Multipliers and Duality for Group Actions

We define operator-valued Schur and Herz--Schur multipliers in terms of module actions, and show that the standard properties of these multipliers follow from well-known facts about these module actions and duality theory for group actions. These results are applied to study the Herz--Schur multipliers of an abelian group acting on its Pontryagin dual: it is shown that a natural subset of these Herz--Schur multipliers can be identified with the classical Herz--Schur multipliers of the direct product of the group with its dual group.


Introduction
Schur multipliers -the scalar-valued functions on N × N for which the entrywise product maps B(ℓ 2 ) into itself -arose from Schur's work on the entrywise product of matrices in the early twentieth century. Their importance was recognised by Grothendieck [10] (see also Pisier [19,Chapter 5]), who used them to formulate his fundamental theorem. These classical Schur multipliers have been extended in several directions; see, for example, [19,Chapter 5].
Herz-Schur multipliers, or completely bounded multipliers of the Fourier algebra of a group, originate in work of Herz [13] where they were viewed as a generalisation of the Fourier-Stieltjes transform. They have proved useful in the study of approximation properties of operator algebras associated to groups; this was first made explicit by De Cannière and Haagerup [7], and has since been exploited by many other authors (see [6,Chapter 12] for further references). This utility has driven the development of several classes of Herz-Schur multipliers, for example the radial multipliers which first appeared in [12]. Bożejko and Fendler [4] linked these two notions, using unpublished work of Gilbert (see also Jolissaint [14]) to give a 'transference' theorem, showing that every Herz-Schur multiplier of G gives rise to a Schur multiplier acting on B(L 2 (G)). Moreover, one can characterise the Herz-Schur multipliers as those Schur multipliers which are invariant, in the sense that they commute with conjugation by the right regular representation of G. We regard the transference and characterisation results as important goals of the generalised theory presented here.
The importance of the theory of multipliers has led to several authors introducing operator-valued versions of Schur multipliers [3,17] and Herz-Schur multipliers [1,2,8,17]. In particular our work with Todorov and Turowska [17] develops and studies C * -algebra-valued versions of Schur and Herz-Schur multipliers, including both transference and characterisation theorems. The present work arose from an attempt to distill the essential features of some of the proofs given in that paper.
Aspects of the theory of Schur and Herz-Schur multipliers have also been generalised to quantum groups. For example, Junge-Neufang-Ruan [15] give a transference theorem in the setting of locally compact quantum groups, and Brannan [5] uses similar ideas when discussing approximation properties of quantum groups. This paper serves two purposes: firstly we show how to obtain the main results of [17] in the von Neumann algebra setting, and secondly we show how the definitions and important properties of (operator-valued) Schur and Herz-Schur multipliers can be obtained from basic properties of group and module actions on operator algebras. More specifically, after preliminaries in Section 2, in Section 3 we define Schur multipliers as completely bounded maps commuting with a particular module action, and obtain a dilation-type characterisation of these multipliers in Theorem 3.3.
Section 4 begins with the definition of a Herz-Schur multiplier of a group action, so that the classical Herz-Schur multipliers are the Herz-Schur multipliers of the trivial action of the group on C. We then prove the main results of the paper: Proposition 4.5 is a version of transference for our multipliers, identifying the Herz-Schur multipliers of a group action with certain Schur multipliers associated to the dual coaction, and a characterisation of the Schur multipliers which arise in this way in Theorem 4.6.
In Section 5 we focus on abelian groups. When G is abelian the algebra B(L 2 (G)) is the crossed product formed by an action of either G or the dual groupĜ, and in Theorem 5.1 we characterise the maps on B(L 2 (G)) which are Herz-Schur multipliers of both actions simultaneously as the Herz-Schur multipliers of G ×Ĝ.
Finally, we note that preliminary investigations have recovered some of the results of this paper for Kac algebras. This will be explored in a future work.

Preliminaries
Throughout, G denotes a locally compact group, and M a von Neumann algebra acting on the Hilbert space H M . The normal spatial tensor product of von Neumann algebras will be denoted by ⊗. The unit of M will be written 1 M , id denotes the identity representation of a von Neumann algebra, and I H the identity operator on the Hilbert space H.
We follow Nakagami-Takesaki [18] (except that we use the left group von Neumann algebra). An action of G on M is a homomorphism α : G → Aut(M ), continuous in the point-weak* topology. Equivalently, there is a normal * -isomorphism π α : M → M ⊗ L ∞ (G) satisfying Here α G denotes the action of G on L ∞ (G), so that which is the coproduct on L ∞ (G). Given an action α the corresponding isomorphism π α is defined by π α (a)ξ(s) := α −1 s (a)ξ(s), a ∈ M, s ∈ G, ξ ∈ L 2 (G, H M ). The crossed product associated to the action α, denoted M ⋊ α G, is the von Neumann algebra on H M ⊗L 2 (G) generated by π α (M ) and C⊗vN(G). Note that vN(G) is the crossed product formed by the trivial action of G on C.
The definitions for a coaction of G are identical to the above, except that the roles of vN(G) and L ∞ (G) are exchanged: a coaction δ of G on M is a normal * -isomorphism π δ : M → M ⊗ vN(G) satisfying Here δ G denotes the coaction of G on itself, so that which is the coproduct on vN(G). The crossed product associated to the coaction δ, denoted M ⋊ δ G, is the von Neumann algebra on H M ⊗ L 2 (G) generated by π δ (M ) and C ⊗ L ∞ (G). Note that the crossed product formed by the trivial coaction of G on C is L ∞ (G). When G is abelian vN(G) can be identified with L ∞ (Ĝ), so in this case a coaction of G is an action ofĜ.
Given an action α of G on M there is a dual coactionα of G on M ⋊ α G, given by πα π α (a)λ r := π α (a)λ r ⊗ λ r , a ∈ M, r ∈ G.
Similarly, given a coaction δ there is a dual actionδ of G on M ⋊ δ G. The Takai duality theorem for abelian groups can be generalised to this setting, and gives isomorphisms We will denote the first of these isomorphisms by Φ; it is given on the where ρ is the right regular representation of G.
We use the basic theory of operator spaces and completely bounded maps, as found in [9] for example, without comment. The space of completely bounded, weak*-continuous maps on a von Neumann algebra M will be written CB σ (M ); if M is also a bimodule over A then the completely bounded, weak*-continuous A-bimodule maps on M will be denoted by CB A σ (M ).

Schur multipliers
In this section we define generalised Schur multipliers. Throughout X = (X, µ) denotes a standard measure space for which the underlying topology is locally compact. Recall from [17] that given k ∈ L 2 (X × X) ⊙ M one can associate a bounded operator T k by and that such operators are norm-dense in M ⊗ min K(L 2 (X)). In [17, Theorem 2.6] we showed that the Schur multipliers defined there correspond to certain symbols ϕ : In this paper we have defined Schur X-multipliers of M , which act on M ⊗ B(L 2 (X)); in the next result, which is based on [17, Theorem 2.6], we show that our definition of a Schur multiplier S determines how S acts on the operators T k defined above, and use this to associate a symbol to S.
, defines a normal representation of B(L 2 (X)) on H θ . As is well known, this implies that we can write H θ = H ρ ⊗ L 2 (X) for another Hilbert space H ρ and identify θ 0 (T ) with I Hρ ⊗ T . Since θ(a ⊗ T ) = θ(a ⊗ I L 2 (X) )θ(I Hρ ⊗ T ) = θ(a ⊗ I L 2 (X) )θ 0 (T ) and for all a ∈ M and T ∈ B(L 2 (X)) we have that θ(a ⊗ I) commutes with C ⊗ B(L 2 (X)), so we obtain a representation ρ of M on H ρ such that θ(a ⊗ T ) = ρ(a) ⊗ T , acting on H ρ ⊗ L 2 (X). We now have that S = W * 0 (ρ ⊗ id)(·)V 0 (identifying the ranges of V 0 and W 0 with H ρ ⊗ L 2 (X)). Arguing as in [17, Theorem 2.6], using that S commutes with C ⊗ L ∞ (X), we can find projections P, Q so that V := P V 0 and W := QW 0 commute with C ⊗ L ∞ (X), so by Takesaki  (ii) =⇒ (i) It is clear that if S = S ϕ with ϕ as in (ii) then S = W * (ρ ⊗ id)(·)V , so S is completely bounded. Since V, W ∈ L ∞ (X, B(H M , H ρ )) it follows that S is an L ∞ (X)-bimodule map, so S is a Schur X-multiplier of M .
To show that ϕ(x, y) is an A-(bi)module map when S is take a ∈ M, b ∈ A, k ∈ L 2 (X × X). Then . Since S is a module map the last two displays are equal, which implies k(y, x) b · ϕ(x, y)(a) = k(y, x)ϕ(x, y)(b · a), and using the fact that they hold for all k ∈ L 2 (X × X), x, y ∈ X we conclude that ϕ(x, y)(b · a) = b · ϕ(x, y)(a) for all a ∈ M and b ∈ A. A similar calculation shows that ϕ respects the right module action when S does; thus ϕ(x, y) is a completely bounded A-(bi)module map on M . The converse follows similarly.
Remarks 3.4. i. The above theorem reduces to a well-known characterisation of classical Schur multipliers when M = C [11,16]. ii. If α is an action of G on M then the crossed product by the dual coaction α is identified with M ⊗ B(L 2 (G)), and Theorem 3.3 above identifies Schur multipliers on this space with functions G × G → CB σ (M ). In the next section we will define Herz-Schur multipliers of α and identify them with a certain subspace of these Schur multipliers on G × G.

Herz-Schur Multipliers
We are now going to define Herz-Schur multipliers for a group action on a von Neumann algebra. Throughout, G denotes a second-countable locally compact group.
We will refer to a map S satisfying condition (3) by writing "S commutes withα".
If α is the trivial action of G on C thenα = δ G , which induces the product on A(G). If T satisfies (4) then, for x ∈ vN(G), u, v ∈ A(G), the calculation shows that T satisfies (3). A similar calculation shows (3) implies (4). iii. Observe that M ⋊ α G carries an A(G)-module structure: for u ∈ A(G) define so that u * (π α (a)λ r ) = u(r)π α (a)λ r . It is easy to see that Definition 4.1 is equivalent to requiring that S commutes with this module action.
iv. Given a Herz-Schur multiplier of α, say S, equation (3) and (iii) above imply that S(π α (a)λ r ) ∈ π α (M )λ r , so there is some a S,r ∈ M with S(π α (a)λ r ) = π α (a S,r )λ r . Setting F (r)(a) := a S,r we obtain a function F on G such that F (r) is a linear map on M for each r ∈ G. Moreover, since S is completely bounded and weak*-continuous F (r) must be so too. This shows that for every Herz-Schur multiplier of α S there is a symbol F : G → CB σ (M ) such that S π α (a)λ r = π α F (r)(a) λ r , a ∈ M, r ∈ G.
v. Suppose that v : G → C is a classical Herz-Schur multiplier of G. For any action α of G on M we can extend S v to a completely bounded, weak*-continuous map on M ⋊ α G by It is easily checked that S v commutes withα, so that S v is a Herz-Schur multiplier of α. vi. Let G be abelian, and consider the canonical action of G on L ∞ (G) = vN(Ĝ). In [17,Section 6] we showed that every element of B(G) ⊙ B(Ĝ) is a Herz-Schur multiplier of this action; moreover, by symmetry, each such multiplier is also a Herz-Schur multiplier ofĜ on L ∞ (Ĝ) = vN(G), and the multipliers of this form are A(G) module maps on vN(G) (and A(Ĝ) module maps on L ∞ (G)). We will study these multipliers further below.
If M has a (left) module structure over A we can introduce an A-module structure on M ⋊ α G by (5) b · π α (a)λ r := π α (b · a)λ r , b ∈ A, a ∈ M, r ∈ G.
It is easy to check that under the additional assumption this module action is the one induced on M ⋊ α G by the canonical module action of A on M ⊗ B(L 2 (G)). If S is a Herz-Schur multiplier with symbol F : G → CB A σ (M ) then S is also an A-module map, since S b · π α (a)λ r = S π α (b · a)λ r = π α b · F (r)(a) λ r = b · S(π α (a)λ r ) .
Recall that for an action α of G on M the crossed product by the dual coactionα of G on M ⋊ α G can be identified with M ⊗ B(L 2 (G)). Given a map R : (1). Observe that R is completely bounded (resp. completely positive) if R is. In the remainder of this section we explain how Herz-Schur multipliers of α interact with the Schur multipliers of M ⊗ B(L 2 (G)).
First we have a version of the transference theorem (see also [4,15,17]). . Then S is a Schur multiplier ofα with symbol ϕ(s, t)(a) = α t −1 (F (ts −1 )(α t (a))). Moreover, if M has an A-module structure satisfying (6) and F (r) is an A-module map for all r ∈ G then ϕ(s, t) is an A-module map for all s, t ∈ G, so S is also an A-module map.
Proof. Let S be a Herz-Schur multiplier of α. For a ∈ M , r ∈ G and φ ∈ L ∞ (G) we have similarly S commutes with left multiplication by L ∞ (G). That S is a Schur multiplier follows by linearity and weak*-continuity.
To calculate the symbol ϕ associated to the Schur multiplier S fix a ∈ M and r ∈ G. For k ∈ L 2 (G × G, M ) we define k r : G → M by k r (p) := k(p, r −1 p). Let (u i ) i∈I and (k i ) i∈I be as in Lemma 4.3. Similarly one checks that (k r i ) i∈I converge to π α (a) in the weak* topology of L ∞ (G, M ). Since S(T k i ) → π α (F (r)(a))λ r we have The claimed identity follows.
The statement about module maps is an easy calculation using (6).
The following result characterises the Herz-Schur multipliers of α among the Schur multipliers ofα. We identifyα with the action α ⊗ Ad ρ as in (1).
Hence the restriction of R to M ⋊ α G is a Herz-Schur multiplier.
(ii) =⇒ (i) Suppose S is a Herz-Schur multiplier of α with symbol F . Then, for any a ∈ M, r ∈ G, φ ∈ L ∞ (G), using the equivalent of (1) for dual actions [18,Theorem 2.7], so the claim follows by linearity and continuity.
If S is a module map then S is also a module map by Proposition 4.5. On the other hand, if S is a module map then S b · π α (a)λ r = S (b ⊗ I L 2 (G) ) · π α (a)λ r = (b ⊗ I L 2 (G) ) · S π α (a)λ r = b · S π α (a)λ r , so S is also a module map.
Remark 4.7. When M = C and α is trivial the above results recover the known fact [4] that a Schur multiplier S on B(L 2 (G)) restricts to a Herz-Schur multiplier on vN(G) if and only if S commutes with the action Ad ρ (the second dual of the trivial action). In this classical case Lemma 4.4 states that every Schur multiplier of B(L 2 (G)) can be identified with a Herz-Schur multiplier of α G .
In this section we have been careful to keep track of multipliers which respect an additional module structure. The reason is that the Herz-Schur multipliers of a semidirect product H ⋊ G have an obvious identification with Herz-Schur multipliers of vN(H) ⋊G, and become A(H)-module maps under this identification. In the next section we will make use of multipliers respecting this extra module structure.

Abelian Groups
We now assume that G is abelian, with dual groupĜ. By Takai duality B(L 2 (G)) is isomorphic to the crossed product formed by the coaction δ G dual to the trivial action of G on C, or the action α G dual to the trivial action ofĜ on C. For a map S on B(L 2 (G)) we write S α G for the corresponding map on vN(Ĝ) ⋊ α G G and S δ G for the corresponding map on vN(G) ⋊ δ GĜ, so for example In [17,Section 6] we raised the question of how the Herz-Schur multipliers of G acting on vN(Ĝ) are related to B(G) ⊙ B(Ĝ); note that the convolution multipliers considered there are precisely those appearing in (i) below.
Theorem 5.1. Let S be a completely bounded, weak*-continuous map on B(L 2 (G)). The following are equivalent: i. S α G is a Herz-Schur multiplier of α G and is an A(Ĝ)-module map; ii. S δ G is a Herz-Schur multiplier of δ G and is an A(G)-module map. Moreover, the set of all S satisfying the equivalent conditions can be identified with the space B(Ĝ × G).
Proof. Observe that under the identifications of each crossed product with B(L 2 (G)) the module action · of A(G) on vN(G) ⋊ δ GĜ (see (5)) is carried to the action * on vN(Ĝ) ⋊ α G G of Remark 4.2(iv), and the corresponding statement holds for the module actions of A(Ĝ). Assume (i) holds, take u ∈ A(G), r ∈ G and γ ∈Ĝ, and write m γ for the multiplication operator on L 2 (G) associated to γ ∈Ĝ. Then S δ G u · π δ G (λ r )m γ = S u(r)λ r m γ = γ, r S u(r)m γ λ r = γ, r S α G u * π α G (m γ )λ r = γ, r u * S α G π α G (m γ )λ r = γ, r γ, r u · S δ G π δ G (λ r )m γ = u · S δ G π δ G (λ r )m γ , so S defines an A(G)-module map on vN(G) ⋊ δ GĜ. Similarly we calculate that S δ G (v * π δ G (λ r )m γ ) = v * S δ G (π δ G (λ r )m γ ) for each v ∈ A(Ĝ), so S δ G is a Herz-Schur multiplier of the actionα by Remark 4.2(iii). We have now shown that (i) implies (ii); by Pontryagin duality the same proof shows (ii) implies (i). Now let S satisfy (i); if F denotes the symbol of S α G then, for any r ∈ G, γ ∈Ĝ, F (r) is an A(Ĝ)-module map, and therefore a Herz-Schur multiplier ofĜ, so we identify F with a mapĜ × G → C. Consider the Schur multiplier S α G ; it will be convenient to regard S α G as acting on L ∞ (G)⊗B(L 2 (G)). The restriction of S α G to L ∞ (G)⊗vN(G) is a completely bounded, weak*-continuous map; to see that it preserves L ∞ (G) ⊗ vN(G) we calculate, using the modularity of S α G , From this calculation we also see that this restriction is an A(Ĝ×G)-module map on L ∞ (G) ⊗ vN(G), and therefore a Herz-Schur multiplier.
That S commutes with the A(Ĝ)-module action also follows easily.