A Few Observations on Weaver’s Quantum Relations
In [Wea2012], Weaver introduced the concept of quantum relation over a von Neumann algebra . When is either finite dimensional or discrete and abelian, is given by an orthogonal projection in . Here, we generalize such result to general von Neumann algebras, proving that quantum relations are in bijective correspondence with weak- closed left ideals inside , where is the extended Haagerup tensor product. The correspondence between the two is given by identifying with -bimodular operators and proving a double annihilator relation
Given an action of a group/quantum group on we give a definition for invariant quantum relations and prove that in the case of group von Neumann algebras , invariant quantum relations are left ideals in the measure algebra . At the end we explore possible applications to noncommutative harmonic analysis, in particular noncommutative Gaussian bounds.
1.1. Weaver’s Quantum Relations
In [Wea2012, KuWea2012] Kuperberg and Weaver introduced the concept of a quantum relation over a von Neumann algebra . They defined a quantum relation to be a weak- closed operator bimodule over , i.e.: a linear weak- closed subset satisfying that . It is easy to see that such notion doesn’t depend on the representation .
In the case acting by multiplication operators we have that . Identifying with matrices indexed by , gives that is a quantum relation whenever
for every , . This in turns easily implies, see [Wea2012, Proposition 1.3], that there is a unique subset such that
and reciprocally every such subset have associated the operator bimodule of all matrices supported on . When is abelian but not atomic we do not have a bijective correspondence between bimodules and measurable subsets of . In that case the natural object to substitute the (discrete) relations will be the, so called, measurable relations, i.e. weak- open subsets satisfying that
The measurable relation associated with a quantum relation is given by
Notice that in the abelian discrete case we have that is just the set of projections such that there are and with . Reciprocally, given any measurable relation we can associate a quantum relation over given by
It is proved in [Wea2012] that the map is injective. Unfortunately it is not surjective in general. This has to do with the fact that all the operator bimodules arising like in 1.3 are not just weak- closed but operator reflexive, see [Er1986, Lars1982] and in particular closed in the weak operator topology, or WOT in short. The way to fix that is to observe that if is any weak- closed linear subspace is operator reflexive. Since is a -bimodule and we have that is a quantum relation over the amplified algebra . This suggests that the right definition for quantum relations as pairs of related projections is given by amplified projections in . The next definition captures this intuition.
([Wea2012, Definition 2.24]) We will say that is an intrinsic quantum relation iff
is weak- open.
If and are sets of families of projections in then
For every we have that
where represents the left (or final) projection of the operator .
Quantum relations over and intrinsic quantum relations (or i.q.r.) over are in bijective correspondence and the adaptations of the maps 1.2 and 1.3 are inverse of each other. Indeed, this correspondence works for every von Neumann algebra not necessarily abelian or discrete, see [Wea2012, Theorem 2.32].
Through this article we are going to employ liberally the language of operator spaces, see [Pi2003, EffRu2000Book, BleMer2004Operator] for more information. An operator space is a closed linear subset . Given two operator spaces and we say that a linear map is completely bounded, or c.b. in short, iff the matrix amplifications are uniformly bounded on . We are going to denote by the space of all completely bounded (or c.b.) operators with the norm given by
The category of operator spaces is the collection of all operator spaces with c.b. maps as morphisms. There is also an intrinsic characterization of operator spaces as Banach spaces endowed with collections of matrix norms satisfying the Ruan’s axioms. Either an isometric injection or a family of compatible matrix norm will be called an operator space structure, or o.s.s. in short.
Let be a discrete measure space with the counting measure and let us identify with matrices indexed by . Given a matrix we define the Schur multiplier of symbol as the operator given by
Whenever is completely bounded we will say that is a c.b. Schur multiplier. We are going to denote by the set of all c.b. Schur multipliers and by the space of all c.b. and normal ones (i.e. weak- continuous for ). Assume that is a finite set, let be a relation and be its associated quantum relation. We have that the ideal given by
contains just the Schur multipliers whose symbol satisfies that if . The reciprocal is also true and we have the following.
Let be a finite set and and be as above. Then if an ideal in we have that
are bijections between the sets of quantum relations and the set of ideals of Schur multipliers. Furthermore, the maps and are inverse of each other.
Such result was generalized to general, not necessarily abelian, finite dimensional von Neumann algebras by Weaver [Wea2012]. For that end recall that is actually equal to the algebra of all completely bounded normal operators that are -bimodular. We are going to denote the the algebras of -bimodular c.b. normal operators on by . It is trivial to see that in the case of finite dimensional we have a bounded, quasi-isometric and multiplicative map given by extension of
To see that, let , so that, is quasi-isometric to and is quasi isometric to . If , we denote by the operator given by . It is clear that is -bimodular iff it belongs to the commutant of but such algebra is isomorphic to as we claimed. If is a quantum relation over we have that is a left ideal and therefore is of the form for some . Furthermore, we have the following.
([Wea2012, Proposition 2.23]) If is finite dimensional the correspondence defined as above is an order-preserving bijection between quantum relations over and projections in .
In the case of infinite dimensional von Neumann algebras the result above fails and not every quantum relation can be associated with a projection in . The reason for that is that although the map is bounded and multiplicative for every finite dimensional algebra it is far from isometric. Indeed its norm explodes with . The problem can be solved by changing the tensor norm from the spatial tensor norm to the Haagerup tensor norm of the two von Neumann algebras. With that tool at hand we will be able to prove a generalization of 1.1 for general algebras in the next section.
1.2. Module Maps and The Haagerup Tensor Product
Let , be two operator spaces. We define the bilinear form by
Of course such definition makes perfect sense with matrices of different sizes just by embedding all matrices inside and restricting. The Haagerup tensor norm for is defined to be
The Haagerup tensor product is defined as the completion under that norm. Similarly can be given an o.s.s by defining:
In the case of two dual operator spaces and the weak- Haagerup tensor product, introduced in [BleSmi1992] by Blecher and Smith, is given by
Since the Haagerup tensor norm is self dual, see [EffRu1991SelfDual], we have that embeds inside isometrically and is weak- dense. This tensor product is a complemented subspace of the normal Haagerup tensor product introduced by Effros and Kishimoto [EffKi1987] and which satisfies that
In [EffRu2003] Effros and Ruan introduced the extended Haagerup tensor product generalizing the weak- Haagerup tensor to (potentially) non-dual operator spaces. Indeed if is a matrix whose entries are, possibly infinite, sums of simple tensors, we say that iff
for every possible index set . It can be seen that it is enough to take to be the smallest cardinality of a dense set in with . Particularly when and are separable von Neumann algebras we can take numerable. In the case of , being dual operator spaces, we have that
The coarser topology in making the pairing with every element in continuous is strictly finer than the weak- topology given by the predual . Since is -closed, , with the topology, is a dual space. Its predual is obtained by a quotient of
When , are von Neumann algebras is a weak- Banach algebra with a jointly completely bounded multiplication, see [EffRu2000Book, pp. 126], given by extension of
When there is also a natural multiplicative involution .
Recall that the space of completely bounded has a natural o.s.s. given by the identification . If and are dual operator spaces we define to be susbspace of all weak- continuous operators. We have a natural identification . When are bimodules over a von Neumann algebra we will denote by and the subspaces of completely bounded and bimodular operators. Such subspaces are easily seen to be norm closed. We will treat mainly the case when . We have, using that and that , see [BleMer2004Operator, (1.28)], that
The identification is given by restriction to and by passage to the second dual. The identity 1.6 allow us to give a predual for by
where is the Fubini tensor product, see [EffKraRu1993], [EffRu2003] or [EffRu2000Book] which is isomorphic to the dual of the (operator space) projective tensor product , see [EffRu2000Book, Chap. 7]. Similarly the predual of is given by . In both cases the pairing is given by linear extension of , for . A subtle point is that the coarser topology in making the paring with all the elements in continuous is, in general, strictly finer than the weak- topology given by the predual . To see that, notice that the following inclusion holds
Indeed, the inclusion above is just a consequence of the fact that and the injectivity of the functor , where is the predual of any hyperfinite von Neumann algebra, see [Pi1998]. Since -closed sets are -closed we have that is -closed and so the topology induces another predual for . Clearly, the topology of pointwise weak- convergence in is coarser than the topology. Analogously, the topology of pointwise (in ) weak- topology is coarser that the topology. In both cases the topologies coincide over bounded sets.
The subspace of bimodular operators is closed in both the and the topologies. Indeed, it is closed in the -pointwise weak- topology which is coarser than both. As a consequence, using the Hanhn-Banach Theorem, we get that inherits two natural preduals topologies
where , are the corresponding preannihilators. Similarly is also a dual space with the topology. The spaces , and are Banach algebras with the composition operation. They have a natural multiplicative involution given by and satisfying that .
Recall that in the case of we have that
For non-discrete measure spaces we have that corresponds to the algebra of measurable Schur multipliers, see [Spronk2004].
Now we are in position of stating the isomorphism between Haagerup tensors and bimodular operators.
The result above is well known to the experts, although their pieces are scattered throughout the literature. We will just give a brief sketch with references. Recall too that the first appearance of such result is credited to be in an unpublished note of Haagerup [HaagSD].
Let us concentrate on 2, which will be the most important for our applications. The fact that is a complete contraction amounts to a trivial calculation. Indeed, if we may define, for every , the matrices
inside , where is a system of matrix units. Then satisfies that
where is the orthogonal projection on the span of . Clearly
and is an -bimodular operator. Taking the supremum over and the infimum over all representations of gives that . To see that it is surjective notice that if by Wittstock’s factorization theorem for c.b. maps, see [Pa1986Book], we have that there is a large enough (we can take the dimension of to be equal to that of for infinite dimensional spaces), a representation and two elements , such that and but we can identify and with a row and a column respectively inside and we have that , where . It only rest to prove that if is -bimodular we can pick , which is the main result in [Smith1991, Theorem 3.1]. The rest of the points are similarly proved, see also [BleSmi1992] for 3. ∎
As a consequence of the preceding theorem we are going to identify at times and its weak- topology with and . The following lemma describe the weak- continuous functionals on for its different preduals.
Let , then
is -continuous iff
is -continuous iff
Furthermore, is pointwise weak- continuous, iff in 1 can be taken in . Similarly, is -pointwise weak- continuous iff we can take .
We will prove 1 first. Since, by Theorem 1.3, the predual for the topology is given by , where is the preannihilator of the -bimodular maps, can be lifted to an element (that we will denote also by ) in inducing the same functional. By definition of the o.s. projective tensor product we have that there are, possibly infinite, index sets , and elements , and , where , such that
The action on is given by
Note that, by [Pi1998, Theorem 1.5], . We have thus that every weak- continuous functional can be expressed as
The other claims in the statement follows by a repetition of the ideas used to prove that SOT-continuous and WOT-continuous functionals coincide over . Indeed, assume is pointwise weak- continuous. Then, there are finite collection and such that whenever for . In particular, taking gives
As a consequence, if , for , we have and so factors through a finite dimensional space. Therefore, can be expressed as a finite combination of simple tensors. ∎
2. The Correspondence Between Ideals and Modules
In this section we are going to prove the correspondence between left ideals in and quantum relations over . We are going to start recalling two easy lemmas that will be thoroughly used in this section. The first asserts that the bilinear form can be extended from to , where is the weak- closed spatial tensor or equivalently, since is a von Neumann algebra, the Fubini tensor product. The second is a stability result for weak- closed left ideals in . In the forthcoming text we are going to denote by the weak- closed tensor product, with respect to the topology. Recall that, using the following identifications
and reasoning like in (1.7), we have that the predual of can be expressed as a quotient of .
The bilinear map is bounded and continuous over bounded sets if is given the product strong operator topology and the topology.
Let be nets in the unit ball satisfying that and in the SOT. Since the SOT and -SOT topologies agree on bounded set we can assume that we have SOT convergence for any given representation of and in particular for its representation on the Hilbert-Schmidt operators . Again since the weak- topology and the pointwise weak- topology of agree on bounded sets it is enough to see that for any and , . But using that and expressing , where are Hilbert-Schmidt operators, gives , where the last paring is just the inner product of . Using the SOT-convergence of and gives
and that concludes the proof. ∎
Let be a -closed left ideal, the following holds
If satisfy that then for every .
if and only if .
Since is a