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Let [MATH] [MATH] ) be a collection of strongly constructible left ideals. Writing [MATH] for the set of unit vectors in [MATH] , for each |
[MATH] there are directed sets [MATH] and nets [MATH] for [MATH] such that an [MATH] belongs to [MATH] if and only if [MATH] By Lemma 4.11 , find a new net [MATH] |
in [MATH] which is eventually in a subset of [MATH] if and only if each of the [MATH] are eventually in that set. Fix [MATH] and let [MATH] be given. Let |
[EQUATION] Clearly [MATH] iff for every [MATH] and every [MATH] [MATH] is eventually in [MATH] This happens iff for every [MATH] [MATH] |
is eventually in [MATH] , which in turn happens iff [MATH] . Thus [MATH] is strongly constructible. Corollary 4.12 Every proper left ideal [MATH] of [MATH] is contained in a smallest strongly constructible left ideal, which we shall call the |
strongly constructible hull of [MATH] Corollary 4.13 The primitive ideals of [MATH] are strongly constructible. Proof. Every primitive ideal is the intersection of the maximal left ideals which contain it |
, §24, Proposition 12 (iv)] The result follows by By Propositiona 4.9 and 4.10 Example 4.14 In particular, the maximal two-sided ideals of [MATH] , being primitive, are strongly constructible. Recall that the strong radical of a unital algebra is the intersection of all its maximal two-sided ideals. In 17 , Theorem 3.2... |
is a continuous nest algebra then any norm-closed, two-sided ideal of [MATH] which contains the strong radical is the intersection of the maximal two-sided ideals which contain it. Thus by Propositions 4.10 and 4.13 all such ideals are strongly constructible. |
Corollary 4.15 All norm-closed, two-sided ideals of a continuous nest algebra which contain the strong radical are strongly constructible. |
Question 4.16 Is every norm-closed left ideal of a nest algebra strongly constructible? Strongly constructible ideals are also characterized by two ostensibly weaker conditions: |
Proposition 4.17 Let [MATH] be a proper left ideal of [MATH] . The following are equivalent: (1) [MATH] is strongly constructible. |
(2) [MATH] can be specified by a net [MATH] where [MATH] for all [MATH] (3) [MATH] can be specified by a net [MATH] where [MATH] is bounded. |
Proof. Clearly [MATH] and so it remains to prove [MATH] Suppose [MATH] specifies [MATH] and [MATH] is bounded. Since [MATH] is proper, by Lemma 4.2 |
[MATH] and so there is an [MATH] such that [MATH] is frequently at least [MATH] . For each [MATH] set [EQUATION] Each [MATH] is a directed set (with the order relation inherited from [MATH] ) and the restricted net [MATH] defines a left ideal [MATH] Since the [MATH] are bounded away from zero on [MATH] , we can normali... |
[MATH] and then the result will follow by Proposition 4.10 Clearly since each [MATH] , also [MATH] and so [MATH] . Suppose [MATH] . Then |
[MATH] and so there is an [MATH] such that [MATH] frequently. Choose [MATH] so that then whenever [MATH] then [EQUATION] and thus [MATH] . It follows that [MATH] |
frequently on [MATH] , and so [MATH] Proposition 4.18 Let [MATH] be a maximal left ideal in [MATH] and let [MATH] be a sequence of pairwise orthogonal projections in [MATH] . There is a subsequence [MATH] such that the projection [MATH] belongs to [MATH] |
Proof. By Proposition 4.9 [MATH] is strongly constructible, say by a net [MATH] where each [MATH] is a unit vector in the range of |
[MATH] . By Kelley’s Theorem, this net has a universal subnet, which specifies a proper ideal containing [MATH] , hence in fact specifies [MATH] itself. Thus we may assume |
[MATH] is universal. The proof now proceeds by means of a fairly routine diagonal argument. For any [MATH] write [MATH] Take [MATH] and split [MATH] into two infinite sets, [MATH] and [MATH] If [MATH] and [MATH] are each eventually greater than [MATH] then |
[MATH] is eventually greater than [MATH] , which is impossible. Since [MATH] is universal that means at least one of [MATH] [MATH] is eventually no greater than [MATH] ; without loss suppose that [MATH] eventually, and set [MATH] |
Now decompose [MATH] in the same way as the union of infinite subsets and, as before, we conclude that at least one of [MATH] [MATH] |
is eventually no greater than [MATH] . Take [MATH] to be one of [MATH] [MATH] for which this holds. Proceeding in this way we obtain a sequence |
[MATH] of infinite subsets of [MATH] such that for each [MATH] , eventually [MATH] . Now take [MATH] to be the [MATH] th element of [MATH] in order, which is a strictly increasing sequence, and let [MATH] . Thus [MATH] is finite for all [MATH] |
Finally, write [MATH] and, given [MATH] , take [MATH] such that [MATH] For all sufficiently large [MATH] [EQUATION] But the sum in the last line is finite and so is eventually less than [MATH] We can conclude [MATH] so that [MATH] |
Corollary 4.19 Let [MATH] be a maximal right ideal in [MATH] and let [MATH] be a sequence of pairwise orthogonal projections in [MATH] . There is a subsequence [MATH] such that the projection [MATH] belongs to [MATH] |
Proof. The result follows on taking adjoints and working in [MATH] 5. Atomic nest algebras In this section we shall focus on atomic nest algebras and relate the character of primitive ideals to the family of diagonal operators they contain. Observe that if [MATH] is a primitive ideal of [MATH] then [MATH] is a norm-clo... |
Proposition 5.1 Let [MATH] be an atomic nest and [MATH] a two-sided ideal in [MATH] Then [MATH] is a maximal two-sided ideal if and only if [MATH] |
is a maximal two-sided ideal of [MATH] Proof. Suppose [MATH] is maximal. Then by 20 , Theorem 3.8] [MATH] contains [MATH] . It follows that [MATH] If [MATH] is not maximal then there is a larger proper ideal [MATH] of [MATH] But then [MATH] is a proper ideal of [MATH] and strictly larger than [MATH] , contrary to fact. |
Suppose on the other hand that [MATH] is maximal. By , Theorem 10.2] [MATH] is generated as a two-sided ideal by a generator which is the sum of three commutators |
[MATH] [MATH] ) where [MATH] and [MATH] is a projection in the core [MATH] of [MATH] (Recall that the core of a nest algebra is the abelian von Neumann algebra |
[MATH] .) Now since [MATH] is a maximal ideal of [MATH] , and the [MATH] are in the centre of [MATH] , it follows that one of [MATH] |
must lie in [MATH] for each [MATH] . Thus in any event the commutators [MATH] belong to [MATH] and so [MATH] contains [MATH] . Thus, again, [MATH] If [MATH] is not maximal then there is a larger proper ideal [MATH] of [MATH] But then since [MATH] also contains [MATH] |
[MATH] and so [MATH] is a proper ideal of [MATH] and larger than [MATH] , contrary to fact. The proof of Proposition 5.1 is deceptively straightforward. In fact the result cited from |
depends on Marcus, Spielman, and Srivastava’s proof of the Paving Theorem. Recall (Definition 2.6 ) that we write [MATH] for the unique diagonal ideal contained by the primitive ideal [MATH] |
Proposition 5.2 Let [MATH] be an atomic nest, let [MATH] be a primitive ideal of [MATH] and suppose [MATH] . Then there are non-zero projections in |
[MATH] Proof. We shall prove the result in the case when [MATH] for some [MATH] in [MATH] If, instead, [MATH] for some [MATH] then we take adjoints and apply the result to [MATH] In this case [MATH] is a right primitive ideal of [MATH] and so we shall take care that our proof accommodates the case when [MATH] is either... |
If [MATH] is a left primitive ideal, let [MATH] be a maximal left ideal such that [MATH] is the kernel of the left regular module action of [MATH] on [MATH] In the case that [MATH] is right primitive, let [MATH] be a maximal right ideal such that |
[MATH] is the kernel of the right regular module action of [MATH] on [MATH] Suppose that [MATH] . Note that [MATH] cannot be finite for if it were then [MATH] would be a maximal ideal of [MATH] and so [MATH] , contrary to hypothesis. If [MATH] then the only proper ideal strictly containing |
[MATH] is [MATH] , which must therefore equal [MATH] . Any finite rank projection of the form [MATH] will serve to establish the result in this case. |
For the remainder of the proof, assume that [MATH] and take [MATH] By Lemma 3.3 there are [MATH] such that [MATH] is block diagonal with respect to some sequence [MATH] of nest projections strictly increasing to [MATH] and each of the blocks has norm greater than [MATH] . Replacing [MATH] with [MATH] we can assume |
[MATH] where the norm of each term is greater than [MATH] Consider the sequence of intervals [MATH] . These are each in [MATH] and so in [MATH] By Proposition 4.18 and Corollary 4.19 , whether [MATH] is assumed to be maximal right or maximal left, there is a subsequence [MATH] |
such that [MATH] contains [MATH] Then for each [MATH] find an atom [MATH] Choose vectors [MATH] such that [MATH] and [MATH] and [MATH] are in the range of [MATH] with |
[MATH] and [MATH] . Thus, [EQUATION] where both of the sums converge strongly and are in [MATH] because [EQUATION] and [EQUATION] |
since [MATH] Thus [MATH] . Let [MATH] which is dominated by a projection in [MATH] and so is also in [MATH] We shall show that [MATH] |
Suppose for a contradiction that [MATH] It follows, as observed in Remark 3.1 that there are [MATH] such that [MATH] . We can assume that [MATH] and [MATH] Write [MATH] where |
[EQUATION] so that [MATH] Likewise, write [MATH] where [EQUATION] so that [MATH] The sums for [MATH] and [MATH] converge strongly because the sequences of terms are norm-bounded and have pairwise orthogonal ranges and cokernels. |
Now set [MATH] and [MATH] . From the following computations we see that [MATH] and [MATH] are in [MATH] since the terms of the sums are in [MATH] |
[EQUATION] Furthermore, since [MATH] and [MATH] we have that [EQUATION] and [EQUATION] Since [MATH] , it now follows that also [MATH] |
Now note that [EQUATION] where [MATH] We can decompose [MATH] in two ways, either as [EQUATION] or as [EQUATION] These two cases are of the form [MATH] and [MATH] respectively where in both cases [MATH] |
is nilpotent. Recall that [MATH] and so, whether [MATH] is a maximal left ideal or a maximal right ideal, we conclude that [MATH] , which is impossible since this is invertible and [MATH] |
is proper. From this contradiction we conclude that [MATH] Theorem 5.3 Let [MATH] be an atomic nest and let [MATH] be a primitive ideal of [MATH] |
(1) If [MATH] is a maximal two-sided ideal of [MATH] then [MATH] is a maximal two-sided ideal of [MATH] (2) If [MATH] is equal to [MATH] for some diagonal ideal [MATH] |
then [MATH] is a diagonal ideal and, in fact, [MATH] Proof. Case ( ) is just Proposition 5.1 . To prove Case ( ), suppose that [MATH] for some diagonal ideal [MATH] . First observe that [MATH] Now, distinct diagonal ideals contain complementary projections (see the proof of 23 , Lemma 4.8] for this fact) and so [MATH] ... |
[MATH] contains projections which are not in [MATH] , contrary to hypothesis. We can now distinguish three classes of primitive ideals based on the diagonal operators they contain. The first class ( [MATH] ) consists of primitive ideals for which [MATH] |
is a maximal ideal of [MATH] , and this consists of the maximal two-sided ideals of [MATH] The second class ( [MATH] ) consists of primitive ideals for which [MATH] |
for some diagonal ideal [MATH] and this class consists of diagonal ideals. The third class ( [MATH] ) consists of the remaining primitive ideals for which [MATH] |
takes neither its minimal nor its maximal values. The maximal ideals of a general nest algebra were completely described in 20 , Corollary 3.10] . In particular when [MATH] is atomic the ideals in [MATH] are precisely the ideals of the form [MATH] |
where [MATH] is a maximal two-sided ideal of [MATH] . The ideals in [MATH] are the primitive ideals which are also diagonal ideals. Trivially all ideals of the form [MATH] where [MATH] (or, equivalently, [MATH] where [MATH] are included in this class. (See the first paragraph of the proof of Theorem 3.7 for details.) B... |
6. The infinite upper triangular operators Throughout this section, let [MATH] and consider the algebra [MATH] of all upper triangular operators with respect to the standard basis of [MATH] Recall that we write [MATH] for the standard basis and let [MATH] |
be the projection onto the span of [MATH] , and [MATH] . Then [MATH] is the algebra of infinite upper triangular operators with respect to the [MATH] and [MATH] |
is simply the ideal of infinite strictly upper triangular operators. Moreover, the diagonal ideals of [MATH] are precisely the ideals |
[MATH] where [MATH] for [MATH] and [MATH] . Note that [MATH] coincides with the compact operators of [MATH] , a fact which we shall develop below. |
6.1. The quasitriangular algebra Let [MATH] be the set of all compact operators in [MATH] and write [MATH] for the quasitriangular algebra [MATH] . By |
and, in more generality, [MATH] is a norm-closed algebra in [MATH] and the canonical isomorphism between [MATH] and [MATH] is isometric. |
Corollary 6.1 Assuming the Continuum Hypothesis, [MATH] is a left (resp. right) primitive algebra. Proof. [MATH] , which is a left primitive ideal by Theorem 3.7 and a right primitive ideal by Corollary 3.8 |
Corollary 6.2 Assuming the Continuum Hypothesis, [MATH] is a left (resp. right) primitive algebra, and [MATH] is a left (resp. right) primitive ideal in [MATH] |
6.2. A catalogue of primitive ideals Clearly [MATH] contains [MATH] . Assuming the Continuum Hypothesis then by Theorem 3.7 [EQUATION] |
By 20 , Corollary 3.10] the ideals of [MATH] are precisely the ideals of the form [MATH] where [MATH] is a maximal ideal of [MATH] . In this case [MATH] is naturally identified with [MATH] |
and its maximal ideal space with the sequences vanishing at points of [MATH] The maximal ideals of [MATH] corresponding to points of [MATH] are precisely the |
[MATH] and so we can write [EQUATION] where [MATH] is the maximal ideal of [MATH] corresponding to sequences in [MATH] vanishing at [MATH] |
There remains the set [MATH] of primitive ideals which are neither diagonal ideals nor maximal ideals. These are the primitive ideals [MATH] where [MATH] is a closed ideal of [MATH] |
corresponding to an ideal of [MATH] which strictly contains [MATH] and is not maximal. We cannot give a complete catalogue of these ideals but we can provide a rich set of examples. |
Consider the following special case of a general construction of epimorphisms between nest algebras, taken from Corollary 5.3 of |
Let [MATH] be integers such that the intervals [MATH] are pairwise disjoint and let [MATH] be a free ultrafilter on [MATH] . Suppose that [MATH] Let [MATH] |
be the partial isometry mapping [MATH] to [MATH] when [MATH] and zero otherwise. For [MATH] define [EQUATION] where convergence is in the weak operator topology and the limit always exists by WOT -compactness of the unit ball. Then by , Corollary 5.3] this map is an epimorphism of [MATH] onto [MATH] Note also that [MAT... |
If [MATH] is such an epimorphism of [MATH] onto [MATH] and [MATH] is an irreducible representation of [MATH] then clearly [MATH] is also an irreducible representation of [MATH] If [MATH] is in [MATH] then so is [MATH] . However, as we shall see, if [MATH] then [MATH] will be in [MATH] and this provides a rich supply of... |
Assuming the Continuum Hypothesis, [MATH] , so consider the primitive ideal [MATH] Note that [MATH] annihilates [MATH] and so [MATH] is the unique diagonal ideal in [MATH] Writing [MATH] for the diagonal expectation [MATH] observe that [MATH] and so [MATH] Thus [MATH] . On the other hand, [MATH] since, by |
20 , Theorem 3.8] , every maximal ideal of [MATH] contains [MATH] , but [MATH] does not contain the unilateral backward shift [MATH] |
since [MATH] . Thus [MATH] and [MATH] and so [MATH] In fact this construction readily yields uncountably many incomparable ideals in [MATH] . For fix projections [MATH] where |
[MATH] and let [MATH] be a fixed free ultrafilter. As is well-known we can find an uncountable collection [MATH] of infinite subsets of [MATH] with the property that distinct members of [MATH] intersect only in finite sets. For [MATH] , list the elements of [MATH] in order as [MATH] and build an ultrafilter epimorphism... |
as above, this time employing the intervals [MATH] and the ultrafilter [MATH] Write [MATH] for the diagonal expectation [MATH] As before, [MATH] Now for any [MATH] |
[MATH] , for otherwise [EQUATION] We can also exhibit infinite chains of ideals in [MATH] for since [MATH] , the ideals [MATH] form a chain of distinct ideals in [MATH] for any fixed epimorphism |
[MATH] 6.3. Some properties of ideals in [MATH] Although the ultrafilter epimorphism construction of ideals in [MATH] is not representative, we can prove some properties which all ideals in [MATH] share with the ultrafilter construction. These results are, however, tightly bound to the case of [MATH] |
(especially Proposition 6.3 and it is unclear how they might be extended. Proposition 6.3 Let [MATH] be a primitive ideal of [MATH] and suppose [MATH] Then there is an increasing sequence of integers |
[MATH] such that [MATH] contains [EQUATION] Proof. Let [MATH] be a maximal left ideal such that [MATH] is the kernel of the left-regular representation on [MATH] By Proposition 5.2 [MATH] contains a projection |
[MATH] . Choose a subsequence of nest projections [MATH] such that [EQUATION] for all [MATH] . We shall show that if [EQUATION] then [MATH] . By Remark 3.1 , since [MATH] is a two-sided ideal of [MATH] , if [MATH] then [MATH] , so suppose for a contradiction that |
[MATH] By maximalty of [MATH] [MATH] and so there is an [MATH] such that [MATH] . Decompose [MATH] as [MATH] where [EQUATION] and [MATH] . Observe that therefore |
[EQUATION] for all [MATH] Now take fixed arbitrary [MATH] in [MATH] and consider two cases. First, if [MATH] does not dominate any [MATH] then there must be a |
[MATH] such that [MATH] , and so [MATH] On the other hand if [MATH] does dominate some [MATH] , take [MATH] to be the largest possible (which exists since [MATH] ) and observe that, by ( ), |
[EQUATION] It follows that in either case [EQUATION] Since the right-hand side is infinite if [MATH] , the inequality is valid for all [MATH] |
in [MATH] . It follows immediately from 21 , Theorem 2.6] that [MATH] factors through [MATH] as [MATH] for some [MATH] , and so [MATH] whence [MATH] |
However since [MATH] the terms of the sum for [MATH] are [EQUATION] so that [MATH] is nilpotent of order [MATH] . Thus [MATH] cannot belong to the proper left ideal [MATH] , which is a contradiction. |
Let [MATH] [MATH] ) be a set of pairwise orthogonal intervals of [MATH] For [MATH] let [MATH] and [MATH] . For convenience write [MATH] for [MATH] . The last result shows that, at least in [MATH] primitive ideals which are not in [MATH] must contain [MATH] for suitable [MATH] . The next two lemmas explore the consequen... |
Lemma 6.4 Let [MATH] be a primitive ideal of [MATH] and suppose [MATH] Then [MATH] is an ultrafilter. Proof. [MATH] itself is non-empty since [MATH] , and the sets in [MATH] are non-empty since [MATH] If [MATH] and [MATH] then |
[MATH] and so [MATH] If [MATH] then [MATH] and so [MATH] . Thus [MATH] is a filter. Let [MATH] be an irreducible representation with [MATH] For any [MATH] and [MATH] [MATH] |
and so [MATH] commutes with [MATH] . Thus [MATH] is an invariant subspace of [MATH] and so [MATH] Suppose that [MATH] and so [MATH] Then for any [MATH] |
[EQUATION] and so [EQUATION] whence [MATH] and [MATH] Likewise, if [MATH] , then [MATH] Thus [MATH] is an ultrafilter. Lemma 6.5 |
Let [MATH] be a primitive ideal of [MATH] and suppose [MATH] Suppose that for each [MATH] we can decompose [MATH] as the sum [MATH] of intervals of [MATH] . Then [MATH] contains one of [MATH] where |
[MATH] Proof. Each [MATH] is decomposed into the sum of two intervals which share a common endpoint. Let [MATH] be the set of [MATH] for which the shared endpoint is the upper endpoint of [MATH] and the lower endpoint of [MATH] . Clearly |
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