Preduals for spaces of operators involving Hilbert spaces and trace-class operators
Hannes Thiel

TL;DR
This paper explores the structure of preduals of operator spaces involving Hilbert spaces and trace-class operators, establishing correspondences and uniqueness results for their isometric preduals.
Contribution
It introduces a natural correspondence between isometric preduals of operator spaces and their underlying spaces, extending the understanding of preduals in the context of Hilbert and trace-class operators.
Findings
Isometric preduals of $\\mathcal{L}(H,Y)$ correspond to those of $Y$.
Preduals of $\\mathcal{L}(\mathcal{S}_1)$ relate to preduals of $\mathcal{S}_1$.
The compact operators are the unique predual of $\mathcal{S}_1$ with separately weak* continuous multiplication.
Abstract
Continuing the study of preduals of spaces of bounded, linear maps, we consider the situation that is a Hilbert space. We establish a natural correspondence between isometric preduals of and isometric preduals of . The main ingredient is a Tomiyama-type result which shows that every contractive projection that complements in its bidual is automatically a right -module map. As an application, we show that isometric preduals of , the algebra of operators on the space of trace-class operators, correspond to isometric preduals of itself (and there is an abundance of them). On the other hand, the compact operators are the unique predual of making its multiplication separately weak* continuous.
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Preduals for spaces of operators involving Hilbert spaces and trace-class operators
Hannes Thiel
Hannes Thiel. Mathematisches Institut, Universität Münster, Einsteinst. 62, 48149 Münster, Germany.
[email protected] www.math.uni-muenster.de/u/hannes.thiel/
Abstract.
Continuing the study of preduals of spaces of bounded, linear maps, we consider the situation that is a Hilbert space. We establish a natural correspondence between isometric preduals of and isometric preduals of .
The main ingredient is a Tomiyama-type result which shows that every contractive projection that complements in its bidual is automatically a right -module map.
As an application, we show that isometric preduals of , the algebra of operators on the space of trace-class operators, correspond to isometric preduals of itself (and there is an abundance of them). On the other hand, the compact operators are the unique predual of making its multiplication separately weak∗ continuous.
Key words and phrases:
Preduals, unique predual, Banach algebras, complemented subspace
2010 Mathematics Subject Classification:
Primary: 47L05, 47L10, Secondary: 47L45, 46B20.
The author was partially supported by the Deutsche Forschungsgemeinschaft (SFB 878 Groups, Geometry & Actions).
1. Introduction
An isometric predual of a Banach space is a Banach space together with an isometric isomorphism . Every predual induces a weak∗ topology. Due to the importance of weak∗ topologies, it is interesting to study the existence and uniqueness of preduals; see the survey [God89] and the references therein.
Given Banach spaces and , let denote the space of operators from to . Every isometric predual of induces an isometric predual of : If , then .
Problem \thepbmCt.
Find conditions on and guaranteeing that every isometric predual of is induced from an isometric predual of .
Given reflexive spaces and , Godefroy and Saphar show that is the strongly unique isometric predual of ; see [GS88, Proposition 5.10]. In particular, in this case every isometric predual is induced from .
The main result of this paper extends this to the case that is a Hilbert space and is arbitrary: Every isometric predual of is induced from an isometric predual of ; see Section 2. In particular, has a (strongly unique) isometric predual if and only if does; see Section 2.
To obtain these results, we use that isometric preduals of naturally correspond to contractive projections that are right -module maps and have weak∗ closed kernel; see [GT16, Theorem 5.7]. Hence, we are faced with:
Problem \thepbmCt.
Find conditions on and guaranteeing that every contractive projection is automatically a right -module map.
It was shown by Tomiyama that every contractive projection from a -algebra onto a sub--algebra is automatically a -bimodule map. We therefore consider a positive solution to Section 1 a Tomiyama-type result. Adapting the proof of Tomiyama’s result, we obtain a positive solution to Section 1 whenever is a Hilbert space; see Section 2.
In Section 3, we show that our results also hold when the Hilbert space is replaced by the space of trace-class operators . It follows that isometric preduals of naturally correspond to isometric preduals of ; see Section 3. We note that - and consequently - has many different isometric preduals. On the other hand, we show that the ‘standard’ predual of compact operators is the unique predual of making its multiplication separately weak∗ continuous; see Section 3.
Acknowledgements
The author would like to thank Eusebio Gardella and Tim de Laat for valuable feedback.
Notation
Given Banach spaces and , an operator from to is a bounded, linear map . The space of such operators is denoted by . The projective tensor product of and is denoted by . We identify with a subspace of its bidual, and we let denote the inclusion. A projection is an operator satisfying for all .
2. Preduals involving Hilbert spaces
Throughout, and denote Banach spaces. For conceptual reasons, it is useful to consider preduals of as subsets of . More precisely, a closed subspace is an (isometric) predual of if for the inclusion map , the transpose map restricts to an (isometric) isomorphism .
The space is said to have a strongly unique isometric predual if there exists an isometric predual and if for every isometric predual . Every reflexive space has a strongly unique isometric predual, namely .
\thepgrCt**.**
The space has a natural --bimodule structure. Given , the action of is given by , , for . Thus, acts by precomposing on the right of . Similarly, the action of is given by postcomposing on the left, that is, by , , for .
We obtain a --bimodule structure on . The left action of on is given by . The right action of on is given by . Similarly, we obtain a --bimodule structure on .
Given a -algebra and with , we have , which is an analog of Bessel’s inequality; see [Bla06, II.3.1.12, p.66]. We first prove two versions of this result in a more general context.
Lemma 2.1**.**
Let be a Hilbert space, let be a Banach space, let satisfying , and let . Then
[TABLE]
Proof.
The equation implies that the ranges of and are orthogonal. Given , it follows that the elements and are orthogonal in , whence
[TABLE]
Using this at the second step, we deduce that
[TABLE]
as desired. ∎
Lemma 2.2**.**
Let be a Hilbert space, let be a Banach space, let satisfying , and let . Then
[TABLE]
Similarly, given , we have .
Proof.
We denote the transpose of an operator by , to distinguish it from the adjoint of an operator in . We have and . Further, , where is given by the left action of on . It follows from that . We have in , and therefore
[TABLE]
Applying Lemma 2.1 at the third step, we compute
[TABLE]
Let us show the second inequality. Let and be the right support projections of and in , respectively. Then , , and . It follows that and .
Using Goldstine’s theorem, we choose nets and in such that converges weak∗ to , such that converges weak∗ to , and such that for all and for all . Then converges weak∗ to . Using this at the second step, we deduce that
[TABLE]
and hence . Analogously, we obtain that .
Using this at the third step, using that the net converges weak∗ to at the first step, and using the first inequality of this lemma at the second step, we deduce that
[TABLE]
Let be a -algebra and let be a sub--algebra. By Tomiyama’s theorem, every contractive projection is automatically a -bimodule map (called a conditional expectation). The next result is in the same spirit. It provides a partial positive solution to Section 1.
Theorem \thethmCt.
Let be a Hilbert space, and let be a Banach space. Then every contractive projection is automatically a right -module map, that is, for every and .
Proof.
First, we show the result for the case that is a projection. Let satisfy , and set . The following argument is adapted from the proof of Tomiyama’s theorem in [Bla06, Theorem II.6.10.2, p.132]. Let . We have . Using this at the first step, using that and at the third step, and using Lemma 2.2 at the fourth step, we get
[TABLE]
It follows that
[TABLE]
Since this holds for every , we deduce that . Adding to this equation, we obtain
[TABLE]
Switching the place of and in the above argument, we get . Adding , we get . We deduce that
[TABLE]
Finally, we use that every operator on a Hilbert space is a linear combination of finitely many projections; see [PT67, Corollary 2.3]. ∎
\thepgrCt**.**
Given , we let denote the evaluation map, given by , for . Let be the operator introduced in [GT16, Definition 3.16]. We have
[TABLE]
for all and ; see [GT16, Lemma 3.18]. The map is always a contractive, right -module map.
Let be a projection. Define by , for . Set . Note that
[TABLE]
for and . The map was considered in [GT16, Section 4], where it is shown that is a right -module projection.
Recall that (concrete) isometric preduals of are in natural bijection with contractive projections that have weak∗ closed kernel; see for example [GT16, Proposition 2.7]. Every predual induces a weak∗ topology. A predual of makes the right action by each element from weak∗ continuous if and only if the associated projection is a right -module map; see for example [GT16, Proposition B.6]. The following result is contained in [GT16, Theorem 4.7].
Theorem \thethmCt.
Let and be Banach spaces with . Given a projection , let be as in Section 2. This assignment defines a natural bijection between the following classes:
- (1)
Contractive projections ; 2. (2)
Contractive, right -module projections .
Moreover, the kernel of is weak∗ closed if and only if the kernel of is. Thus, the above correspondence restricts to a natural bijection between isometric preduals of and isometric preduals of that make the right action by weak∗ continuous.
Combining Theorems 2 and 2, we obtain the main result of this paper:
Theorem \thethmCt.
Let be a Hilbert space with , and let be a Banach space. Assigning to a projection the projection , as in Section 2, establishes a natural bijection between the following classes.
- (1)
Contractive projections ; 2. (2)
Contractive projections .
Restricted to projections with weak∗ closed kernel, we obtain a natural bijection between isometric preduals of and isometric preduals of .
Corollary \thecorCt.
Let be a Hilbert space with , and let be a Banach space. Then is -complemented in its bidual if and only if is. Further, has an isometric predual if and only if does. Moreover, if has a strongly unique isometric predual if and only if does.
Corollary \thecorCt.
Let be a Hilbert space, let be a Banach space, and let be an isometric predual. Then for each , the right action of on is continuous for the weak∗ topology induced by .
Remark \thermkCt.
There is a canonical isometric isomorphism . Given an isometric isomorphism , we obtain isometric isomorphisms
[TABLE]
Hence, is an isometric predual of . Given a Hilbert space , Section 2 states that every isometric predual of occurs this way. In particular, given an isometric isomorphism for some Banach space , there is an isometric isomorphism , for some isometric predual of .
Remark \thermkCt.
By [GS88, Proposition 5.10], is the strongly unique isometric predual of if and satisfy the Radon Nikodým property (RNP). Every reflexive space (in particular, every Hilbert space) satisfies the RNP. Thus, if has an isometric predual satisfying the RNP, then Section 2 follows from [GS88]. However, not every Banach spaces with strongly unique isometric predual occurs as the dual of a space satisfying the RNP; see Section 2.
Example \theexaCt.
Let be a von Neumann algebra. By Sakai’s theorem, has a strongly unique isometric predual, denoted by . It follows from Section 2 that is the strongly unique isometric predual of . By [Chu81, Theorem 4], satisfies the RNP if and only if is a direct sum of type factors.
Let denote the hyperfinite -factor. Then does not have the RNP, yet is the strongly unique isometric predual of .
Question \theqstCt.
Does Section 2 hold when the Hilbert space is replaced by a general Banach space satisfying the RNP? More modestly, if is an -space, do isometric preduals of correspond to isometric preduals of ?
Remark \thermkCt.
Note that every positive solution of Section 1 leads to an analog of Section 2. Therefore, Section 2 has a positive answer if the following instance of Section 1 has a positive solution: Given a measure space and a Banach space , is every contractive projection automatically a right -module map?
Consider the space of bounded sequences. Since is a von Neumann algebra, it has a strongly unique isometric predual. Thus, if Section 1 had a positive solution for , then would have a strongly unique isometric predual. However, it was noted in [GS88, Remark 5.12] that has nonisomorphic isometric preduals. It follows in particular that there exists a contractive projection that is not a right -module map.
3. Preduals involving trace-class operators
Throughout this section denotes a Hilbert space with . We let and denote the compact and trace-class operators on , respectively.
An operator belongs to if and only if for some (equivalently, every) orthonormal basis of the sum is finite. Given and an orthonormal basis of , the sum converges absolutely. Moreover, it is independent of the choice of a orthonormal basis and we call
[TABLE]
the trace of . We set . This defines a norm on , turning the trace-class operators into a Banach space. Note that is a (non-closed) two-sided ideal in . Moreover, we have .
\thepgrCt**.**
Given , the map , , is a bounded, linear functional on . This induces an isometric isomorphism . It is also well known that is an isometric predual of . Thus, we have isometric isomorphisms
[TABLE]
Let us say that a Banach space satisfies () if for every Banach space , every contractive projection is automatically a right -module map. If satisfies (), then it follows from Section 2 that every contractive projection is of the form for a unique contractive projection . Section 2 states that Hilbert spaces satisfy ().
Lemma 3.1**.**
If two Banach spaces and satisfy (), then so does .
Proof.
Assume that and satisfy (). Let be another Banach space, and let be a contractive projection.
Banach spaces form a closed monoidal category for the projective tensor product and with adjoint to . Thus, there is a natural isometric isomorphism
[TABLE]
An operator is identified with the operator that sends to the operator given , for .
The projection corresponds to a contractive projection . Applying that satisfies to the projection , there exists a unique contractive projection such that . The situation is shown in the following diagram:
[TABLE]
Let F\in{\mathcal{L}}\big{(}X\hat{\otimes}Y,E\big{)}^{**}, which corresponds to \tilde{F}\in{\mathcal{L}}\big{(}X,{\mathcal{L}}(Y,E)\big{)}^{**}. Then
[TABLE]
for and .
Applying that satisfies to the projection , there exists a unique contractive projection such that
[TABLE]
for every and . We claim that is the desired projection to verify that satisfies .
Given with corresponding element , note that . It follows that . Using this at the last step, we deduce that
[TABLE]
for every and . Thus, we have , for every simple tensor in . It follows from linearity and continuity of the involved maps that the same equation holds for every , as desired. ∎
Lemma 3.2**.**
Let be a Banach space. Then every contractive projection from to is automatically a right -module map.
Proof.
It is well known that the trace-class operators on are isometrically isomorphic to . Therefore, the statement follows from Lemma 3.1. ∎
Using Lemma 3.2 instead of Section 2, we obtain the analog of Section 2 with the space of trace-class operators in place of the Hilbert space.
Proposition \theprpCt.
Let be a Banach space. Assigning to a projection the projection , as in Section 2, established a natural bijection between the following classes.
- (1)
Contractive projections ; 2. (2)
Contractive projections .
Restricted to projections with weak∗ closed kernel, we obtain a natural bijection between isometric preduals of and isometric preduals of .
Corollary \thecorCt.
Let be a Banach space, and let be an isometric predual. Then for each , the right action of on is continuous for the weak∗ topology induced by .
Remark \thermkCt.
Using Lemma 3.1 inductively, Section 3 and Section 3 hold for any projective tensor power in place of .
Remark \thermkCt.
We have isometric isomorphisms
[TABLE]
and we consider as the ‘standard’ predual of . Right multiplication by an element from is continuous for the induced weak∗ topology; see Section 3. However, this is not the case for left multiplication. Indeed, a Banach space is reflexive if and only if left multiplication by elements from is weak∗ continuous (for any predual); see [GT16, Corollary 6.4].
Example \theexaCt.
By Section 3, there is a natural correspondence between isometric preduals of and isometric preduals of . The compact operators on form the canonical isometric predual of .
However, if , then has also many other isometric preduals. Indeed, the diagonal operators in form an isometric copy of that is closed for the ‘standard’ weak∗ topology induced by the compact operators. Since does not have a strongly unique isometric predual, neither does .
Theorem \thethmCt.
The ‘standard’ predual of makes multiplication in separately weak∗ continuous. Moreover, is the only such predual: If is a (not necessarily isometric) predual that makes multiplication in separately weak∗ continuous, then . In particular, every predual of making multiplication separately weak∗ continuous is automatically an isometric predual.
Proof.
Given , we let denote the functional on given by , for . This identifies with .
The multiplication on induces a -bimodule structure on its dual; see [GT16, Paragraph A.3]. Let us recall some details. Given , let be given by and , for . Then the left action of on is given by , and the right action is given by . Thus, given and , we have
[TABLE]
for , and therefore . Similarly, we obtain .
Let be a predual. Then left (right) multiplication on is -continuous if and only if is a right (left) -submodule of ; see [GT16, Corollary B.7]. Given and , we have . Thus, the predual is a -sub-bimodule of , which shows that it makes multiplication in separately weak∗ continuous.
Conversely, let be a predual making multiplication in separately weak∗ continuous. Then is invariant under the left and right action of on . We have shown above that the left (right) action of on is simply given by right (left) multiplication with .
Claim: The set is a closed, two-sided ideal in . To verify the claim, let , and let . Given a finite-dimensional subspace , let be the orthogonal projection onto . We order the finite-dimensional subspaces of by inclusion. Since , we have and therefore
[TABLE]
For each , we have . Since is invariant under right multiplication by , it follows that . Since is norm-closed, we deduce that . Analogously, one shows that is a left ideal in , which proves the claim.
The only closed, two-sided ideals of are and . It is easy to see that . Thus, . Since both and are preduals of , it follows that , as desired. ∎
Corollary \thecorCt.
Let be the trace-class operators on a Hilbert space . Then every Banach algebra isomorphism is weak∗ continuous (for the ‘standard’ predual .)
Remark \thermkCt.
A dual Banach algebra is a Banach algebra together with a predual making the multiplication in separately weak∗ continuous. This concept was introduced by Runde, [Run02, Definition 4.4.1, p.108], and extensively studied by Daws; see [Daw11] and the references therein. Section 3 states that the trace-class operators with their ‘standard’ predual of compact operators form a dual Banach algebra. Moreover, the compact operators are the only predual making the trace-class operators into a dual Banach algebra.
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