in the road! All four wheelprints are ALWAYS considered to be lying in a PERFECTLY FLAT plane!
Fortunately, there was a large blank space at the end of the chapter, so I added some more calculations. I imagined that the road is very "smooth", but it is also cambered in the usual manner so that
the road surface has a cylindrical shape, which in "end-view" has a radius of about 40 metres. So, if the two edges of the road are 10 metres (30 ft) apart, then the centreline of the road is 0.3
metres (1 ft) higher than the edges (quite typical of real roads).
Driving parallel to the centreline of this road introduces no Twist into the suspension, even if the road curves around a bend. But a car with ~3 metre wheelbase and ~1.5 metre track, driving
diagonally across this road at an angle of about 15 degrees to the centreline, has about 7 mm (1/4") of Twist-mode between its four wheelprints (ie. one diagonal pair of wheelprints are up 7 mm, and
the other diagonal pair down 7 mm, wrt car-body). Please do the calcs to assure yourselves of this.
Furthermore, if the car is doing 100 mph (~45 m/s) while following this diagonal line from the outside of the road towards the inside "apex", then it will spend almost a full second with its
suspension constantly "Twisted" by 7 mm. So the Twist is effectively "steady-state". But when exiting the corner, from inner apex to outside of road, the Twist will be in the opposite direction!
And even furthermore, if the road surface is smoothly cambered "concave up", as is common with banked corners, then the Twist introduced by a diagonal driving line is of the same magnitude as above,
but of opposite sign.
So, the big question:
What does this twist-in-the-road do to your precisely calculated wheel loads?
Based on the (quite soft) corner-spring and ARB rates in the Milliken example, the 1/4" Twist changes the wheel loads by about +/-160 lbs! And depending on which way the Twist is, the LLTD ends up
being either 74%F, 26%R (for corner entry of convex-up road), or 34%F, 66%R (corner exit, convex-up road). Put simply, the handling balance changes from massive understeer on corner entry, to massive
oversteer on corner exit. Yippeeee!!!
Anyway, there are a whole lot of other effects which should also be considered, some of which lessen the above changes, others which exacerbate them. But the bottom line is that with conventional
suspensions, all your precise "handling balance" calculations get tossed out the window as soon as you put the car on a real road. And THE STIFFER THE SPRINGS, especially the Roll and Twist-mode
stiffening Lateral-U-Bars (= ARBs), THE WORSE! Please do the calcs.
Finally, it is worth noting that FSAE's short-wheelbase-small-track cars don't feel the above sort of twist-in-the-road as much as larger cars (because the further the wheelprints are apart, the
further the road surface moves out of a flat-plane). But any "twist-in-the-road" will still change the wheel loads of your FSAE car.
How much? Easy to measure! Put your car on its four corner scales, on FLAT ground. Adjust your spring-mounts so that the corner-weights are symmetrical side-to-side. Now slip a 6 mm thick piece of
plywood under two diagonally opposite wheels (or a single 12 mm piece under one wheel). This represents a Twist-mode of 3 mm (1/8"), which might represent some parts of some of the "wilder" FSAE
tracks. Write down the changes in the four wheel loads.
Now ask yourselves why you bothered doing all those precise "handling balance" calculations in the first place. Because, with conventional suspensions, the road decides what the LLTD is, not you! | {"url":"http://www.fsae.com/forums/archive/index.php/t-8950-p-2.html?s=3f07565c65405f0c5ff2fd2e59ff52ec","timestamp":"2024-11-11T21:18:32Z","content_type":"application/xhtml+xml","content_length":"77697","record_id":"<urn:uuid:823a42ef-253f-4769-bdea-da6b6a34152f>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00484.warc.gz"} |
The Stacks project
Lemma 15.13.2. Let $(A, I)$ be a henselian pair. The functor $B \to B/IB$ determines an equivalence between finite étale $A$-algebras and finite étale $A/I$-algebras.
Proof. Let $B, B'$ be two $A$-algebras finite étale over $A$. Then $B' \to B'' = B \otimes _ A B'$ is finite étale as well (Algebra, Lemmas 10.143.3 and 10.36.13). Now we have $1$-to-$1$
correspondences between
idempotents $e$ of $B''$ such that $B' \to B'' \to eB''$ is an isomorphism.
The bijection between (2) and (3) sends $\sigma : B'' \to B'$ to $e$ such that $(1 - e)$ is the idempotent that generates the kernel of $\sigma $ which exists by Algebra, Lemmas 10.143.8 and
10.143.9. There is a similar correspondence between $A/I$-algebra maps $B/IB \to B'/IB'$ and idempotents $\overline{e}$ of $B''/IB''$ such that $B'/IB' \to B''/IB'' \to \overline{e}(B''/IB'')$ is an
isomorphism. However every idempotent $\overline{e}$ of $B''/IB''$ lifts uniquely to an idempotent $e$ of $B''$ (Lemma 15.11.6). Moreover, if $B'/IB' \to \overline{e}(B''/IB'')$ is an isomorphism,
then $B' \to eB''$ is an isomorphism too by Nakayama's lemma (Algebra, Lemma 10.20.1). In this way we see that the functor is fully faithful.
Essential surjectivity. Let $A/I \to C$ be a finite étale map. By Algebra, Lemma 10.143.10 there exists an étale map $A \to B$ such that $B/IB \cong C$. Let $B'$ be the integral closure of $A$ in
$B$. By Lemma 15.11.5 we have $B'/IB' = C \times C'$ for some ring $C'$ and $B'_ g \cong B_ g$ for some $g \in B'$ mapping to $(1, 0) \in C \times C'$. Since idempotents lift (Lemma 15.11.6) we get
$B' = B'_1 \times B'_2$ with $C = B'_1/IB'_1$ and $C' = B'_2/IB'_2$. The image of $g$ in $B'_1$ is invertible. Then $B_ g = B'_ g = B'_1 \times (B_2)_ g$ and this implies that $A \to B'_1$ is étale.
We conclude that $B'_1$ is finite étale over $A$ (integral étale implies finite étale by Algebra, Lemma 10.36.5 for example) and the proof is done. $\square$
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How To Do Radix Sort in C++
What is Radix Sort?
Radix Sort is a non-comparative sorting algorithm that sorts numbers by processing individual digits. It sorts the numbers in multiple passes, from the least significant digit (LSD) to the most
significant digit (MSD). Radix Sort is particularly effective for sorting integers and is especially useful when dealing with a large volume of data that can fit into the same range.
Time Complexity
The time complexity of Radix Sort depends on the number of digits in the maximum number and the number of elements in the array. The complexities are as follows:
• Best Case: O(d * (n + k)), where d is the number of digits in the maximum number, n is the number of elements, and k is the input range.
• Average Case: O(d * (n + k))
• Worst Case: O(d * (n + k))
Space Complexity
The space complexity of Radix Sort is O(n + k), which accounts for the storage of elements in counting arrays used for each digit.
Radix Sort Pseudocode with Explanation
function radixSort(arr):
maxValue = getMax(arr) // Step 1: Find the maximum value in the array
numDigits = maxDigitLength(maxValue) // Step 2: Determine the number of digits in the maximum value
for digitPosition from 1 to numDigits: // Step 3: Loop through each digit position
countingSort(arr, digitPosition) // Step 4: Sort the array based on the current digit
function getMax(arr):
maxVal = arr[0]
for each num in arr:
if num > maxVal:
maxVal = num
return maxVal
function maxDigitLength(num):
count = 0
while num > 0:
num = num // 10 // Divide by 10 to remove the last digit
count += 1 // Increment digit count
return count
function countingSort(arr, digitPosition):
const int base = 10
output = array of size arr.length // Create an output array to hold the sorted order
count = array of size base initialized to 0 // Initialize count array for digits 0-9
// Step 1: Count occurrences of each digit
for each num in arr:
index = (num // digitPosition) % base // Find the digit in the current position
count[index] += 1 // Increment the count for this digit
// Step 2: Change count[i] to contain the actual position of this digit in output[]
for i from 1 to base - 1:
count[i] += count[i - 1] // Cumulative count
// Step 3: Build the output array
for i from arr.length - 1 down to 0: // Process elements in reverse order for stability
index = (arr[i] // digitPosition) % base
output[count[index] - 1] = arr[i] // Place element in its sorted position
count[index] -= 1 // Decrement the count for the digit
// Step 4: Copy the output array back to arr[]
for i from 0 to arr.length - 1:
arr[i] = output[i]
1. getMax(arr): Finds the maximum value in the array to determine the number of digits.
2. maxDigitLength(maxValue): Calculates the total number of digits in the maximum value.
3. countingSort(arr, digitPosition): Sorts the array based on the current digit using Counting Sort, which is stable and efficient for small ranges.
Radix Sort Implementation in C++
Here is a complete implementation of Radix Sort in C++:
#include <iostream>
#include <vector>
#include <algorithm>
// Function to get the maximum value in the array
int getMax(const std::vector<int>& arr) {
return *std::max_element(arr.begin(), arr.end());
// Counting sort based on the digit at digitPosition
void countingSort(std::vector<int>& arr, int digitPosition) {
const int base = 10;
std::vector<int> output(arr.size());
std::vector<int> count(base, 0);
// Store count of occurrences in count[]
for (int num : arr) {
count[(num / digitPosition) % base]++;
// Change count[i] to contain the actual position of this digit in output[]
for (int i = 1; i < base; i++) {
count[i] += count[i - 1];
// Build the output array
for (int i = arr.size() - 1; i >= 0; i--) {
output[count[(arr[i] / digitPosition) % base] - 1] = arr[i];
count[(arr[i] / digitPosition) % base]--;
// Copy the output array to arr[]
for (int i = 0; i < arr.size(); i++) {
arr[i] = output[i];
// Main radix sort function
void radixSort(std::vector<int>& arr) {
int maxVal = getMax(arr);
for (int digitPosition = 1; maxVal / digitPosition > 0; digitPosition *= 10) {
countingSort(arr, digitPosition);
int main() {
std::vector<int> arr = {170, 45, 75, 90, 802, 24, 2, 66};
std::cout << "Initial array: ";
for (int num : arr) {
std::cout << num << " ";
std::cout << "\nSorted array: ";
for (int num : arr) {
std::cout << num << " ";
return 0;
Initial array: 170 45 75 90 802 24 2 66
Sorted array: 2 24 45 66 75 90 170 802
Step-by-Step Process of Radix Sort
1. Initial Setup
• Input Array: Start with an unsorted array of integers, e.g., {170, 45, 75, 90, 802, 24, 2, 66}.
• Find Maximum Value: Determine the maximum value in the array using the getMax function. This helps in deciding how many digits the largest number has.
2. Determine the Number of Digits
• Count Digits: Use the maxDigitLength function to find out how many digits are in the maximum value. For example, if the maximum value is 802, it has 3 digits.
3. Sorting by Each Digit
The core of Radix Sort involves sorting the array multiple times based on each digit, from the least significant to the most significant.
First Pass (Least Significant Digit – LSD)
• Digit Position: Start with the least significant digit (1s place).
• Counting Sort: Call the countingSort function to sort the array based on the current digit.
□ Counting Occurrences: Count how many times each digit (0-9) appears at this position.
□ Cumulative Count: Update the count array to determine the position of each digit in the output.
□ Build Output Array: Construct the output array by placing elements in their correct positions based on the digit’s count.
□ Copy to Original Array: Copy the output array back to the original array.
Second Pass (Next Significant Digit – Tens Place)
• Digit Position: Move to the next digit (10s place).
• Counting Sort: Repeat the counting sort process for this digit.
□ Count occurrences for the current digit.
□ Update the cumulative count and build the output array.
□ Copy the output back to the original array.
Third Pass (Most Significant Digit – Hundreds Place)
• Digit Position: Now sort by the most significant digit (100s place).
• Counting Sort: Again, use counting sort for this digit.
□ Count occurrences, update cumulative counts, build the output, and copy it.
4. Final Sorted Array
• Sorted Result: After processing all digit positions, the original array will be sorted. For our example, the final sorted array will be {2, 24, 45, 66, 75, 90, 170, 802}.
Summary of Steps
1. Create Buckets: Initialize the necessary structures for counting occurrences of digits (using a counting array).
2. Distribute Elements: For each digit position, determine which bucket (count index) each element belongs to based on its current digit.
3. Sort Buckets: Sort the elements within each bucket using a stable sorting method (Counting Sort).
4. Concatenate: After sorting each digit, concatenate the sorted buckets to form the original array again, ready for the next digit.
5. Repeat: Continue this process for each digit until all digits have been processed.
Performance test for Radix Sort
Radix Sort is known for its efficiency, particularly when sorting large datasets. Unlike comparison-based sorting algorithms, which have a lower bound of O(n log n) time complexity, Radix Sort can
achieve a time complexity of O(d * (n + k)), where:
• d is the number of digits in the maximum number,
• n is the number of elements to be sorted,
• k is the range of the input values.
This makes Radix Sort particularly advantageous when sorting integers or fixed-length strings, especially when the number of digits (d) is significantly smaller than the number of elements (n).
Performance Considerations
In this performance test section, we will analyze how Radix Sort scales with different array sizes and configurations. We will specifically look at:
• Execution Times: By measuring execution times with high precision using the std::chrono library, we can observe how Radix Sort performs across various scenarios—random, sorted, and reverse-sorted
• Scalability: We will assess how the algorithm handles increasing data and whether its performance aligns with the theoretical time complexity.
#include <iostream>
#include <vector>
#include <algorithm>
#include <chrono>
#include <random>
void countingSort(std::vector<int>& arr, int digitPosition);
void radixSort(std::vector<int>& arr);
int getMax(const std::vector<int>& arr);
int maxDigitLength(int num);
void measureRadixSortPerformance(int arraySize, const std::string& configuration);
// Counting sort based on the digit at digitPosition
void countingSort(std::vector<int>& arr, int digitPosition) {
const int base = 10;
std::vector<int> output(arr.size());
std::vector<int> count(base, 0);
// Step 1: Count occurrences of each digit
for (int num : arr) {
count[(num / digitPosition) % base]++;
// Step 2: Change count[i] to contain the actual position of this digit in output[]
for (int i = 1; i < base; i++) {
count[i] += count[i - 1];
// Step 3: Build the output array
for (int i = arr.size() - 1; i >= 0; i--) {
output[count[(arr[i] / digitPosition) % base] - 1] = arr[i];
count[(arr[i] / digitPosition) % base]--;
// Step 4: Copy the output array back to arr[]
for (int i = 0; i < arr.size(); i++) {
arr[i] = output[i];
// Main radix sort function
void radixSort(std::vector<int>& arr) {
int maxVal = getMax(arr);
for (int digitPosition = 1; maxVal / digitPosition > 0; digitPosition *= 10) {
countingSort(arr, digitPosition);
// Function to get the maximum value in the array
int getMax(const std::vector<int>& arr) {
return *std::max_element(arr.begin(), arr.end());
// Function to determine the number of digits in the maximum value
int maxDigitLength(int num) {
int count = 0;
while (num > 0) {
num /= 10;
return count;
// Function to generate an array filled with random integers
std::vector<int> generateRandomArray(int size) {
std::vector<int> arr(size);
std::mt19937 gen(std::random_device{}()); // Random number generator
std::uniform_int_distribution<> dis(0, 1000000); // Range of random numbers
for (int i = 0; i < size; ++i) {
arr[i] = dis(gen);
return arr;
// Function to generate a sorted array
std::vector<int> generateSortedArray(int size) {
std::vector<int> arr(size);
for (int i = 0; i < size; ++i) {
arr[i] = i; // Sorted array from 0 to size-1
return arr;
// Function to generate a reverse sorted array
std::vector<int> generateReverseSortedArray(int size) {
std::vector<int> arr(size);
for (int i = 0; i < size; ++i) {
arr[i] = size - i - 1; // Reverse sorted array
return arr;
// Function to measure the performance of Radix Sort
void measureRadixSortPerformance(int arraySize, const std::string& configuration) {
std::vector<int> arr;
// Generate the array based on the specified configuration
if (configuration == "random") {
arr = generateRandomArray(arraySize);
} else if (configuration == "sorted") {
arr = generateSortedArray(arraySize);
} else if (configuration == "reverse_sorted") {
arr = generateReverseSortedArray(arraySize);
// Measure execution time
auto start = std::chrono::high_resolution_clock::now();
auto end = std::chrono::high_resolution_clock::now();
std::chrono::duration<double, std::milli> duration = end - start; // Duration in milliseconds
std::cout << "Configuration: " << configuration
<< ", Array Size: " << arraySize
<< " - Radix Sort took " << duration.count() << " ms." << std::endl;
int main() {
std::vector<int> sizes = {1000, 10000, 100000, 1000000}; // Array sizes to test
std::vector<std::string> configurations = {"random", "sorted", "reverse_sorted"};
for (const auto& size : sizes) {
for (const auto& config : configurations) {
measureRadixSortPerformance(size, config);
return 0;
Results and Analysis
Configuration Array Size Time (ms)
Random 1000 0.456256
Sorted 1000 0.235989
Reverse Sorted 1000 0.233782
Random 10000 4.52651
Sorted 10000 2.9801
Reverse Sorted 10000 2.96853
Random 100000 47.8618
Sorted 100000 34.8683
Reverse Sorted 100000 31.5582
Random 1000000 438.375
Sorted 1000000 375.626
Reverse Sorted 1000000 389.305
Here is the graphical representation of the data:
Analysis of the Radix Sort Performance Results
The plot above illustrates the performance of Radix Sort across different array sizes and configurations (random, sorted, and reverse sorted). Here are some key observations and conclusions drawn
from the data:
1. Scalability:
□ The execution time increases significantly as the array size grows, consistent with Radix Sort’s expected behavior. This aligns with its theoretical time complexity of O(d * (n + k)), where
larger datasets lead to longer sorting times.
2. Configuration Impact:
□ Random Arrays: Radix Sort takes the longest time with random arrays, especially noticeable at larger sizes (e.g., 438.375 ms for 1,000,000 elements). This is expected due to the
unpredictability of the digit distributions, which can lead to more sorting passes.
□ Sorted Arrays: The algorithm performs well with already sorted arrays, exhibiting the shortest execution times across all sizes.
□ Reverse-Sorted Arrays: The performance of reverse-sorted arrays is slightly slower than that of sorted arrays but better than that of random arrays. This indicates that while Radix Sort is
generally efficient, the initial arrangement of the data still affects its performance.
3. Time Complexity Insights:
□ The differences in execution time highlight the algorithm’s efficiency, particularly in scenarios with sorted or nearly sorted data. The relative constancy of time increases (although
exponential) suggests that Radix Sort remains a strong choice for large datasets, especially when the digit length remains manageable.
4. Practical Applications:
□ Given the performance characteristics observed, Radix Sort is particularly suitable for applications that involve sorting integers or fixed-length strings where the input size is large but
the digit length is relatively small. For instance, sorting numerical IDs or fixed-length strings in databases would be ideal scenarios for Radix Sort.
Summary of Radix Sort Execution Time Behavior
Execution Time Growth: The overhead from sorting multiple digits results in significant time increases, especially in larger datasets, leading to exponential-like growth in execution time.
Linear Time Complexity: Radix Sort operates with a time complexity of O(d * (n + k)), where d is the number of digits, n is the number of elements, and k is the range of values. Execution time
increases linearly with n, influenced by d and k.
Array Size Impact: Larger arrays lead to more comparisons and increased execution time.
Overhead in Each Pass: Each digit processed involves counting sort (O(n + k)), causing cumulative execution time to rise with larger arrays.
Digit Distribution: Non-uniform digit distributions in random arrays can increase the number of operations needed for sorting.
Strengths of Radix Sort
One of Radix Sort’s primary strengths is its ability to sort data in linear time relative to the number of elements, especially when the number of digits in the maximum value is low. Additionally, it
is stable, meaning it preserves the relative order of records with equal keys, which can be advantageous in specific applications.
Weaknesses of Radix Sort
It requires additional memory for counting occurrences and can be less efficient for small datasets or data types with large ranges of values. The overhead of processing multiple digits can lead to
increased execution time when working with larger numbers or non-integer data types.
When to Use Radix Sort
Radix Sort is well-suited for applications such as sorting numerical data, processing keys in databases, and organizing fixed-length strings in text processing. It excels in scenarios where the
dataset is large and the range of values is known, making it an excellent choice for tasks like sorting large lists of IDs or processing large datasets in computational applications.
Radix Sort is a robust non-comparison sorting algorithm that stands out for its efficiency when handling large datasets, particularly those consisting of integers or fixed-length strings. Its time
complexity of O(d * (n + k)) allows it to outperform traditional comparison-based algorithms like Quick Sort and Merge Sort in specific scenarios, especially when the range of input values is
Congratulations on reading to the end of this tutorial!
To implement Radix Sort in Python, read the article How To Do Radix Sort in Python.
For further reading on sorting algorithms in C++, go to the articles:
Have fun and happy researching!
Suf is a senior advisor in data science with deep expertise in Natural Language Processing, Complex Networks, and Anomaly Detection. Formerly a postdoctoral research fellow, he applied advanced
physics techniques to tackle real-world, data-heavy industry challenges. Before that, he was a particle physicist at the ATLAS Experiment of the Large Hadron Collider. Now, he’s focused on bringing
more fun and curiosity to the world of science and research online. | {"url":"https://researchdatapod.com/radix-sort-cpp/","timestamp":"2024-11-13T18:35:54Z","content_type":"text/html","content_length":"128975","record_id":"<urn:uuid:ee227fbc-404d-4584-a9f2-6a9994efec20>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00114.warc.gz"} |
Unwrapping the Mysteries of NumPy: A Comprehensive Guide - Adventures in Machine Learning
Introduction to NumPy unwrap
As an essential mathematical function in programming, NumPy unwrap is an operation that plays a significant role in unwrapping an array. NumPy is an open-source library in Python that deals with
operations on multi-dimensional arrays and matrices.
Among its vast collection of mathematical functions, the “unwrap” function is one of the essential methods. It is used to handle the phase angle (an angle represented in radians, between – to ) of an
array in its most straightforward form.
Unwrapping an array enables a smoother and continuous phase angle plot in scientific studies, signal processing, and other fields. In this article, we will explore the various aspects of NumPy
unwrap, including its syntax, working, and importance.
Syntax of NumPy unwrap Function
Before delving into the details of the NumPy unwrap function, it is essential to understand its syntax. The code snippet below shows how to write the syntax of NumPy unwrap:
numpy.unwrap(p, discont=3.14, axis=-1)
As we can observe, the NumPy unwrap function takes three parameters.
The “p” parameter is the input phase angle array in radians. The “discont” parameter is an optional attribute, representing the numerical discontinuity in the phase angle.
The default value is 3.14. The “axis” parameter denotes the array axis along which to unwrap.
Its default value is -1.
Working with NumPy unwrap
Importance of Unwrapping an Array
In signal processing, we often encounter multiple discontinuities in the data, which may hamper our analysis. Numeric discontinuities reveal a considerable gap between the phase angle value,
introducing irregularities in the plot.
Therefore, it becomes necessary to unwrap an array to ensure smooth and continuous phase angle representation. Unwrapping an array adjusts the phase angle values by adding or subtracting a multiple
of 2 radians or a custom value to create a continuous sequence.
Default NumPy Unwrap Without Attributes
The default NumPy unwrap function modifies phase angle values by adding or subtracting 2 radians (if it exceeds or falls below the range between – to ) to create a continuous sequence. It implements
this process using a simple unwrap method that ignores the discontinuity points between consecutive phase angles.
import numpy as np
p = np.array([0, 1, 6.28, 5.28, 3.14, -3.14, -2.14, -6.28, -7.28])
The output of the above code will be:
array([ 0. , 1. , 0. , -0.99999998, -3.1399999 , 3.14159265, 2.14159265, -0.0015931, 0.9984069 ])
It modifies the phase angle for index 2, 3, 7, 8, adding or subtracting 2 radians to the original value.
This adjustment ensures that the output is a continuous sequence.
NumPy Unwrap with Discont Attribute
This method unwraps an array similarly to the default method by adding or subtracting radians from a problematic point. Still, it takes into account the discontinuity points in the data.
If the input sequence jumps by more than the “discont” value between two consecutive phase angle values, it adds or subtracts 2 radians from the problematic point. > import numpy as np
import numpy as np
p = np.array([0, 1, 6.28, 5.28, 3.14, -3.14, -2.14, -6.28, -7.28])
print(np.unwrap(p, discont=4))
The output of the above code will be:
array([ 0. , 1. , 6.28, 5.28, 3.14, -3.14, -2.14, -2.94, -3.94])
It modifies the phase angles for indices 7 and 8, subtracting 2 radians from the original value to ensure a continuous sequence.
Here, we have set the “discont” parameter to 4, which implies that any jump greater than four radians between two consecutive values is considered a discontinuity.
In conclusion, we have seen the importance of NumPy unwrap in creating smooth and continuous phase angle plots. We have also understood the differences between default unwrapping and specialized
unwrapping using the “discont” parameter.
NumPy unwrap with its array and mathematical operations provides a set of customized functionalities to tackle real-world problems. Therefore, knowledge of NumPy unwrap can be of great advantage to
developers in mathematical computation and data science.
In this article, we have explored the various aspects of NumPy unwrap, an essential mathematical function that unwraps an array. We have discussed the syntax of NumPy unwrap and its importance in
creating smooth and continuous phase angle plots.
We have also covered the differences between default and specialized unwrapping using the “discont” parameter. Lastly, we have learned how NumPy unwrap provides a set of customized functionalities to
tackle real-world problems.
For developers looking to apply NumPy unwrap in their work, the official NumPy documentation provides a comprehensive guide with several examples. The documentation provides detailed information on
the arguments, return values, and examples with sample input and output.
Furthermore, it includes practical examples and sample code to facilitate the learning process. Therefore, consulting the NumPy official documentation can be helpful in understanding the full
potential of NumPy unwrap and other NumPy functions.
Reference to NumPy Official Documentation for More Examples
NumPy unwrap function is one of the most widely used mathematical functions in programming due to its vast array of applications, particularly in signal processing, scientific studies, and other
fields. It enables data scientists and developers to process complex data and gain insights into data trends and patterns.
However, its functionality can range from simple unwrapping to more complex operations that require extensive domain knowledge. The NumPy official documentation provides an extensive guide on NumPy
unwrap, enabling developers to understand the function’s full potential and apply it in their projects.
The documentation includes several directives on how to use NumPy unwrap, its parameters, and return values. Moreover, it also includes practical examples, sample code, and explanations, with outputs
to facilitate the learning process.
Here is an example from the NumPy documentation:
import matplotlib.pyplot as plt
import numpy as np
phase = np.linspace(0, 10*np.pi, 51)
phase[-1] -= 0.1
cos_wave = np.cos(phase)
plt.plot(phase, np.angle(np.exp(1j*phase)), '.')
plt.title('Original phase angle (wrapped)')
plt.plot(phase, np.unwrap(np.angle(np.exp(1j*phase))), '.')
plt.title('Unwrapped phase angle')
The output of the above code generates two figures, with the second figure showcasing the original phase unwrapped, resulting in a smoother and continuous phase angle representation. In conclusion,
NumPy unwrap is an essential mathematical function in programming, with vast applications in signal processing, scientific studies, and other fields.
The NumPy official documentation provides a comprehensive guide to apply NumPy unwrap in practical scenarios. Moreover, its examples and explanatory code are helpful tools for developers to grasp the
full potential of NumPy unwrap and use it to tackle real-world problems.
NumPy unwrap is a crucial mathematical function that handles the phase angle of an array, and it finds its application in signal processing, scientific studies, and other fields. It adjusts the phase
angle values to create smooth and continuous plots by adding or subtracting a multiple of 2 radians or a custom value.
We have discussed the syntax and demonstrated how NumPy unwrap works using practical examples for default and specialized unwrapping. The official NumPy documentation is a valuable resource for
developers looking to apply NumPy unwrap and other NumPy functions.
Understanding NumPy unwrap’s full potential can aid developers in mathematical computation and data science and enable them to process complex data and gain insights into data trends and patterns. | {"url":"https://www.adventuresinmachinelearning.com/unwrapping-the-mysteries-of-numpy-a-comprehensive-guide/","timestamp":"2024-11-05T21:39:01Z","content_type":"text/html","content_length":"77093","record_id":"<urn:uuid:b6e965db-81da-4d19-b8fd-5930bb3cac13>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00346.warc.gz"} |
Transactions Online
Hiroshi NAGAMOCHI, Toshimasa ISHII, Toshihide IBARAKI, "A Simple Proof of a Minimum Cut Algorithm and Its Applications" in IEICE TRANSACTIONS on Fundamentals, vol. E82-A, no. 10, pp. 2231-2236,
October 1999, doi: .
Abstract: For the correctness of the minimum cut algorithm proposed in [H. Nagamochi and T. Ibaraki, Computing edge-connectivity of multigraphs and capacitated graphs, SIAM J. Discrete Mathematics,
5, 1992, pp. 54-66], several simple proofs have been presented so far. This paper gives yet another simple proof. As a byproduct, it can provide an O(m log n) time algorithm that outputs a maximum
flow between the pair of vertices s and t selected by the algorithm, where n and m are the numbers of vertices and edges, respectively. This algorithm can be used to speed up the algorithm to compute
DAG[s,t] that represents all minimum cuts separating vertices s and t in a graph G, and the algorithm to compute the cactus Γ(G) that represents all minimum cuts in G.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/e82-a_10_2231/_p
author={Hiroshi NAGAMOCHI, Toshimasa ISHII, Toshihide IBARAKI, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={A Simple Proof of a Minimum Cut Algorithm and Its Applications},
abstract={For the correctness of the minimum cut algorithm proposed in [H. Nagamochi and T. Ibaraki, Computing edge-connectivity of multigraphs and capacitated graphs, SIAM J. Discrete Mathematics,
5, 1992, pp. 54-66], several simple proofs have been presented so far. This paper gives yet another simple proof. As a byproduct, it can provide an O(m log n) time algorithm that outputs a maximum
flow between the pair of vertices s and t selected by the algorithm, where n and m are the numbers of vertices and edges, respectively. This algorithm can be used to speed up the algorithm to compute
DAG[s,t] that represents all minimum cuts separating vertices s and t in a graph G, and the algorithm to compute the cactus Γ(G) that represents all minimum cuts in G.},
TY - JOUR
TI - A Simple Proof of a Minimum Cut Algorithm and Its Applications
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 2231
EP - 2236
AU - Hiroshi NAGAMOCHI
AU - Toshimasa ISHII
AU - Toshihide IBARAKI
PY - 1999
DO -
JO - IEICE TRANSACTIONS on Fundamentals
SN -
VL - E82-A
IS - 10
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - October 1999
AB - For the correctness of the minimum cut algorithm proposed in [H. Nagamochi and T. Ibaraki, Computing edge-connectivity of multigraphs and capacitated graphs, SIAM J. Discrete Mathematics, 5,
1992, pp. 54-66], several simple proofs have been presented so far. This paper gives yet another simple proof. As a byproduct, it can provide an O(m log n) time algorithm that outputs a maximum flow
between the pair of vertices s and t selected by the algorithm, where n and m are the numbers of vertices and edges, respectively. This algorithm can be used to speed up the algorithm to compute DAG
[s,t] that represents all minimum cuts separating vertices s and t in a graph G, and the algorithm to compute the cactus Γ(G) that represents all minimum cuts in G.
ER - | {"url":"https://global.ieice.org/en_transactions/fundamentals/10.1587/e82-a_10_2231/_p","timestamp":"2024-11-09T03:07:46Z","content_type":"text/html","content_length":"60732","record_id":"<urn:uuid:5b09d3e9-e603-4e3f-90aa-54d1db2a8a0a>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00559.warc.gz"} |
An actively stabilized random number generator includes a random number generator and a feedback controller. The random number generator includes a chaotic physical circuit realizing an iterated
function. The iterated function is configured to produce a trajectory of iterates and has an operating parameter β and a desired Markov operating point. A binary bit converter has a symbol function
configured to produce binary symbols from the trajectory of iterates and a maximal kneading sequence. The feedback controller is configured to observe the maximal kneading sequence within the
trajectory of iterates and adjust the operating parameter to the desired Markov operating point.
The present disclosure generally relates to cryptography, and more specifically, to random number generation via chaotic physical systems.
Random numbers are used in a multitude of fields such as encryption, computer simulations, artificial scene generation, and gambling. Most devices that are used to generate random numbers are
pseudo-random number generators (PRNGS). However, PRNGS are not true-random because they are based on systems and/or algorithms with inherent, and thus determinable, patterns. For applications such
as encryption, true-random generation is important to securing communication. Otherwise, deviation from true-random generation creates potential for said encryption to be “broken” (i.e., the random
number generator possesses exploitable vulnerability). For example, private communication, data sharing, and commerce are just a few applications where unbreakable encryption is becoming increasingly
Currently, devices generating true-random numbers often rely on special equipment based on quantum uncertainty or complex chaotic classical dynamics. However, devices based on quantum uncertainty are
costly, slow, and are therefore not suitable for large-scale integration, and devices based on complex chaotic classical dynamics defy complete analysis and can exhibit vulnerabilities due to
non-ideal realization. All generators based on physical properties are sensitive to changes in environmental parameters, such as ambient temperature.
Therefore, there is a need for true-random generators without vulnerabilities that have the ability for large-scale integration and are tolerant to environmental change.
An actively stabilized random number generator includes a random number generator and a feedback controller. The random number generator includes a chaotic physical circuit designed to realize an
iterated function. The iterated function of the random number generator has a single input state on an interval. The iterated function further includes a single output state being on the same
interval. The iterated function is configured to have a trajectory of iterates, wherein each iterate is within the interval. The iterated function having a positive entropy rate, wherein the entropy
rate defines a rate at which random information is produced. The iterated function defines a unimodal map, with the unimodal map having a slope configuration. The iterated function has an operating
parameter. The operating parameter is on the interval 1<β≤2. The operating parameter corresponds with the slope configuration of the unimodal map. The operating parameter corresponding to the entropy
rate of the iterated function. The operating parameter has a desired Markov operating point.
For a better understanding of the nature and objects of the disclosure, reference should be made to the following detailed description taken in conjunction with the accompanying drawings, in which:
FIG. 1 is a diagram of a chaotic circuit realizing a unimodal map and a feedback controller.
FIG. 2 is a diagram of a unimodal map.
FIG. 3 is a flow chart showing functions performed by the actively stabilized random number generator.
Reference is made in the following detailed description of preferred embodiments to accompanying drawings, which form a part hereof, wherein like numerals may designate like parts throughout that are
corresponding and/or analogous. It will be appreciated that the figures have not necessarily been drawn to scale, such as for simplicity and/or clarity of illustration. For example, dimensions of
some aspects may be exaggerated relative to others. Further, it is to be understood that other embodiments may be utilized. Furthermore, structural and/or other changes may be made without departing
from claimed subject matter. References throughout this specification to “claimed subject matter” refer to subject matter intended to be covered by one or more claims, or any portion thereof, and are
not necessarily intended to refer to a complete claim set, to a particular combination of claim sets (e.g., method claims, apparatus claims, etc.), or to a particular claim.
The present disclosure provides an actively stabilized random number generator (ASRNG) 10, as schematically illustrated in FIG. 1. The ASRNG 10 of the present disclosure generates true random outputs
for applications such as, but not limited to, encryption, computer simulations, and compressive sensing. As one skilled in the art would know, “true random” outputs are unpredictable whereas
“pseudo-random” outputs are deterministic function outputs with suitable statistics designed to mimic true random numbers. As schematically illustrated in FIG. 1, the actively stabilized random
number generator 10 has a random number generator, a binary bit converter 13, and a feedback controller 16.
As seen in FIGS. 1 and 2, the random number generator of the ASRNG 10 has a chaotic physical circuit 18, wherein the chaotic physical circuit 18 is constructed to have an iterated function. The
iterated function defines a unimodal map 22, the unimodal map being a tent map, and the iterated function has a single input state, a single output state, a trajectory of iterates 28, a positive
entropy rate, and an operating parameter.
As seen in FIG. 2, the unimodal map 22 of the iterated function has a reduced-slope configuration 30. As one of ordinary skill in the art would know, a unimodal map 22 may have a full-slope
configuration 31, but for the purpose of this disclosure the reduced-slope configuration 30 is realized because the full-slope configuration is structurally unstable. The operating parameter is
denoted using the symbol “β”. The operating parameter corresponds with the full-slope configuration 31 and the reduced-slope configuration 30 of the unimodal map 22 and is on the interval 1<β<2. The
operating parameter also has a desired Markov operating point. The desired Markov operating point being defined as a corresponding value at which the iterates are considered Markov, or finitely
In the equation below, the single input state is denoted by the variable “x[n]”, with the subscript “n” denoting an iteration number. Generally, the iteration number represents time. The iterated
function is configured to input the single input state and output the single output state. The single input state is on an interval 0<x[n]<1. Each of the single output states create an iterate 32 of
the trajectory of iterates 28 of the unimodal map 22, as seen in FIG. 2. The trajectory of iterates 28 are within the same interval of the single input state.
$x n + 1 = β · { x n , x n ≤ 1 / β 1 - x n , x n > 1 / β$
As one skilled in the art would know, a different trajectory of iterates 28 can be produced given different initial input states. For example, given two scenarios, the first scenario initial input
state has a small difference between an initial input of the second scenario, eventually resulting in vastly differing trajectories. The iterated function results in the trajectory of iterates 28 as
the iteration number is increased, wherein the trajectory of iterates 28 define an overall shape of the unimodal map 22, as shown in FIG. 2.
As seen below by the following equation, a rate at which random information is produced by each trajectory of iterates 28 defines the positive entropy rate, denoted by “h”, which has units of bits
per iteration, and is calculated using the operating parameter:
In the present disclosure, the value of the operating parameter of the actively stabilized random number generator (ASRNG) 10 is β<2, corresponding with the positive entropy rate being h<1 bit per
iteration by the equation above.
Each one of the trajectory of iterates 28 are then processed by the binary bit converter 13 of the ASRNG 10. The binary bit converter 13 is configured to have a symbol function defined by the
following equation:
$S n = { 0 , x n ≤ 1 / β 1 , x n > 1 / β$
The symbol function produces a binary symbol, which is a bit, for each iterate 32 of the trajectory of iterates 28, denoted in the equation above by “S[n]”. Generally, a sequence of bits is produced
as the symbol function translates a plurality of iterates of the iterated function as the iterated function increases in iteration number. A natural partition, relative to a critical point 33 of the
unimodal map 22, the critical point 33 being configured by the operating parameter, is utilized within the symbol function, resulting in the symbol function having a symbol rate, which is one bit per
iteration. The symbol rate being defined as the rate at which bits are produced via the symbol function. As one skilled in the art would know, the entropy rate cannot exceed the symbol rate, and the
amount by which the symbol rate exceeds the entropy rate is due to bias and interdependence in binary symbols produced by the binary bit converter 13. Methods to remove bias from the binary symbols
are known to one skilled in the art. Interdependence, which is defined as a dependence on prior outcomes of the iterated function, results in vulnerability within the ASRNG 10. The vulnerability is
mitigated using the feedback controller 16, as described in further detail below.
To be able to generate independent random binary symbols, the feedback controller 16 is integrated into the chaotic physical circuit 18 to correct for interdependence within the ASRNG 10, as a result
of the positive entropy rate being h<1 bit per symbol, and to automatically correct for physical deviations due to circuit implementation (e.g., manufacturing defects or environmental changes). The
feedback controller 16 exploits the Markov properties associated with the operating parameter utilizing kneading theory. As one skilled in the art would know, kneading theory states that the desired
Markov operating point of the operating parameter is defined by a unique maximal kneading sequence. As a result, each iterate 32 of the iterated function is considered Markov and emits random
sequences that order less than or equal to its corresponding maximal kneading sequence. The feedback controller 16 observes the maximal kneading sequence within the trajectory of iterates 28 to
determine if the operating parameter needs to be increased or decreased in order to maintain the iterates at the desired Markov operating point, and to compensate for the physical deviations of the
chaotic physical circuit 18 that may cause the operating parameter to change. One skilled in the art would know that methods exist for extracting random bit sequences without bias and interdependence
from the system with the desired Markov operating point.
Described above is the mathematical construction of the actively stabilized random number generator 10. As previously mentioned, the actively stabilized random number generator 10 can be realized
using the chaotic physical circuit 18, as seen in FIG. 1. As one skilled in the art would know, the chaotic physical circuit 18 contains both analog and digital circuitry. The analog circuitry
contains a capacitor C 34 and a negative resistor −R 36, as seen in FIG. 1. The practical realization of an electronic negative resistor is known to one skilled in the art. The digital circuitry has
a RS flip flop 38 and a clocked D flip flop 40. In addition, the chaotic physical circuit 18 has a comparator 42 for detecting if voltage above the capacitor C 34 exceeds a threshold V[T ]44 and a
buffer circuit that converts digital to analog voltages where a digital F 46 generates a zero 48, and a digital T 50 generates twice the threshold 2V[T ]52. An input clock signal 54 is a periodic
impulse of a short duration that resets an output of the RS flip flop 38 and loads data into the clocked D flip flop 40. As one skilled in the art would understand, a short delay τ 56 is included
between the RS flip flop 38 and the clocked D flip flop 40 to allow the D flip flop to latch the RS flip flop output prior to the RS flip flop's reset via the clock signal 54. As seen in FIG. 1, the
chaotic physical circuit 18 outputs a digital signal S 58 and the clock signal 54 which are synchronized, relative to a feedback controller portion 60 of the chaotic physical circuit. The digital
signal S 58 exhibits a new random bit with each iterate 32 of the iterated function. Random numbers of the ASRNG system 10 are derived from the unimodal map 22 utilizing the operating parameter. The
operating parameter value “β” is derived using the following equation:
$β = exp { T RC }$
In the equation above, “exp” indicates an exponential function, “T” is a clock period, “R” is a magnitude of a negative resistance, and “C” is a capacitance. The chaotic circuit 18 is designed such
that “T” of the clock period is within a parameter of 0<T<RC ln(2). When “R” is the magnitude of the negative resistance is multiplied by “C” is the capacitance and the natural logarithm 2; such that
the parameters for the unimodal map 22 are 1<β<2.
FIG. 1 further shows a schematic illustration of the feedback controller 16 integrated into the chaotic physical circuit 18, forming the feedback controller portion 60 of the chaotic physical
circuit. The outputs of the chaotic physical circuit 18 relative to the feedback controller 16 are S 58 and the synchronized clock signal 54 signals, which are then entered into a shift register 62
of the feedback controller portion 60. Random bits that are conveyed on S 58 are converted to a parallel representation using the shift register 62 driven by the clock signal 54. A fixed register 64
stores the maximal kneading sequence corresponding to the desired Markov operating point. A size of the shift register 62 in bits is matched to a size of the fixed register 64 that is required to
store the maximal kneading sequence. A parallel output of the shift register 62 is bit-wise compared to the maximal kneading sequence in a XOR/AND block 66 of the feedback controller portion 60. A
digital output from a comparison from the XOR/AND block 66 indicates a true state, if and only if, contents of the shift register 62 exactly match the maximal kneading sequence stored in the fixed
register 64.
The digital output of the comparison from the XOR/AND block 66 is converted to two analog voltage level signals, with a logical F translating to voltage V1 68 and a logical T to voltage V2 70. These
analog voltage level signals are the equivalent of an analog signal 72. The analog signal 72 passes through a low-pass filter 74 with a time constant that is sufficiently long to yield an output
voltage V3 signal 76 that is inversely proportional to a relative occurrence of the maximal kneading sequence within a random bit stream. The output voltage V[3 ]signal 76 is added to a fixed voltage
V[4 ]78 that provides a coarse operating point to aid targeting the desired Markov operating point using an output signal V[F ]80. The feedback controller 16 output signal V[F ]80 can be used to
control the operating parameter via voltage control of either the negative resistance −R 36, the capacitance C 34, or a clock period. For purposes of clarity, FIG. 3 schematically illustrates the
steps of the ASRNG 10 in accordance with the present disclosure.
The present disclosure provides the chaotic physical circuit 18 that generates random numbers and may be built using readily available electronic parts. Detection of the maximal kneading sequence
within the trajectory of iterates 28 of the iterated function provides a built-in diagnostic for monitoring the positive entropy of the random number generator, forming the ASRNG 10.
The foregoing description has been presented for the purposes of illustration and description. It is not intended to be exhaustive or to limit the disclosure to the precise form disclosed. Many
modifications and variations are possible in view of this disclosure. Indeed, while certain features of this disclosure have been shown, described and/or claimed, it is not intended to be limited to
the details above, since it will be understood that various omissions, modifications, substitutions and changes in the apparatuses, forms, method, steps and system illustrated and in its operation
can be made by those skilled in the art without departing in any way from the spirit of the present disclosure.
Furthermore, the foregoing description, for purposes of explanation, used specific nomenclature to provide a thorough understanding of the disclosure. However, it will be apparent to one skilled in
the art that the specific details are not required in order to practice the disclosure. Thus, the foregoing descriptions of specific embodiments of the present disclosure are presented for purposes
of illustration and description. They are not intended to be exhaustive or to limit the disclosure to the precise forms disclosed, many modifications and variations are possible in view of the above
teachings. The embodiments were chosen and described in order to best explain the principles of the disclosure and its practical applications, to thereby enable others skilled in the art to best
utilize the disclosed system and method, and various embodiments with various modifications as are suited to the particular use contemplated.
1. An actively stabilized random number generator comprising:
a random number generator; and
a feedback controller,
wherein the random number generator comprises: a chaotic physical circuit that realizes an iterated function, the iterated function having a single input state on an interval and a single output
state within the interval, the iterated function configured to have a trajectory of iterates, wherein each iterate is within the interval, the iterated function having a positive entropy rate
that defines a rate at which random information is produced; wherein the iterated function defines a unimodal map, the iterated function having an operating parameter (β) on an interval of 1<β≤2,
the operating parameter determining a slope configuration of the unimodal map, the operating parameter corresponding to the entropy rate of the iterated function, the operating parameter having a
desired Markov operating point; and
wherein the feedback controller is configured to observe a maximal kneading sequence within the trajectory of iterates and to adjust the operating parameter to the desired Markov operating point.
2. The actively stabilized random number generator of claim 1, wherein the feedback controller is further configured to maintain the operating parameter at the desired Markov operating point.
3. The actively stabilized random number generator of claim 1, wherein the actively stabilized random number generator further comprises a binary bit converter, the binary bit converter realizing a
symbol function, the symbol function having a natural partition and a symbol rate, the natural partition being configured to generate binary symbols, the symbol rate being one binary symbol generated
per iteration, the symbol rate being greater than or equal to the entropy rate.
4. The actively stabilized random number generator of claim 1, wherein the slope configuration of the unimodal map is a reduced slope configuration.
5. The actively stabilized random number generator of claim 4, wherein the symbol rate is less than the entropy rate of the chaotic physical circuit.
6. The actively stabilized random number generator of claim 4, wherein the reduced slope configuration has the condition β<2.
7. The actively stabilized random number generator of claim 1, wherein the slope configuration of the unimodal map is a full height configuration.
8. The actively stabilized random number generator of claim 7, wherein the full slope configuration has the condition β=2.
Patent History
Publication number
: 20220308836
: Mar 23, 2021
Publication Date
: Sep 29, 2022
Patent Grant number
12086570 Inventors
Ned J. Corron
(Madison, AL),
Jonathan N. Blakely
(Madison, AL)
Application Number
: 17/209,680
International Classification: G06F 7/58 (20060101); | {"url":"https://patents.justia.com/patent/20220308836","timestamp":"2024-11-06T12:40:48Z","content_type":"text/html","content_length":"79329","record_id":"<urn:uuid:9777c6bf-9be7-4e09-8b8d-10d11b0f19de>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00328.warc.gz"} |
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Unsteady Couette Flow through a Porous Medium in a Rotating System
Unsteady Couette Flow through a Porous Medium in a Rotating System ()
1. Introduction
The flow between two parallel plates is a classical problem that has many applications in accelerators, aerodynamic heating, electrostatic precipitation, polymer technology, petroleum industry,
purification of crude oil, fluid droplets and sprays. Such a flow model is of great interest, not only for its theoretical significance, but also for its wide applications to geophysics and
engineering. A lot of research work concerning the flow between two parallel plates studied in a rotating system have appeared, for example, Batchelor [1], Ganapathy [3], Gupta [4] and Mazumder [5].
The flows through porous medium are very much prevalent in nature and therefore, the study of such flows has become of principal interest in many scientific and engineering applications. This type of
flows has shown their great importance in petroleum engineering to study the movements of natural gas, oil and water through the oil reservoirs; in chemical engineering for the filtration and water
purification processes. Further, to study the underground water resources and seepage of water in river beds one need the knowledge of the fluid flow through porous medium. Therefore, there are
number of practical uses of the fluid flow through porous media. Rotation has an immense importance in various phenomena such as in cosmical fluid dynamics, meteorology, geophysical fluid dynamics,
gaseous and nuclear reactors and many engineering applications, that is why, the study of Couette flow through porous medium in a rotating system enhances an interest to the researchers due to its
applications in the aforesaid area. Such a study has a greater importance in the design of turbines and turbo mechanics, in estimating the flight path of rotating wheels and spin-stabilized missiles.
A large number of investigations has been made on the flow through a porous medium in a rotating system. In general, most of solutions for unsteady flows of viscous fluids are in a series form. These
series may be rapidly convergent for large values of the time but slowly convergent for small values of the time or vice versa. Sometimes, it can be difficult to obtain the solution for small values
of the time but it can be easy to obtain it for large values of the time and the opposite can also be true. Vidyanidhi and Nigam [6] studied the channel flow between rotating parallel plates under
constant pressure gradient. Jana and Dutta [7] studied the steady Couette flow of a viscous incompressible fluid between two infinite parallel plates, one stationary and the other moving with uniform
velocity, in a rotating frame of reference. Singh and Sharma [8] have presented the three dimensional Couette flow through porous media. A periodic solution of oscillatory Couette flow through a
porous medium in rotating system has been obtained by Singh et al. [9]. Guria et al. [10] have described the unsteady Couette flow in a rotating system. Das et al. [10] have studied the unsteady
Couette flow with an oscillatory velocity of one of the plates in a rotating system. The unsteady MHD Couette flow in a rotating system has been investigated by Das et al. [12]. Attia [13] has
studied the effect of porosity on unsteady Couette flow with heat transfer in the presence of uniform suction and injection. Israel-Cookey et al. [14] have presented the MHD oscillatory Couette flow
of a radiating viscous fluid in a porous medium with periodic wall temperature. The unsteady hydromagnetic Couette flow through a porous medium in a rotating system have been presented Prasad and
Kumar [15]. Das et al. [16] have studied the Couette flow through porous medium in a rotating system.
In the present paper, we have studied the unsteady Couette flow between two infinite horizontal parallel plates in a porous medium in a rotating system when one of the plate moving with uniform
velocity and the other one held at rest. The fluid and plates are in a state of rigid body rotation with uniform angular velocity
2. Mathematical Formulation and Its Solution
Consider the unsteady flow of a viscous incompressible fluid between two infinite parallel porous plates embedded in a porous medium. The plates are separated by a distance h. The fluid and channel
rotate in unison about an axis normal to the planes of the plates with a uniform angular velocityFigure 1). Flow within the channel is induced due to the motion of the upper plate at
The equation of continuity gives
The x-, yand z-components of Navier-Stokes equation are
The initial and boundary conditions are
Introducing the non-dimensional variables
Equations (1) and (3) become
Combing Equations (6) and (7), we have
The initial and boundary conditions for
Taking the Laplace transform, Equation (8) becomes
The boundary conditions for
The solution of Equation (12) subject to the boundary conditions (14) is
The inverse Laplace’s transform of Equation (15) is
On separating into a real and imaginary parts, we get
The solution given by Equations (19) and (20) exists for both Re < 0 (corresponding to
Solutions for Small Time
Following Carslaw and Jaegar [17], for small time, the solution of (12) subject to the boundary conditions (14) is obtained by Laplace transform technique in the following form
The solution (22) can be written as
On separating into a real and imaginary parts, we get the velocity distributions for the primary and the secondary flow as
Equations (25) and (26) describe the fluid velocities for small time.
3. Results and Discussion
To study the effects of rotation, Reynolds number and porosity parameter on the velocity distributions we have presented the non-dimensional velocity components Figure 2 that the primary velocity
Figure 3 reveals that both the primary velocity Figure 4 that both the primary velocity
Figure 2. Velocities u[1] and w[1] for different K^2 when Re = 2, σ = 0.1 and τ = 0.2.
Figure 3. Velocities u[1] and w[1] for different Re only when K^2 = 2, σ = 0.1 and τ = 0.2.
Figure 4. Velocities u[1] and w[1] for different σ when K^2 = 2, Re = 2 and τ = 0.2.
flow field decreases as the porosity parameter Figure 5 that both the primary velocity
The non-dimensional shear stresses at the stationary plate
On separating into a real and imaginary parts, we get the shear stress components due to the primary and secondary flows at the stationary plate
Figure 5. Velocities u[1] and w[1] for different time τ when K^2 = 2, Re = 2 and σ = 0.1.
Figure 6. Velocity u[1] for general solution and solution for small time when K^2 = 2, Re = 2 and σ = 0.1.
Figure 7. Velocity w[1] for general solution and solution for small time when K^2 = 2, Re = 2 and σ = 0.1.
The numerical values of the non-dimensional shear stresses Figure 8 that both the absolute value of the shear stresses
For small times, the non-dimensional shear stresses due to the primary and secondary flows at the stationary plate
On separating into a real and imaginary parts, we get the shear stress components due to the primary and secondary flows as
For small time, the numerical values of the shear stress components calculated from Equations (29), (30), (33) and (34) are given in Tables 1 and 2 for several values of Re and
We shall now discuss the asymptotic behavior of the solutions (25) and (26) for small and large values of
Case 1): When
When Re is large and
Figure 8. Shear stresses τ[x] and τ[y] for different K^2 when σ = 0.1 and τ = 0.2.
Figure 9. Shear stresses τ[x] and τ[y] for different σ when K^2 = 2 and τ = 0.2.
Figure 10. Shear stresses τ[x] and τ[y] for different τ when K^2 = 2 and σ = 0.1.
Table 1. Shear stress 10τ[x] due to primary flow when K^2 = 2 and σ = 0.1.
Table 2. Shear stress −10^2τ[y] due to secondary flow when K^2 = 2 and σ = 0.1.
It is evident from Equations (37) and (38) that there exists a single-deck boundary layer of thickness of order
Case 2): When
In this case, the velocity distributions are obtained from the Equations (36) as
Equations (40) and (41) show that there exists a singledecker boundary layer of thickness of order
4. Conclusion
The unsteady Couette flow of a viscous incompressible fluid through a porous medium in a rotating system has been investigated. It is found that the primary velocity decreases and the magnitude of
the secondary velocity increases with an increase in rotation parameter. The fluid velocity components decrease with an increase in Reynolds number. An increase in the porosity of the medium both the
primary and the secondary velocities increase. That is, the porosity of the medium has an accelerating influence on the flow field. In turn, it can control the velocity field by introducing porous
medium in a rotating system. It is also found that the solution for small time converges more rapidly than the general solution. For steady state, the asymptotic behavior of the solution is analyzed
for small as well as large values of rotation parameter and Reynolds number. It is observed that a thin boundary layer is formed near the moving plate of the channel and the thicknesses of the layer
increases with an increase in porosity parameter. | {"url":"https://www.scirp.org/journal/paperinformation?paperid=25720","timestamp":"2024-11-12T18:31:26Z","content_type":"application/xhtml+xml","content_length":"130212","record_id":"<urn:uuid:e11fd8ee-bed5-40da-b563-8a3bba6ffc3b>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00706.warc.gz"} |
Learn how to display incoming audio data as a spectrogram by using the FFT class of the DSP module. Understand the benefits of using a Fast Fourier Transform.
Windows , macOS , Linux
dsp::FFT, Image, Colour, FloatVectorOperations
Getting started
Download the demo project for this tutorial here: PIP | ZIP . Unzip the project and open the first header file in the Projucer.
If your operating system requires you to request permission to access the microphone (currently iOS, Android and macOS Mojave) then you will need to set the corresponding option under the
relevant exporter in the Projucer and resave the project.
If you need help with this step, see Tutorial: Projucer Part 1: Getting started with the Projucer.
The demo project
When completed, the demo project will display the incoming audio data as a three-dimensional spectrogram in the time (x-axis), frequency (y-axis) and amplitude (colour) domains. The values displayed
on the screen will be updated 60 times a second and the window at any time frame may look something like this:
The code presented here is broadly similar to the SimpleFFTExample from the JUCE Examples.
The Fast Fourier Transform
A time or space domain signal can be converted to the frequency domain by using a transformation formula called the Fourier transform. A common efficient implementation of this transformation
function is the Fast Fourier Transform or FFT, which is included in the JUCE DSP module and which we will use in this tutorial.
The FFT allows us to decompose an audio signal into its frequencies and represent the magnitude and phase information for each of these frequencies. Using its inverse function, we can revert the
signal into its original domain thus making it really useful to process individual frequency components such as for filtering.
Since this tutorial only deals with displaying the audio data without actual processing for output, we focus on the forward FFT rather than the inverse FFT.
Processing Audio Data
Currently our application does not display nor process any incoming audio signals so let's start by implementing the FFT.
FFT Initialisation
In the SpectrogramComponent class, start by defining some useful constants for the FFT implementation:
static constexpr auto fftOrder = 10; // [1]
static constexpr auto fftSize = 1 << fftOrder; // [2]
• [1] : The FFT order designates the size of the FFT window and the number of points on which it will operate corresponds to 2 to the power of the order. In this case, let's use an order of 10
which will produce an FFT with 2 ^ 10 = 1024 points.
• [2] : To calculate the corresponding FFT size, we use the left bit shift operator which produces 1024 as binary number 10000000000.
Next, declare private member variables required for the FFT implementation as shown below:
juce::dsp::FFT forwardFFT; // [3]
juce::Image spectrogramImage;
std::array<float, fftSize> fifo; // [4]
std::array<float, fftSize * 2> fftData; // [5]
int fifoIndex = 0; // [6]
bool nextFFTBlockReady = false; // [7]
• [3] : Declare a dsp::FFT object to perform the forward FFT on.
• [4] : The fifo float array of size 1024 will contain our incoming audio data in samples.
• [5] : The fftData float array of size 2048 will contain the results of our FFT calculations.
• [6] : This temporary index keeps count of the amount of samples in the fifo.
• [7] : This temporary boolean tells us whether the next FFT block is ready to be rendered.
Now let's initialise these variables in the member initialisation list of our constructor like so:
: forwardFFT (fftOrder),
spectrogramImage (
::RGB, 512, 512, true)
The FFT object has to be explicitly initialised with the correct order at this point.
In the overriden getNextAudioBlock() function, we simply push all the samples contained in our current audio buffer block to the fifo to be processed at a later time:
void getNextAudioBlock (const juce::AudioSourceChannelInfo& bufferToFill) override
if (bufferToFill.buffer->getNumChannels() > 0)
auto* channelData = bufferToFill.buffer->getReadPointer (0, bufferToFill.startSample);
for (auto i = 0; i < bufferToFill.numSamples; ++i)
pushNextSampleIntoFifo (channelData[i]);
To push the sample into the fifo, implement the pushNextSampleIntoFifo() function as described below:
void pushNextSampleIntoFifo (float sample) noexcept
// if the fifo contains enough data, set a flag to say
// that the next line should now be rendered..
if (fifoIndex == fftSize) // [8]
if (! nextFFTBlockReady) // [9]
std::fill (fftData.begin(), fftData.end(), 0.0f);
std::copy (fifo.begin(), fifo.end(), fftData.begin());
nextFFTBlockReady = true;
fifoIndex = 0;
fifo[(size_t) fifoIndex++] = sample; // [9]
• [8] : If the fifo contains enough data in this case 1024 samples, we are ready to copy the data to the fftData array for it to be processed by the FFT. We also set a flag to say that the next
line should now be rendered and always reset the index to 0 to start filling the fifo again.
• [9] : Every time this function gets called, a sample is stored in the fifo and the index is incremented.
The fifo data now occupies the first half of the FFT input array and is ready to be processed and displayed.
Displaying the Spectrogram
In the drawNextLineOfSpectrogram() function, insert the pixel drawing implementation as explained below:
void drawNextLineOfSpectrogram()
auto rightHandEdge = spectrogramImage.getWidth() - 1;
auto imageHeight = spectrogramImage.getHeight();
// first, shuffle our image leftwards by 1 pixel..
spectrogramImage.moveImageSection (0, 0, 1, 0, rightHandEdge, imageHeight); // [1]
// then render our FFT data..
forwardFFT.performFrequencyOnlyForwardTransform (fftData.data()); // [2]
// find the range of values produced, so we can scale our rendering to
// show up the detail clearly
auto maxLevel = juce::FloatVectorOperations::findMinAndMax (fftData.data(), fftSize / 2); // [3]
juce::Image::BitmapData bitmap { spectrogramImage, rightHandEdge, 0, 1, imageHeight, juce::Image::BitmapData::writeOnly }; // [4]
auto y
= 1;
< imageHeight; ++
// [5]
skewedProportionY = 1.0f - std::exp (std::log ((
/ (
) imageHeight) * 0.2f);
auto fftDataIndex = (size_t) juce::jlimit (0, fftSize / 2, (int) (skewedProportionY * fftSize / 2));
auto level = juce::jmap (fftData[fftDataIndex], 0.0f, juce::jmax (maxLevel.getEnd(), 1e-5f), 0.0f, 1.0f);
bitmap.setPixelColour (0,
, juce::Colour::fromHSV (level, 1.0f, level, 1.0f));
// [6]
• [1] : First, shuffle the image leftwards by 1 pixel using the moveImageSection() function on the Image object. Specify the image section as the whole width minus one pixel and the whole height.
• [2] : Then, render the FFT data using the performFrequencyOnlyForwardTransform() function on the FFT object with the fftData array as an argument.
• [3] : Find the range of values produced, so that we can scale our rendering to show up the detail clearly. We can do so using the FloatVectorOperations::findMinAndMax() function.
• [4] : Create a BitmapData instance that refers to the rightmost column of pixels in the spectrogram image. When reading or writing to multiple pixels in an image, a BitmapData instance can be
used to buffer pixel values internally, reading or writing them all at once. This approach is normally faster than using the member functions of Image to access individual pixels.
• [5] : Now in the for loop for every pixel in the spectrogram height, calculate the level proportionally to the sample set. To do this, we first need to skew the y-axis to use a logarithmic scale
to better represent our frequencies. We can then feed this scaling factor to retrieve the correct array index and use the amplitude value to map it to a range between 0.0 .. 1.0 .
• [6] : Finally set the appropriate pixel with the correct colour to display the FFT data.
As a final step, update the spectrogram using the timer callback function by calling the drawNextLineOfSpectrogram() only when the next FFT block is ready, reset the flag and update the GUI using the
repaint() function:
void timerCallback() override
if (nextFFTBlockReady)
nextFFTBlockReady = false;
Try to increase the resolution of the FFT and change the rate at which the spectrogram updates.
The source code for this modified version of the code can be found in the SimpleFFTTutorial_02.h file of the demo project.
In this tutorial, we have learnt how to use an FFT function to display audio data in a spectrogram. In particular, we have:
• Learnt the basics of a fast fourier transform function.
• Processed audio sample by sample using a fifo.
• Displayed the data in an Image object pixel by pixel.
See also | {"url":"https://docs.juce.com/master/tutorial_simple_fft.html","timestamp":"2024-11-14T06:45:42Z","content_type":"application/xhtml+xml","content_length":"45586","record_id":"<urn:uuid:464b60e7-b438-4d55-8742-e41e978f0c5c>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00514.warc.gz"} |
Percentage 18663 - math word problem (18663)
Percentage 18663
Of the 1,500 light bulbs produced, 21 were defective. What percentage of light bulbs were flawless?
Correct answer:
Did you find an error or inaccuracy? Feel free to
write us
. Thank you!
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You need to know the following knowledge to solve this word math problem:
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2135 (number)
Interesting facts about the number 2135
• (2135) Aristaeus is asteroid number 2135. It was discovered by S. J. Bus; E. F. Helin from Mount Palomar Observatory on 4/17/1977.
• There is a 2,135 miles (3,435 km) direct distance between Alexandria (Egypt) and Sale (Morocco).
• There is a 1,327 miles (2,135 km) direct distance between Ashgabat (Turkmenistan) and Voronezh (Russia).
• There is a 2,135 miles (3,435 km) direct distance between Bareilly (India) and Ho Chi Minh City (Viet Nam).
• There is a 2,135 miles (3,435 km) direct distance between Barquisimeto (Venezuela) and New York City (USA).
• More distances ...
• There is a 1,327 miles (2,135 km) direct distance between Beirut (Lebanon) and Saratov (Russia).
• There is a 2,135 miles (3,435 km) direct distance between Brooklyn (USA) and Maracay (Venezuela).
• There is a 1,327 miles (2,135 km) direct distance between Budta (Philippines) and Malang (Indonesia).
• There is a 1,327 miles (2,135 km) direct distance between Guadalupe (Mexico) and Charlotte (USA).
• There is a 1,327 miles (2,135 km) direct distance between Hohhot (China) and Tainan (Taiwan).
• There is a 1,327 miles (2,135 km) direct distance between Karaj (Iran) and Ufa (Russia).
• There is a 1,327 miles (2,135 km) direct distance between La Paz (Bolivia) and Quito (Ecuador).
• There is a 2,135 miles (3,435 km) direct distance between Malang (Indonesia) and Yunfu (China).
• There is a 1,327 miles (2,135 km) direct distance between Nagoya-shi (Japan) and Xinyang (China).
• There is a 1,327 miles (2,135 km) direct distance between Ra’s Bayrūt (Lebanon) and Saratov (Russia).
• There is a 2,135 miles (3,435 km) direct distance between Singapore (Singapore) and Wuhan (China).
History and politics
• United Nations Security Council Resolution number 2135, adopted 30 January 2014. Situation in Cyprus, extends mandate of UNFICYP. Resolution text.
• 2135 is a value of n for which σ(n-1) + σ(n+1) = σ(2n).
What is 2,135 in other units
The decimal (Arabic) number
converted to a
Roman number
Roman and decimal number conversions
The number 2135 converted to a Mayan number is
Decimal and Mayan number conversions.
Length conversion
2135 kilometers (km) equals to
miles (mi).
2135 miles (mi) equals to
kilometers (km).
2135 meters (m) equals to
feet (ft).
2135 feet (ft) equals 650.756 meters (m).
Power conversion
2135 Horsepower (hp) equals to 1570.08 kilowatts (kW)
2135 kilowatts (kW) equals to 2903.19 horsepower (hp)
Time conversion
(hours, minutes, seconds, days, weeks)
2135 seconds equals to 35 minutes, 35 seconds
2135 minutes equals to 1 day, 11 hours, 35 minutes
Number 2135 morse code:
..--- .---- ...-- .....
Sign language for number 2135:
Number 2135 in braille:
Gregorian, Hebrew, Islamic, Persian and Buddhist Year (Calendar)
Gregorian year 2135 is Buddhist year 2678.
Buddhist year 2135 is Gregorian year 1592 .
Gregorian year 2135 is Islamic year 1559 or 1560.
Islamic year 2135 is Gregorian year 2692 or 2693.
Gregorian year 2135 is Persian year 1513 or 1514.
Persian year 2135 is Gregorian 2756 or 2757.
Gregorian year 2135 is Hebrew year 5895 or 5896.
Hebrew year 2135 is Gregorian year 1625 a. C.
The Buddhist calendar is used in Sri Lanka, Cambodia, Laos, Thailand, and Burma. The Persian calendar is the official calendar in Iran and Afghanistan.
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Advanced math operations
Is Prime?
The number 2135 is not a
prime number
. The closest prime numbers are
The 2135th prime number in order is
Factorization and factors (dividers)
The prime factors of 2135 are
5 * 7 * 61 The factors of
2135 are
, 2135.
Total factors 8.
Sum of factors 2976 (841).
Prime factor tree
The second power of 2135
is 4.558.225.
The third power of 2135
is 9.731.810.375.
The square root √
is 46,20606.
The cube root of
is 12,876543.
The natural logarithm of No. ln 2135 = log
2135 = 7,666222.
The logarithm to base 10 of No. log
2135 = 3,329398.
The Napierian logarithm of No. log
2135 = -7,666222.
Trigonometric functions
The cosine of 2135 is 0,283836.
The sine of 2135 is -0,958873.
The tangent of 2135 is -3,378269.
Number 2135 in Computer Science
Code type Code value
PIN 2135 It's recommended that you use 2135 as your password or PIN.
2135 Number of bytes 2.1KB
Unix time Unix time 2135 is equal to Thursday Jan. 1, 1970, 12:35:35 a.m. GMT
IPv4, IPv6 Number 2135 internet address in dotted format v4 0.0.8.87, v6 ::857
2135 Decimal = 100001010111 Binary
2135 Decimal = 2221002 Ternary
2135 Decimal = 4127 Octal
2135 Decimal = 857 Hexadecimal (0x857 hex)
2135 BASE64 MjEzNQ==
2135 MD5 9a49a25d845a483fae4be7e341368e36
2135 SHA1 ff075d214040a5d1fa56101542a4aaca51f73f3e
2135 SHA224 7dd828c0fb69cac7d20f5a237783c08d706db5f03a58abbe840cdc71
2135 SHA256 fb48aaf65ec04e4bcfcb01a417e4a0c20297982b5fc67845b287707c0a95c465
2135 SHA384 24cb2c5a0d759c4ead08e3b08b3a730587cf84a98b261bf6ca0234d284cbbef99cc085bc3050175ca1d57433c8a00b5d
More SHA codes related to the number 2135 ...
If you know something interesting about the 2135 number that you did not find on this page, do not hesitate to write us here.
Numerology 2135
The meaning of the number 5 (five), numerology 5
Character frequency 5: 1
The number five (5) came to this world to achieve freedom. You need to apply discipline to find your inner freedom and open-mindedness. It is about a restless spirit in constant search of the truth
that surrounds us. You need to accumulate as much information as possible to know what is happening in depth. Number 5 person is intelligent, selfish, curious and with great artistic ability. It is a
symbol of freedom, independence, change, adaptation, movement, the search for new experiences, the traveling and adventurous spirit, but also of inconsistency and abuse of the senses.
More about the the number 5 (five), numerology 5 ...
The meaning of the number 3 (three), numerology 3
Character frequency 3: 1
The number three (3) came to share genuine expression and sensitivity with the world. People associated with this number need to connect with their deepest emotions. The number 3 is characterized by
its pragmatism, it is utilitarian, sagacious, dynamic, creative, it has objectives and it fulfills them. He/she is also self-expressive in many ways and with good communication skills.
More about the the number 3 (three), numerology 3 ...
The meaning of the number 2 (two), numerology 2
Character frequency 2: 1
The number two (2) needs above all to feel and to be. It represents the couple, duality, family, private and social life. He/she really enjoys home life and family gatherings. The number 2 denotes a
sociable, hospitable, friendly, caring and affectionate person. It is the sign of empathy, cooperation, adaptability, consideration for others, super-sensitivity towards the needs of others.
The number 2 (two) is also the symbol of balance, togetherness and receptivity. He/she is a good partner, colleague or companion; he/she also plays a wonderful role as a referee or mediator. Number 2
person is modest, sincere, spiritually influenced and a good diplomat. It represents intuition and vulnerability.
More about the the number 2 (two), numerology 2 ...
The meaning of the number 1 (one), numerology 1
Character frequency 1: 1
Number one (1) came to develop or balance creativity, independence, originality, self-reliance and confidence in the world. It reflects power, creative strength, quick mind, drive and ambition. It is
the sign of individualistic and aggressive nature.
More about the the number 1 (one), numerology 1 ...
№ 2,135 in other languages
How to say or write the number two thousand, one hundred and thirty-five in Spanish, German, French and other languages. The character used as the thousands separator.
Spanish: 🔊 (número 2.135) dos mil ciento treinta y cinco
German: 🔊 (Nummer 2.135) zweitausendeinhundertfünfunddreißig
French: 🔊 (nombre 2 135) deux mille cent trente-cinq
Portuguese: 🔊 (número 2 135) dois mil, cento e trinta e cinco
Hindi: 🔊 (संख्या 2 135) दो हज़ार, एक सौ, पैंतीस
Chinese: 🔊 (数 2 135) 二千一百三十五
Arabian: 🔊 (عدد 2,135) ألفان و مائة و خمسة و ثلاثون
Czech: 🔊 (číslo 2 135) dva tisíce sto třicet pět
Korean: 🔊 (번호 2,135) 이천백삼십오
Danish: 🔊 (nummer 2 135) totusinde og ethundrede og femogtredive
Hebrew: (מספר 2,135) אלפיים מאה שלשים וחמש
Dutch: 🔊 (nummer 2 135) tweeduizendhonderdvijfendertig
Japanese: 🔊 (数 2,135) 二千百三十五
Indonesian: 🔊 (jumlah 2.135) dua ribu seratus tiga puluh lima
Italian: 🔊 (numero 2 135) duemilacentotrentacinque
Norwegian: 🔊 (nummer 2 135) to tusen, en hundre og tretti-fem
Polish: 🔊 (liczba 2 135) dwa tysiące sto trzydzieści pięć
Russian: 🔊 (номер 2 135) две тысячи сто тридцать пять
Turkish: 🔊 (numara 2,135) ikibinyüzotuzbeş
Thai: 🔊 (จำนวน 2 135) สองพันหนึ่งร้อยสามสิบห้า
Ukrainian: 🔊 (номер 2 135) дві тисячі сто тридцять п'ять
Vietnamese: 🔊 (con số 2.135) hai nghìn một trăm ba mươi lăm
Other languages ...
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Frequently asked questions about the number 2135
• How do you write the number 2135 in words?
2135 can be written as "two thousand, one hundred and thirty-five".
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The value of ∑m=02013∑n=0m(2013m)(mn) equals :... | Filo
Question asked by Filo student
The value of equals :
a. 2014
d. 2013
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Question Text The value of equals :
Updated On Dec 15, 2022
Topic Algebra
Subject Mathematics
Class Class 12
Answer Type Video solution: 1
Upvotes 87
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understand it and essentially learn it.
Topics I hope to have guide’s on before long:
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Database vs. Data Science
1. Database vs. Data Science
Database vs. Data Science
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Because, let’s face it, every project claims to re-invent Big Data. Hadoop and MapReduce being something like the founding fathers of Big Data, other’s projects have since appeared. Most notably,
there are stream processing projects like Twitter’s Storm who move from batch-oriented processing to event-based processing which is more suited for real-time, low-latency processing. Spark is yet
something different, a bit like Hadoop, but puts greater emphasis on iterative algorithms, and in-memory processing to achieve that landmark “100x faster than Hadoop” every current project seems to
need to sport. Twitter’s summingbird project tries to bridge the gap between MapReduce and stream processing by providing us with a high-level set of operators which can then either run on MapReduce
or Storm.
However, both Spark or summingbird leave me sort of flat because you can see that they come from a database background, which means that there will still be a considerable gap to serious machine
So, what exactly is the difference? In the end, it’s the difference between relational and linear algebra. In the database world, you model relationships between objects, which you encode in tables,
and foreign keys to link up entries between different tables. Probably the most important insight of the database world was to develop a query language, a declarative description of what you want to
extract from your database, leaving the optimization of the query and the exact details of how to perform them efficiently to the database guys.
The machine learning community, on the other hand, has its roots in linear algebra and probability theory. Objects are usually encoded as a feature vector, that is, a list of numbers describing
different properties of an object. Data is often collected in matrices where each row corresponds to an object, and each column to a feature, not much unlike a table in a database.
However, the operations you perform in order to do data analysis are quite different from the data base world. Take something as basic as linear regression: your try to learn a linear function f(x)=d
i=1wixi in a d-dimensional space (that is, where your objects are described by a d-dimensional vector) given n examples Xi, and Yi, where Xi are the features describing your objects and Yi is the
real number you attach to Xi. One way to “learn” w is to tune it such that the quadratic error on the training examples is minimal. The solution can be written in closed form as
is the matrix built from the
(putting the
in the columns of
), and
is the vector of outputs
In order to solve this, you need to solve the linear equation (XXT)w=XY which can be done by one of a large number of algorithms, starting with Gaussian elimination, which you’ve probably learned in
your undergrad studies, or the conjugate gradient algorithm, or by first computing a Cholesky decomposition. All of these algorithms have in common that they are iterative. They go through a number
of operations, for example O(d3) for the Gaussian elimination case. They also need to store intermediate results. Gaussian elimination and Cholesky decomposition have rather elementary operations
acting on individual entries, while the conjugate gradient algorithm performs a matrix-vector multiplication in each iteration.
Most importantly, these algorithms can only be expressed very badly in SQL! It’s certainly not impossible, but you’d need to store your data in much different ways than you would in idiomatic
database usage.
So, it’s not about whether or not your framework can support iterative algorithms without significant latency, it’s about understanding that joins, group bys, and count() won’t get you far, but you
need scalar products, matrix-vector and matrix-matrix multiplications. You don’t need indices for most ML algorithms, maybe except for being able to quickly find the k-nearest neighbors, because most
algorithms tend to either take in the whole data set in each iteration or otherwise stream the whole set by some model which is iteratively updated like in stochastic gradient descent. I’m not sure
projects like Spark or Stratosphere have fully grasped the significance of this yet.
Database infrastructure-inspired Big Data has it’s place when it comes to extracting and preprocessing data, but eventually, you move from database land to machine learning land, which invariably
means linear algebra land (or probability theory land, which often also reduces to linear algebra like computations). What often happens today is that you either painstakingly have to break down your
linear algebra into MapReduce jobs, or you actively look for algorithms which fit the database view better.
I think we’re still at the beginning of what is possible. Or, to be a bit more aggressive, claims that existing (infrastructure, database, parallelism inspired) frameworks provide you with sophistic
data analytics are widely exaggerated. They take care of a very important problem by giving you a reliable infrastructure to scale your data analysis code, but there’s still a lot of work that needs
to be done on your side. High-level DSLs like Apache Hive or Pig are a first step in this direction but still too much rooted in the database world IMHO.
In summary, one should be aware of the difference between a framework which mostly is concerned with scaling and a tool which actually provides some piece of data analysis. And even if it comes with
basic database-like analytics mechanisms, there is still a long way to go to do some serious data science.
That’s why we’re also thinking that streamdrill occupies an interesting spot, because it is a bit of infrastructure, allowing you to process a serious amount of event data, but it also provides
valuable analysis based on algorithms you wouldn’t want to implement yourself, even if you had some Big Data framework like Hadoop at hand. That’s an interesting direction I also would like to see
more of in the future.
Note: Just saw that Spark has a logistic regression example on their landing page. Well, doing matrix operations explicitly via map() on collections doesn’t count in my view ;)
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