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Therefore, by Lemma 3.3 , there exists a constant [MATH] such that [EQUATION] Also, by Remark 3.5 , if [MATH] , then [MATH] is adjacent to some [MATH] with [MATH] and, by Proposition 3.4 , it follows that for every such [MATH] there exist at most [MATH] vertices adjacent to it. Hence, |
[EQUATION] Since [MATH] is [MATH] -regular and [MATH] [MATH] is also [MATH] -regular and there exists [MATH] such that [MATH] Therefore, by inequalities ( 3.6 ) and ( 3.7 ), it follows that |
[EQUATION] Thus, since [MATH] satisfies LII, there is some constant [MATH] such that [MATH] , and so, [EQUATION] Hence, [MATH] satisfies LII on every geodesic domain, finishing the proof. |
Remark 3.8 Note that the hypothesis of regularity in Theorem 3.7 is not restrictive at all, since Theorem 2.3 gives that non-regular surfaces do not have LII. |
4. Gromov boundary and LII We present in this section an application of Theorem 3.7 relating Gromov boundary and LII (see Theorem 4.12 ). First of all, we need some background on Gromov hyperbolicity. |
Let [MATH] be a metric space. Fix a base point [MATH] and for [MATH] let [EQUATION] The number [MATH] is non-negative and it is called the Gromov product of [MATH] with respect to [MATH] |
Definition 4.1 A metric space [MATH] is (Gromov) hyperbolic if it satisfies the [MATH] -inequality [EQUATION] for some [MATH] , for every base point [MATH] and all [MATH] |
If [MATH] is a metric space and [MATH] is an interval, we say that the curve [MATH] is a geodesic if we have [MATH] for every [MATH] |
(then [MATH] is equipped with an arc-length parametrization). geodesic ray is a geodesic defined on the interval [MATH] geodesic metric space is a metric space such that for every couple of points there exists a geodesic joining them. |
Let us recall the following from Definition 4.2 A geodesic metric space [MATH] has a pole in a point [MATH] if there exists [MATH] such that each point of [MATH] lies in an [MATH] -neighborhood of some geodesic ray emanating from [MATH] |
Let [MATH] be a hyperbolic space and [MATH] a base point. The relative geodesic boundary of [MATH] with respect to the base point [MATH] is the set of equivalence classes |
[EQUATION] where two geodesic rays [MATH] are equivalent if there exists some [MATH] such that [MATH] , for every [MATH] In fact, the definition above is independent from the base point. Therefore, the set of classes of geodesic rays is called geodesic boundary of [MATH] [MATH] . Herein, we do not distinguish between t... |
A sequence of points [MATH] converges to infinity if [EQUATION] This property is independent of the choice of [MATH] since [EQUATION] |
for any [MATH] Two sequences [MATH] that converge to infinity are equivalent if [EQUATION] Using the [MATH] -inequality, we easily see that this defines an equivalence relation for sequences in [MATH] converging to infinity. The sequential boundary at infinity |
[MATH] of [MATH] is defined to be the set of equivalence classes of sequences converging to infinity. Note that given a geodesic ray [MATH] , the sequence [MATH] converges to infinity and two equivalent rays induce equivalent sequences. Thus, in general, [MATH] |
We say that a metric space is proper if every closed ball is compact. Every uniform graph and every complete Riemannian manifold are proper geodesic metric spaces. |
Proposition 4.3 , Chapter III.H, Proposition 3.1] If [MATH] is a proper hyperbolic geodesic metric space, then the natural map from [MATH] to [MATH] is a bijection. |
For every [MATH] , its Gromov product with respect to the base point [MATH] is defined as [EQUATION] where the infimum is taken over all sequences [MATH] [MATH] |
A metric [MATH] on the sequential boundary at infinity [MATH] of [MATH] is said to be visual , if there are [MATH] [MATH] and positive constants [MATH] [MATH] , such that |
[EQUATION] for all [MATH] . In this case, we say that [MATH] is a visual metric with respect to the base point [MATH] and the parameter [MATH] |
Theorem 4.4 11 , Theorem 2.2.7] Let [MATH] be a hyperbolic space. Then for any [MATH] there is [MATH] such that for every [MATH] there exists a metric [MATH] on |
[MATH] , which is visual with respect to [MATH] and [MATH] Remark 4.5 Notice that for any visual metric, [MATH] is bounded and complete. |
Definition 4.6 Given a metric space [MATH] and a constant [MATH] [MATH] is [MATH] -uniformly perfect if there exists some [MATH] such that for every [MATH] and every [MATH] there exists a point [MATH] such that |
[MATH] [MATH] is uniformly perfect if there exists some [MATH] such that [MATH] is [MATH] -uniformly perfect. Theorem 4.7 27 , Theorem 4.15] |
Given a hyperbolic uniform graph [MATH] with a pole, then [MATH] has LII if and only if [MATH] is uniformly perfect for some visual metric. |
Let [MATH] be a regular non-exceptional Riemann surface and consider some constants [MATH] such that Theorem 3.7 holds. Then, for every collar [MATH] of every simple closed geodesic [MATH] with [MATH] , consider [MATH] (where [MATH] are Jordan curves) and let us define the (non-smooth) surface [MATH] where [MATH] is re... |
[EQUATION] where [MATH] denotes the restriction to [MATH] of the Poincaré length in [MATH] One can check that the infimum in this definition is, in fact, a minimum, and that [MATH] is a proper geodesic metric space. |
Theorem 4.8 Given a non-exceptional Riemannian surface [MATH] [MATH] and [MATH] , then [MATH] and [MATH] are quasi-isometric. Proof. |
Let [MATH] be the graph defined by [MATH] where vertices [MATH] and [MATH] are removed and a new edge is defined between any pair of vertices adjacent to the same vertex [MATH] Notice that the vertex set of [MATH] is [MATH] Let us denote by [MATH] the path distance in [MATH] and by [MATH] the inner distance in [MATH] |
Since [MATH] are pairwise disjoint sets by Remark 3.1 , in order to prove that [MATH] is quasi-isometric to [MATH] , it suffices to check that there is a constant [MATH] such that given any two vertices, [MATH] , adjacent to some [MATH] in [MATH] , then [MATH] |
Claim: Given any pair of points [MATH] [MATH] , where [MATH] is the constant in Lemma 3.3 Consider any geodesic path [MATH] in [MATH] joining [MATH] to [MATH] and let [MATH] It is clear that [MATH] covers [MATH] and [MATH] Denote by [MATH] the upper integer part of [MATH] , i.e., the smallest integer greater than or eq... |
If [MATH] are adjacent to some [MATH] , it follows by Lemma 2.11 that [MATH] Then, by the claim above [MATH] and [MATH] is quasi-isometric to [MATH] |
Let us see now that [MATH] is quasi-isometric to [MATH] Let [MATH] be the inclusion map. Now, consider any two vertices [MATH] and let [MATH] be the shortest path in [MATH] joining them. For [MATH] , if [MATH] is also an edge in [MATH] , then [MATH] If [MATH] , then |
[MATH] and [MATH] are adjacent to the same vertex [MATH] in [MATH] ; therefore, since the length of each connected component of [MATH] is at most [MATH] by Lemma 2.11 [MATH] Thus, [MATH] for every [MATH] and [MATH] |
By the claim above, [MATH] . Hence, [EQUATION] and [MATH] is a [MATH] -quasi-isometric embedding. It is trivial to check that [MATH] is [MATH] -full. |
Theorem 4.9 22 , p.88] If [MATH] is a quasi-isometry between geodesic metric spaces, then [MATH] is hyperbolic if and only if [MATH] is hyperbolic. Furthermore, if [MATH] is a [MATH] -full [MATH] -quasi-isometry and [MATH] (respectively, [MATH] ) is [MATH] -hyperbolic, then [MATH] (respectively, [MATH] ) is [MATH] -hyp... |
[MATH] [MATH] [MATH] and [MATH] Proposition 4.10 27 , Proposition 5.6] Suppose [MATH] are proper hyperbolic geodesic metric spaces and [MATH] is a quasi-isometry. If [MATH] has a pole in [MATH] , then [MATH] has a pole in [MATH] |
The following result follows from 11 , Theorem 5.2.15] and 27 , Proposition 5.10] Theorem 4.11 Let [MATH] be a quasi-isometric map of proper hyperbolic geodesic metric spaces. Then [MATH] is uniformly perfect if and only if [MATH] is uniformly perfect (with respect to any visual metrics). |
Theorem 4.12 Let [MATH] be a [MATH] -regular non-exceptional Riemann surface, [MATH] and [MATH] Assume that [MATH] is hyperbolic and has a pole. Then [MATH] has LII if and only if [MATH] is uniformly perfect. |
Proof. Since [MATH] is hyperbolic and has a pole, Theorems 4.8 and 4.9 , and Proposition 4.10 give that [MATH] is hyperbolic and has a pole. |
Theorems 4.8 and 4.11 give that [MATH] is uniformly perfect if and only if [MATH] is uniformly perfect. By Theorem 4.7 , this holds if and only if [MATH] has LII. Finally, by Theorem 3.7 [MATH] has LII if and only if [MATH] has LII. |
To determine if a geodesic metric space has a pole is not a difficult task. There are many results that allow to determine if a non-exceptional Riemann surface is hyperbolic (see, e.g., |
). Thus, Theorem 4.17 below is useful in order to apply Theorem 4.12 since it characterizes the hyperbolicity of [MATH] in terms of the hyperbolicity of [MATH] |
In order to prove Theorem 4.17 we need some technical results. The following three propositions appear in 37 , Theorems 2.1 and 2.4] and 34 , Lemma 3.1] |
Proposition 4.13 Let [MATH] be a geodesic metric space, [MATH] compact subsets of [MATH] and [MATH] the quotient space obtained from [MATH] by identifying the points of each |
[MATH] in a single point [MATH] Assume that there are positive constants [MATH] such that [MATH] and [MATH] if [MATH] and that [MATH] is a geodesic metric space. Then [MATH] and [MATH] are quasi-isometric, and [MATH] is hyperbolic if and only if [MATH] is hyperbolic. Furthermore, if [MATH] (respectively, [MATH] ) is [M... |
[MATH] [MATH] and [MATH] Let us consider a geodesic metric space [MATH] , a family of geodesic metric subspaces [MATH] such that [MATH] [MATH] |
are compact sets, and positive constants [MATH] We say that [MATH] is a ( [MATH] )- decomposition of [MATH] if [MATH] is not connected for each non-empty set [MATH] |
[MATH] for every [MATH] and [MATH] for every [MATH] and [MATH] with non-empty sets [MATH] and [MATH] Proposition 4.14 Let [MATH] be a geodesic metric space and [MATH] a family of geodesic metric spaces which is a [MATH] -decomposition of [MATH] Then [MATH] is hyperbolic if and only if there exists a constant [MATH] suc... |
Let [MATH] be a non-exceptional Riemann surface. For each choice of doubly connected domains [MATH] in [MATH] we define [EQUATION] |
Proposition 4.15 Let [MATH] be a non-exceptional Riemann surface, [MATH] doubly connected domains in [MATH] and [MATH] positive constants with |
[MATH] for every [MATH] , and [MATH] for every [MATH] If [MATH] , then [MATH] is not hyperbolic. Lemma 4.16 Let [MATH] be a non-exceptional Riemann surface and [MATH] Then there exists a universal constant [MATH] such that [MATH] and [MATH] are [MATH] -hyperbolic for every [MATH] and [MATH] |
Proof. Given [MATH] , let us consider a geodesic ray [MATH] in [MATH] joining [MATH] with the cusp [MATH] , and the inclusion [MATH] Given [MATH] , let us consider a geodesic [MATH] in [MATH] joining the two connected components of [MATH] , and the inclusion [MATH] If [MATH] is a connected component of [MATH] or [MATH]... |
For a non-exceptional Riemann surface [MATH] , let us define [EQUATION] Theorem 4.17 Let [MATH] be a non-exceptional Riemann surface, |
[MATH] and [MATH] Then [MATH] is hyperbolic if and only if [MATH] is hyperbolic and [MATH] Proof. Lemma 2.11 gives that if [MATH] is a connected component of [MATH] , then [MATH] and so, [MATH] For each collar [MATH] of a simple closed geodesic [MATH] with [MATH] , consider [MATH] (where [MATH] are Jordan curves). We h... |
[EQUATION] since [MATH] Assume first that [MATH] Since [MATH] , we have [EQUATION] and we conclude, by Proposition 4.15 , that [MATH] is not hyperbolic. |
Consequently, it suffices to prove that if [MATH] , then [MATH] is hyperbolic if and only if [MATH] is hyperbolic. Denote by [MATH] the connected components of [MATH] |
Lemma 2.14 gives that if [MATH] are connected components of [MATH] and they belong to the closure of different collars, then [MATH] |
Therefore, if we define [MATH] , we have that [MATH] [MATH] [MATH] is a ( [MATH] )-decomposition of [MATH] By Proposition 4.14 and Lemma 4.16 [MATH] is hyperbolic if and only if there exists a constant [MATH] such that [MATH] is [MATH] -hyperbolic for every [MATH] |
For each [MATH] , let [MATH] be the geodesic metric space obtained from [MATH] by identifying the points in the collar [MATH] in a single point [MATH] for every [MATH] with [MATH] Let us define the (non-smooth) surface [MATH] |
obtained from [MATH] by removing the collar [MATH] and by gluing [MATH] and [MATH] for every [MATH] with [MATH] consider on [MATH] the inner metric inherited from [MATH] Let [MATH] be the geodesic metric space obtained from [MATH] |
by identifying the points in [MATH] in a single point [MATH] for every [MATH] with [MATH] Since [MATH] , we have [EQUATION] and so, |
[EQUATION] This fact and Proposition 4.13 give that [MATH] is hyperbolic if and only if [MATH] is hyperbolic, and [MATH] is hyperbolic if and only if [MATH] is hyperbolic, with uniform hyperbolicity constants. Since [MATH] we conclude that there exists a constant [MATH] such that [MATH] is [MATH] -hyperbolic for every ... |
Since [MATH] is a ( [MATH] )-decomposition of [MATH] Proposition 4.14 gives that [MATH] is hyperbolic if and only if there exists a constant [MATH] such that [MATH] is [MATH] -hyperbolic for every [MATH] |
Hence, [MATH] is hyperbolic if and only if [MATH] , and the conclusion holds. Our last result shows that the hypotheses on Theorem 4.12 do not depend on [MATH] |
Proposition 4.18 Let [MATH] be a non-exceptional Riemann surface, [MATH] and [MATH] The following statements hold: [MATH] [MATH] is hyperbolic if and only if [MATH] is hyperbolic. |
[MATH] [MATH] has a pole if and only if [MATH] has a pole. Proof. By Theorem 4.9 and Proposition 4.10 , it suffices to prove that |
[MATH] and [MATH] are quasi-isometric. Without loss of generality we can assume that [MATH] Thus, [MATH] Let [MATH] be the geodesic metric space obtained from [MATH] |
by identifying the points in the closure of the collar [MATH] in a single point [MATH] for every [MATH] , and [MATH] the geodesic metric space obtained from [MATH] |
by identifying the points in [MATH] in a single point [MATH] for every [MATH] Thus, [MATH] Lemma 2.14 gives [MATH] for every [MATH] Also, Lemmas 2.10 and 2.11 give |
[EQUATION] for every [MATH] These inequalities and Proposition 4.13 give that [MATH] and [MATH] are quasi-isometric, and [MATH] and [MATH] are quasi-isometric. Since [MATH] we conclude that [MATH] and [MATH] are quasi-isometric. |
# Source: arxiv 1806.04638 # Title: On the Primitive Ideals of Nest Algebras # Sections: all # Downloaded: 2026-03-03T05:17:59.190624+00:00 |
On the primitive ideals of nest algebras Abstract. We show that Ringrose’s diagonal ideals are primitive ideals in a nest algebra (subject to the Continuum Hypothesis). This provides for the first time concerete descriptions of enough primitive ideals to obtain the Jacobson radical as their intersection. Separately, we... |
Key words and phrases: nest algebra, primitive ideals, nets, continuum hypothesis 1991 Mathematics Subject Classification: 47L35, 47L75 |
1. Introduction The Jacobson radical has been a frequent object of study in non-selfadjoint algebras, and considerable effort has been expended to identify the radical in the context of various classes of non-selfadjoint algebras, e.g., |
Why is this? At fist glance it might seem that since many non-selfadjoint algebras are modelled more or less on the algebra of finite-dimensional upper triangular matrices, the desire is to obtain Wedderburn-type structure theorems for the algebras. In fact, however, the Jacobson radical is rarely the right ideal for s... |
Thus knowledge about the Jacobson radical rather points towards more general structural information about the algebra and, in particular, when the radical is small, indicates the presence of a rich supply of irreducible representations, even in algebras which have a strong heuristic connection with the upper triangular... |
The nest algebras are one such case. Indeed the main result of Ringrose’s paper , which introduced the class of nest algebras, was to describe the Jacobson radical [MATH] of a nest algebra [MATH] (see Section below for precise definitions of terms). However, except in the trivial case of a finite nest, there is no Wedd... |
as the sum of the diagonal algebra and the Jacobson radical. In fact by 18 , Theorem 4.1] , a decomposition [MATH] for some ideal [MATH] is only possible if [MATH] is Larson’s ideal [MATH] |
, and then only if the nest has no continuous part. At issue here is the fact that unless the nest is finite [MATH] is much bigger than the Jacobson radical; in the case of upper triangular matrixes on [MATH] [MATH] is the collection of all strictly upper triangular operators, while |
[MATH] is the set of compact strictly upper triangular operators. Thus, the comparatively small Jacobson radical in nest algebras indicates that there must be many irreducible representations other than the trivial ones obtained as the compression to an atom of the nest. |
However, up to now, the only other primitive ideals which could be identified explicitly were the maximal two-sided ideals. (Maximal two-sided ideals are primitive; see Remark 3.1 for a review of this and other ring-theoretic facts.) In |
we described the maximal two-sided ideals of a continuous nest algebra and in we extended the description to cover all nest algebras. (It should be noted these results rest on deep foundations; between them, they require the similarity theory of nests and the Paving Theorem.) Even so, however, these ideals alone do not... |
and in fact the two coincide when the nest is atomic. The goal of this paper is to identify enough examples of primitive ideals of nest algebras to account for the small Jacobson radical, by which we mean that their intersection should equal the Jacobson radical. The key examples have been in plain view all along; they... |
23 , Theorem 5.3] . We shall show in Theorem 3.7 that the diagonal ideals are primitive. This answers an open question of Lance (repeated in |
). Interestingly, this result relies on assuming a positive answer to the Continuum Hypothesis. See the excellent survey paper for other recent results in operator algebras which make use of nonstandard foundational considerations. |
After this, we turn to an analysis of the left ideals of nest algebras in Section . We establish a standard form for all left ideals, and also a stronger form which holds for many norm-closed left ideals, including the maximal left ideals. In Section |
we explore the primitive ideals of atomic nest algebras in more depth. We identify three classes of primitive ideals (the smallest, the largest, and the intermediate ones), and we show that they are distinguished by their behaviour on the diagonal. Section focusses on the infinite upper-triangular matrices, where we ca... |
2. Preliminaries Throughout this paper the underlying Hilbert spaces are always assumed separable. nest is a set of projections on a Hilbert space which is linearly ordered, contains 0 and I, and is weakly closed (or, equivalently, order-complete). The |
nest algebra [MATH] , of a nest [MATH] is the set of bounded operators leaving invariant the ranges of [MATH] The diagonal algebra [MATH] , is the set of operators having the ranges of projections in [MATH] as reducing subspaces; equivalently, the commutant of [MATH] An interval of [MATH] is the difference [MATH] of tw... |
[EQUATION] Conventionally [MATH] and [MATH] If [MATH] then [MATH] is an atom of [MATH] , and all atoms are of this form. Conversely, if [MATH] then there is a strictly increasing sequence of projections in [MATH] which converge to [MATH] . Similar remarks apply for [MATH] We shall make continual use of the fact that th... |
[MATH] , which we write as [MATH] , belongs to [MATH] if and only if there is an [MATH] such that [MATH] and [MATH] See for further properties of nest algebras. |
Example 2.1 Let [MATH] and let [MATH] be the standard basis. For [MATH] , let [MATH] be the projection onto the span of [MATH] and let [MATH] . This is a nest, and [MATH] is the nest algebra of all infinite upper triangular operators with respect to the standard basis. By slight abuse of notation, we write [MATH] for t... |
We now recall Ringrose’s description of the Jacobson radical of a nest algebra, in terms of diagonal seminorms and diagonal ideals: |
Definition 2.2 Let [MATH] be a nest and fix [MATH] in [MATH] . The diagonal seminorm function [MATH] is defined for [MATH] by [EQUATION] |
Likewise, for [MATH] the diagonal seminorm function [MATH] is [EQUATION] It is straightforward to see that the functions [MATH] are submultiplicative seminorms on [MATH] and dominated by the norm, and so their kernels are norm closed two-sided ideals of [MATH] |
Definition 2.3 Let [MATH] be a nest. The diagonal ideals are the ideals [EQUATION] and [EQUATION] The diagonal ideals can be viewed as generalizations of those ideals of upper-triangular |
[MATH] matrices consisting of all the matrices which vanish at a particular diagonal entry. Indeed if [MATH] then [EQUATION] However if [MATH] , then [MATH] is the set of operators asymptotically |
vanishing close to [MATH] (from below). More precisely, in the case of [MATH] [MATH] is of the form ( ) for all [MATH] and [MATH] is the compact operators of [MATH] . See Section |
for a detailed discussion of the primitive ideals in this algebra. Ringrose gave the following description of the Jaconson radical in terms of these diagonal ideals. |
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