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Theorem 2.4 , Theorem 5.3) The Jacobson radical of [MATH] is the intersection of the diagonal ideals of [MATH] A key point to bear in mind is that although the diagonal ideals are related to the primitive ideals, as the next result quoted shows, they were not known to be primitive. Lance |
asked whether the diagonal ideals are primitive and, in his study of the diagonal ideals and their quotients and proved a number of results which are entailed by primitivity. In Theorem 3.7 |
we show that the diagonal ideals are in fact primitive ideals. The following useful result shows that each primitive ideal of a nest algebra is associated with a unique diagonal ideal. |
Theorem 2.5 , Theorem 4.9) Every primitive ideal of [MATH] contains exactly one diagonal ideal. Based on this result we adopt the following notation: |
Definition 2.6 If [MATH] is a primitive ideal of the nest algebra [MATH] , write [MATH] for the unique diagonal ideal contained in [MATH] |
Finally we close the section by recalling Larson’s ideal [MATH] Definition 2.7 Let [MATH] be the set of [MATH] such that, given [MATH] we can find a collection [MATH] of pairwise orthogonal intervals of [MATH] |
which sum to [MATH] and such that [MATH] Ringrose 23 , Theorem 5.4] provides an alternate description of the Jacobson radical which is formally very similar to Larson’s ideal. The only difference is the requirement that the collections of pairwise orthogonal intervals must be finite . However this makes an enormous dif... |
Example 2.8 Let [MATH] be the canonical nest on [MATH] . Then [MATH] is the set of zero-diagonal compact operators in [MATH] and [MATH] is the set of all zero-diagonal operators in [MATH] . Note in particular that |
[MATH] but that [MATH] (for example, the right-hand side fails to contain the unilateral backward shift). 3. The diagonal ideals are primitive |
The main result of this section is Theorem 3.7 in which we prove that the diagonal ideals of a nest algebra are primitive. We start by recalling some basic facts about primitive ideals which can be found in many standard texts of ring theory or Banach algebras. See, e.g., , Chapter III] |
Remark 3.1 Let [MATH] be a unital Banach algebra. The (left) primitive ideals of [MATH] are the annihilators of left [MATH] -modules, or, equivalently, the kernels of the irreducible representations of [MATH] . If [MATH] is any primitive ideal of [MATH] then there is a maximal left ideal [MATH] of [MATH] such that [MAT... |
[MATH] contained in [MATH] , and is equal to [EQUATION] From this, together with the maximality of [MATH] , it follows easily that [MATH] |
if and only if there are [MATH] such that [MATH] (where [MATH] is the unit of [MATH] ). Finally, of course, the Jacobson radical is, by definition, the intersection of all the primitive ideals of [MATH] . Analogously, the right primitive ideals are the kernels of right [MATH] -modules and each right primitive ideal is ... |
by some maximal right ideal. The intersection of the maximal right primitive ideals is also the (same) Jacobson radical. Lemma 3.3 will enable us to convert arbitrary upper triangular operators to block diagonal form. It relies on the following useful technical lemma which we quote in full. |
Lemma 3.2 19 , Lemma 2.2] Let [MATH] and let [MATH] [MATH] ) be sequences of projections such that [MATH] for all [MATH] , where [MATH] denotes the set of operators of rank not greater than [MATH] . Then there are orthonormal sequences |
[MATH] and [MATH] such that [MATH] for all [MATH] and [MATH] is real and greater than [MATH] for all [MATH] Lemma 3.3 Suppose [MATH] but [MATH] for some [MATH] in [MATH] . Then there are [MATH] |
and a sequence [MATH] of nest projections strictly increasing to [MATH] such that [EQUATION] and each of the terms [MATH] has norm greater than [MATH] |
Proof. Rescaling if necessary, assume [MATH] Choose a sequence [MATH] which increases strictly to [MATH] . We shall inductively construct a subsequence [MATH] such that |
[MATH] for all [MATH] , and the result will follow from an easy application of Lemma 3.2 . Take [MATH] and suppose [MATH] to have been chosen with the desired property. |
Suppose for a contradiction that [MATH] for all [MATH] Fix an [MATH] with [MATH] and for each [MATH] find [MATH] such that [EQUATION] |
The sequence [MATH] is norm-bounded and so has a [MATH] -convergent subsequence, [MATH] . But [MATH] since [MATH] is [MATH] -closed and, by the the lower semicontinuity of the norm, |
[EQUATION] which is a contradiction. Thus we find [MATH] with which to continue the induction. With [MATH] chosen, apply Lemma 3.2 to obtain unit vectors [MATH] in the range of [MATH] such that |
[MATH] for all [MATH] , and [MATH] for all [MATH] Set [EQUATION] Then [MATH] since the terms of both sums are of the form [MATH] , and |
[MATH] , so that [MATH] and each of the terms of the sum has norm greater than [MATH] The following, unfortunately rather technical, definition is central to our analysis in this section. |
Definition 3.4 Fix a nest [MATH] and a projection [MATH] Say that a set [MATH] of operators in [MATH] are of Type-S if there exists a strictly increasing sequence [MATH] in [MATH] which converges to [MATH] , and a sequence of unit vectors [MATH] |
such that for each [MATH] both [MATH] and [MATH] Clearly if [MATH] is of Type-S, then it lies in both a proper left ideal of [MATH] |
and in a proper right ideal of [MATH] . Note, however that it need not lie in a proper two-sided ideal; for example consider the singleton [MATH] where [MATH] is the unilateral backward shift on [MATH] . This is Type-S with respect to the sequences |
[MATH] and [MATH] but does not lie in a proper two-sided ideal of [MATH] In fact this example is the prototype of the analysis which follows and an analogous sequence is at the heart of the proof of the next lemma. Note also that, strictly speaking, “Type-S” is a property which a set has with respect to a particular [M... |
Lemma 3.5 Fix a nest [MATH] and a projection [MATH] in [MATH] and let [MATH] be a set of Type-S. Let [MATH] but [MATH] Then there are [MATH] such that [MATH] |
is also of Type-S. Proof. Take a sequence [MATH] which increases strictly to [MATH] and unit vectors [MATH] such that [MATH] for all [MATH] |
By Lemma 3.3 there are [MATH] in [MATH] and a sequence of nest projections strictly increasing to [MATH] such that [MATH] is block diagonal with respect to these projections and each of the blocks has norm greater than [MATH] Since [MATH] and [MATH] demonstrate the Type-S property, so does any subsequence of theirs and... |
[MATH] with the resulting operator we may now assume that [MATH] is block diagonal with respect to [MATH] and that all the blocks [MATH] have norm greater than [MATH] |
We shall inductively construct a new sequence of unit vectors [MATH] for a subsequence [MATH] , together with contractions [EQUATION] |
and [EQUATION] in [MATH] such that [EQUATION] and [MATH] for all [MATH] The result will then follow by taking [MATH] and [MATH] To perform the induction, fix [MATH] and suppose [MATH] [MATH] [MATH] , and [MATH] have been chosen for all [MATH] . (To get the induction started when [MATH] , define [MATH] and observe that ... |
Note that, for all sufficiently large [MATH] [EQUATION] for all [MATH] . Thus, taking [MATH] we can pick [MATH] such that [MATH] , each [MATH] , and for all [MATH] and [MATH] |
[MATH] Set [MATH] and [MATH] , which is a unit vector since the [MATH] are pairwise orthogonal. For each [MATH] the interval [MATH] dominates a diagonal block of [MATH] |
which has norm greater than 1. Thus, we can choose vectors [MATH] and [MATH] in [MATH] with [MATH] and [MATH] and set [EQUATION] |
Since [EQUATION] each of the terms of the sums are in [MATH] and the ranges and cokernels of the terms are pairwise orthogonal, so that both sums converge strongly. Now clearly for each [MATH] |
[MATH] and, likewise [MATH] Further, [MATH] , so that [EQUATION] and [MATH] so that [EQUATION] Note also that each of the [MATH] for [MATH] lie in the range of [MATH] . Thus |
[MATH] [MATH] , and [MATH] Having met all the requirements, the induction proceeds as stated, and we let [MATH] and [MATH] . Clearly for any fixed [MATH] |
[EQUATION] for all sufficiently large [MATH] and so [MATH] Moreover since [MATH] and [MATH] are block diagonal with respect to [MATH] , as is [MATH] , it follows that [MATH] |
and [MATH] and we are done. Lemma 3.6 Fix a nest [MATH] and a projection [MATH] and let [MATH] [MATH] ) be a countable collection of countable sets of Type-S which form a chain (i.e. for any [MATH] , either [MATH] or [MATH] ). Then [MATH] is also of Type-S. |
Proof. The proof is a routine countability argument. Recall that the strong operator topology on [MATH] is metrizable; let [MATH] be a metric for it. Enumerate [MATH] and let the sets [MATH] [MATH] ) consist of the first [MATH] terms of that enumeration. Fix [MATH] and suppose [MATH] and [MATH] |
have been chosen for [MATH] so that [MATH] [MATH] , and [MATH] for all [MATH] Each [MATH] belongs to some [MATH] and since [MATH] is finite and [MATH] is a chain, |
[MATH] is contained in some [MATH] . Therefore [MATH] is of Type-S. Using this fact, we can find [MATH] with [MATH] and [MATH] such that [MATH] for all [MATH] Continue this inductively to construct a strictly increasing sequence [MATH] |
and [MATH] for all [MATH] such that [MATH] for all [MATH] (taking [MATH] to get the induction started). Each [MATH] belongs to [MATH] for all sufficiently large [MATH] , and so the result follows with the vectors so chosen. |
Theorem 3.7 Assume the Continuum Hypothesis and let [MATH] be a nest. Then the diagonal ideals of [MATH] are primitive ideals. Proof. |
The result is trivial when the diagonal ideal is of type [MATH] with [MATH] or [MATH] with [MATH] . For in either case the diagonal ideal is the kernel of the representation [MATH] where [MATH] is an atom of [MATH] and whose range is therefore all of [MATH] , and so is irreducible. For the remainder of the proof, consi... |
Next, let [MATH] be a diagonal ideal of [MATH] and suppose that [MATH] for some [MATH] in [MATH] It is enough to construct operators [MATH] for each operator [MATH] such that the collection [MATH] generates a proper left ideal of [MATH] . For then there is a maximal left ideal [MATH] which contains this family of opera... |
Now consider the case when [MATH] for some [MATH] in [MATH] . By the same reasoning, it is enough to find [MATH] such that [MATH] |
is contained in a proper left ideal of [MATH] . To do this, we take adjoints and seek [MATH] such that [MATH] is contained in a proper right ideal of [MATH] . Since [MATH] is an arbitrary nest, we can replace [MATH] with [MATH] , to recast this as a second problem about [MATH] in [MATH] namely, to find [MATH] for each ... |
[MATH] generates a proper right ideal of [MATH] . We shall show in fact that the choice can be made so that the same set of operators [MATH] serves to generate both a proper left ideal and a proper right ideal. We shall construct these operators using transfinite recursion. |
The cardinality of [MATH] is equal to the cardinality of the contiuum since every operator can be represented as a countable array of complex numbers. Since we are assuming the Continuum Hypothesis, [MATH] has cardinality [MATH] |
and so it can be put in bijective correspondence with the set of ordinals [MATH] (where [MATH] denotes the first uncountable ordinal). Write this correspondence as [MATH] [MATH] ). To run the transfinite recursion, we suppose that for some [MATH] we have operators [MATH] in [MATH] for all [MATH] , and describe how to o... |
[MATH] is also of Type-S. On the other hand, if it happens that [MATH] is not of Type-S then set [MATH] . (This is a sink terminal state which we shall prove momentarily is never in fact reached.) |
Note that formally Lemma 3.5 assumes a countably infinite collection of predecessors. However the case of finite [MATH] , or even [MATH] , can be covered by padding the collection of predecessors with countably many repeated zeros. Note also the recursion step involves an arbitrary choice of operators, which can easily... |
Having described a rule to construct [MATH] with [MATH] given, we apply the principle of transfinite recursion to obtain [MATH] where the transition rule from the previous paragraph applies for every [MATH] . We next note that for every [MATH] [MATH] is of Type-S. For if this were not true, then we could find the least... |
[MATH] generates a proper left ideal and a proper right ideal, and the result follows. Corollary 3.8 Assuming the Continuum Hypothesis, the diagonal ideals of [MATH] are also right-primitive ideals, that is to say, the annihilators of simple right modules. |
Proof. The conjugate-linear anti-isomorphism [MATH] maps [MATH] to [MATH] maps diagonal ideals to diagonal ideals, and converts left modules into right modules. |
We remark in passing that Theorem 3.7 does provide a new proof of Ringrose’s characterization of the Jacobson radical of a nest algebra. For in view of Theorem 2.5 |
[EQUATION] and the reverse inclusion follows from Theorem 3.7 Insofar as our result assumes the Continuum Hypothesis and also assumes |
[MATH] is separable, this is, of course, substantially less general than Ringrose’s original proof. 4. The left ideals of a nest algebra |
In this section we study the left ideals of nest algebras. Definition 4.1 gives a method of specifying left ideals and in Theorem 4.4 |
we shall see that every left ideal can be specified in this way. We then introduce (Definition 4.7 a stronger property which specifies many closed left ideals, including the maximal left ideals. This leads to insights into the structure of left ideals (Proposition 4.18 which we apply in the following sections. |
Definition 4.1 Let [MATH] be a left ideal of [MATH] . Say that [MATH] is constructible if there is a net indexed by a directed set [MATH] consisting of pairs |
[MATH] of projections [MATH] and vectors [MATH] such that [EQUATION] for every [MATH] Lemma 4.2 [MATH] is itself a constructible ideal and, in general, the constructible ideal, |
[MATH] , specified by the net [MATH] is equal to [MATH] if and only if [MATH] Proof. If [MATH] then, for any fixed [MATH] [MATH] |
and so [MATH] . Conversely, if [MATH] , then [MATH] and so [MATH] is proper. Note that if [MATH] is a net in [MATH] and [MATH] then [MATH] for all [MATH] and so without loss we can always assume that |
[MATH] The following interpolation result of Katsoulis, Moore, and Trent enables us to see that all left ideals are constructible. In this context we remark that the results of |
have a precursor in Lance’s 13 , Theorem 2.3] , introduced to study the radical and diagonal ideals. Theorem 4.3 11 , Theorem 4] Let [MATH] and [MATH] be in [MATH] . Then there are [MATH] |
in [MATH] such that [EQUATION] if and only if [EQUATION] (where [MATH] is interpreted as [MATH] ). Theorem 4.4 Every left ideal of a nest algebra is constructible. |
Proof. Let [MATH] be a fixed left ideal of the nest algebra [MATH] and take [MATH] to be the set of all 4-tuples [MATH] where [MATH] is a finite subset of [MATH] |
[MATH] [MATH] , and [MATH] , subject to the constraint that [MATH] for all [MATH] This is a directed set if we say [MATH] when [MATH] and [MATH] . For the relation is clearly reflexive and transitive, and any pair of members of [MATH] [MATH] and [MATH] is dominated by [MATH] . Define a net on [MATH] with values in |
[MATH] by the mapping which takes [MATH] to [MATH] where [MATH] , and [MATH] We shall see that this net specifies [MATH] exactly. |
On the one hand, trivially, if [MATH] then for any [MATH] , the tuple [MATH] belongs to [MATH] and so for any [MATH] [MATH] . So, next, suppose on the other hand that |
[MATH] Let an arbitrary [MATH] in [MATH] be given. Since [MATH] , there do not exist any [MATH] in [MATH] such that [MATH] Thus by Theorem 4.3 , the supremum ( ) is infinite, and so we can find [MATH] and [MATH] such that |
[EQUATION] for each [MATH] . Rescaling [MATH] , we obtain [MATH] and [MATH] such that [MATH] and [MATH] Thus [MATH] is in [MATH] and we have [MATH] and [MATH] In other words, the net |
[MATH] is frequently equal to 1, and so [MATH] Example 4.5 The set [MATH] of finite rank operators in [MATH] is a two-sided ideal of [MATH] but is not norm-closed. We can specify this with the following net. Let [MATH] consist of the set of pairs [MATH] where [MATH] is a finite-dimensional subspace of [MATH] and [MATH]... |
belongs to [MATH] if and only if there is a finite-dimensional space [MATH] such that [MATH] vanishes on [MATH] . Since the vectors in the pairs are unbounded, the condition [MATH] for all [MATH] is equivalent to [MATH] vanishing on [MATH] |
Example 4.6 The set [MATH] of compact operators in [MATH] is a norm-closed two-sided ideal of [MATH] . We can specify it with the following net, which is similar to the previous example. Let [MATH] consist of the set of pairs [MATH] where [MATH] is a finite-dimensional subspace of [MATH] and [MATH] is a unit vector whi... |
belongs to [MATH] if and only if for any [MATH] there is a finite-dimensional space [MATH] such that [MATH] , which is readily seen to be equivalent to [MATH] |
The contrast between the last two examples, in which the net was unbounded in one case and bounded in the other, motivates the following definition. |
Definition 4.7 Let [MATH] be a left ideal of [MATH] . Say that [MATH] is strongly constructible if it is constructible and a net [MATH] specifying [MATH] |
can be found in which all the vectors [MATH] have norm [MATH] Proposition 4.8 Strongly constructible ideals are norm-closed. Proof. |
Let [MATH] be strongly constructible and specified by [MATH] where [MATH] for all [MATH] . Suppose the sequence of [MATH] converges in norm to [MATH] . Given [MATH] , find a fixed [MATH] such that |
[MATH] and [MATH] such that [MATH] for all [MATH] . Then [EQUATION] Proposition 4.9 The maximal left ideals of [MATH] are strongly constructible. |
Proof. Let [MATH] be a maximal left ideal which we suppose to be specified by the net [MATH] . Without loss, assume that each [MATH] . By Lemma 4.2 |
[MATH] and so there is an [MATH] such that [MATH] is frequently at least [MATH] . Let [MATH] and [MATH] for [MATH] Now [MATH] is a directed set and [MATH] is a net on it. Again by Lemma 4.2 the net [MATH] specifies a proper ideal which, furthermore, contains [MATH] since for [MATH] |
[EQUATION] for all [MATH] and the net on the right converges to zero since [MATH] is a subnet of [MATH] By maximality, the ideal which [MATH] specifies must equal [MATH] |
Proposition 4.10 Arbitrary intersections of strongly constructible ideals are strongly constructible. The proof is a consequence of the following simple result about nets. |
Lemma 4.11 Fix a set [MATH] and suppose that we have a family of nets in [MATH] indexed by a set [MATH] which we denote by [MATH] . Then we can find a net [MATH] in [MATH] with the property that for any |
[MATH] [MATH] is eventually in [MATH] if and only if for each [MATH] [MATH] is eventually in [MATH] Proof. Define [MATH] to be set the set of pairs [MATH] where [MATH] is a section map on the fibre bundle of [MATH] over [MATH] (i.e., for each [MATH] [MATH] ), and [MATH] is an arbitrary member of [MATH] . Put a relation... |
[MATH] if [MATH] for all [MATH] (the relation [MATH] is the directed relation defined on [MATH] ). This is a symmetric and transitive relation. Moreover, if [MATH] and [MATH] are in [MATH] then for each [MATH] we can find an element of [MATH] which dominates both [MATH] and [MATH] . By the Axiom of Choice there is ther... |
[MATH] for all [MATH] . Taking an arbitrary [MATH] , then [MATH] dominates both [MATH] and [MATH] in [MATH] . Thus [MATH] is a directed set, and we define the net |
[MATH] by [MATH] Now, on one hand, suppose that [MATH] is eventually in [MATH] Thus there is a [MATH] such that [MATH] for all [MATH] Fix [MATH] and consider [MATH] . If [MATH] |
then define [MATH] for all [MATH] and [MATH] Then [MATH] and so [MATH] This shows that for each [MATH] [MATH] is eventually in [MATH] |
Conversely, let [MATH] and suppose that for every [MATH] [MATH] is eventually in [MATH] . That is to say, for each [MATH] , we can can find an [MATH] such that |
[MATH] for all [MATH] in [MATH] . Again by the Axiom of Choice we pick one such [MATH] for each [MATH] and obtain a section [MATH] such that for each [MATH] and |
[MATH] in [MATH] , we have [MATH] . Pick an arbitrary [MATH] and then suppose [MATH] . This means that, in particular, [MATH] , so that |
[MATH] . We conclude that the net [MATH] is eventually in [MATH] The proof of Proposition 4.10 now follows straightforwardly. Proof (of Proposition 4.10 ). |
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