Exactness and SOAP of Crossed Products via Herz--Schur multipliers

Given a $C^*$-dynamical system $(A,G,\alpha)$, with $G$ a discrete group, Schur $A$-multipliers and Herz--Schur $(A,G,\alpha)$-multipliers are used to implement approximation properties, namely exactness and the strong operator approximation property (SOAP), of $A \rtimes_{\alpha , r} G$. The resulting characterisations of exactness and SOAP of $A \rtimes_{\alpha , r} G$ generalise the corresponding statements for the reduced group $C^*$-algebra.


Introduction
Recently in [11] the notion of classical Schur multipliers, which has been intensively studied in the literature (see e.g. [6,8,13,14]), was generalised to the operator-valued setting: for a C * -algebra A ⊂ B(H) on a Hilbert space H, and a set X, Schur A-multipliers were defined as functions ϕ : X × X → CB(A, B(H)) such that the associated map S ϕ : K(ℓ 2 (X)) ⊗ A → K(ℓ 2 (X)) ⊗ B(H) is completely bounded. Among many applications of classical Schur multipliers in Operator Theory is Ozawa's characterisation of exactness of the reduced C * -algebra of a discrete group G [12]: for a certain family of Schur multipliers (ϕ i : G × G → C) i the associated maps S ϕ i implement exactness of C * r (G). In this paper we will use Schur A-multipliers to generalise this result to the setting of reduced crossed products.
Schur multipliers are related to the notion of Herz-Schur multipliers associated to groups. The latter are functions ψ : G → C on a locally compact group G that give rise to completely bounded maps on the reduced C *algebra. It is known that they constitute the "invariant" part of the Schur multipliers on G × G. Herz-Schur multipliers for the reduced crossed product were defined in [1] for discrete groups and in [11] for general locally compact groups; in the latter they were also related to Schur A-multipliers in a similar manner as for the group case. Herz-Schur multipliers of special type are known to encode such approximation properties as nuclearity, the Haagerup approximation property, the completely bounded approximation property (CBAP) and the (strong) operator approximation property ((S)OAP) of the reduced C * -algebra of a discrete group. Recently it was shown that Herz-Schur (A, G, α)-multipliers do the same for the reduced crossed product A ⋊ α,r G and the first three approximation properties, see [9,10]. Bédos-Conti [1,Section 4] have also used similar techniques to investigate regularity of C * -dynamical systems: they give a condition involving Herz-Schur (A, G, α)-multipliers which implies the full and reduced crossed products are canonically isomorphic. In this note we shall give a characterisation of SOAP for A ⋊ α,r G in terms of Herz-Schur multipliers for crossed products.
The paper is organised as follows. In Section 2 we fix notation and recall the notions of Schur A-multipliers and Herz-Schur multipliers of a C *dynamical system. The main result of Section 3 is a characterisation of exactness of a reduced crossed product in terms of the existence of certain Schur A-multipliers; from this characterisation we are able to deduce the known result that exactness is preserved by amenable actions. We also deduce Ozawa's characterisation of exact discrete groups by combining the main result of Section 3 with a characterisation of nuclearity of reduced crossed products from [10]. Finally, in Section 4 we give a characterisation of when a reduced crossed product has the strong operator approximation property in terms of the existence of certain Herz-Schur multipliers of the dynamical system.

Preliminaries
Throughout G is a discrete group and A is a unital C * -algebra acting on the Hilbert space H. Standard notation for tensor products will be used: the minimal tensor product of C * -algebras will be written A⊗B, and the normal spatial tensor product of von Neumann algebras will be written M ⊗ N ; the Hilbert space tensor product will be written H ⊗ K. We will make heavy use of the theory of operator spaces, complete boundedness and complete positivity; a suggested reference for background on these topics is [5].
2.1. Dynamical Systems. Take a discrete group G and a group homomorphism α : G → Aut(A); the triple (A, G, α) is called a C * -dynamical system and α is called an action of G on A. Using the representation A ⊆ B(H) we define representations of A and G on ℓ 2 (G) ⊗ H ∼ = ℓ 2 (G, H) by π(a)ξ (s) := α −1 s (a)ξ(s), λ r ξ(s) := ξ(r −1 s), a ∈ A, r, s ∈ G, ξ ∈ ℓ 2 (G, H). The pair (π, λ) satisfies the covariance condition λ r π(a)λ * r = π α r (a) , a ∈ A, r ∈ G, and therefore defines a * -representation of The reduced crossed product associated to the system (A, G, α) is the C *algebra defined as the closure of π ⋊ λ(ℓ 1 (G, A)) in the operator norm of B(ℓ 2 (G, H)), and denoted A ⋊ α,r G. We refer to [2, Section 4.1] for the details of this construction, including the fact that the resulting C * -algebra A ⋊ α,r G does not depend on the initial choice of faithful representation A ⊆ B(H).
Note that when A = C and the action α is trivial the reduced crossed product C * -algebra constructed above is the reduced group C * -algebra C * r (G). Identify ℓ 2 (G) ⊗ H with ⊕ s∈G H, so that each element x ∈ B(ℓ 2 (G) ⊗ H) can be identified in the usual way with a matrix (x s,t ) s,t∈G , where each entry comes from B(H). One can check that the main diagonal of x ∈ A ⋊ α,r G, that is {x s,t : s −1 t = e}, is given by {α −1 p (a e ) : p ∈ G} for some a e ∈ A. The map E which takes x ∈ A ⋊ α,r G to a e ∈ A is a conditional expectation [2, Proposition 4.1.9], and is equivariant in the sense that 2.2. Multipliers. In this section we summarise the definitions and results on Schur and Herz-Schur multipliers from [11] needed for this paper (though we use some conventions from [10] which are more convenient). Note that because we work only with discrete spaces and counting measure the separability assumptions of [11] are not required here. Given k ∈ ℓ 2 (X × X, A) define an operator T k : ℓ 2 (X, H) → ℓ 2 (X, H) by It is easily checked that T k is a bounded operator and T k ≤ k 2 . The collection of all such T k forms a dense subset of K(ℓ 2 (X)) ⊗ A. Let ϕ : X × X → CB(A, B(H)) be a bounded function and, for k ∈ ℓ 2 (X × X, A), define ϕ · k(x, y) := ϕ(x, y) k(x, y) , x, y ∈ X. Clearly the map S ϕ : T k → T ϕ·k is bounded with respect to the norm · 2 ; we say ϕ is a Schur A-multiplier if S ϕ extends to a completely bounded map from K(ℓ 2 (X)) ⊗ A to K(ℓ 2 (X)) ⊗ B(H); in this case we write ϕ S := S ϕ cb . Those functions ϕ which are Schur A-multipliers are characterised by Stinespring-type dilation results -see [11,Theorem 2.6] and [10, Theorem 2.6] for a similar characterisation of when S ϕ is completely positive, in the latter case we say that ϕ is a positive Schur-A-multiplier. We recall those results here. Suppose that ϕ is a Schur A-multiplier. Then the map S ϕ : K(ℓ 2 (X)) ⊗ A → K(ℓ 2 (X))⊗B(H) has a canonical extension to a map from B(ℓ 2 (X))⊗A into the von Neumann tensor product B(ℓ 2 (X)) ⊗ B(H) defined as follows. Consider the second dual of the map S ϕ : Writing E for the conditional expectation from B(ℓ 2 (G)) ⊗ B(H) * * to the space B(ℓ 2 (G)) ⊗ B(H) we obtain a completely bounded map we shall also write S ϕ for the restriction of Φ to B(ℓ 2 (X)) ⊗ A and this is the required extension. It is easy to see that if ρ, V , W are as in Theorem 2.1(ii), and considering V and W as operators from ℓ 2 (X) ⊗ H to ℓ 2 (X) ⊗ H ρ we have that the extension satisfies In what follows writing S ϕ we shall often mean the extension to B(ℓ 2 (X))⊗A. Now consider a C * -dynamical system (A, G, α) and let F : We say that F is a Herz-Schur (A, G, α)-multiplier if the map S F : π ⋊ λ(f ) → π ⋊ λ(F · f ) extends to a completely bounded map on A ⋊ α,r G; in this case we write F m := S F cb . The collection of Herz-Schur (A, G, α)multipliers will be denoted S(A, G, α).
The functions F which are Herz-Schur (A, G, α)-multipliers can be characterised in terms of Schur A-multipliers: Then F is a Herz-Schur (A, G, α)-multiplier if and only if N (F ) is a Schur A-multiplier, and in this case F m = N (F ) S [11, Theorem 3.8].

Exactness
Recall that a C * -algebra A is called exact if it has a faithful nuclear representation π : A → B(H), i.e. there exists a net (k i ) i∈I of natural numbers and completely positive contractions ϕ i : A → M k i (C) and ψ i : The notion is independent of the representation π: if A ⊂ B(H) is a concretely represented exact C * -algebra then the identity representation id : A → B(H) is nuclear. It follows from Kirchberg's characterisation of exactness that one can replace the existence of completely contractive positive maps in the definition of exactness by completely bounded maps with uniformly bounded cb norms (see [2, Proposition 3.7.8, Theorem 3.9.1]).
Recall also that a bounded linear map φ : A → B between two C *algebras is called decomposable if there exist two completely positive maps is completely positive. In this case one defines where the infimum is taken over all ψ 1 , ψ 2 as above. We have The following result is inspired by Dong-Ruan [4, Theorem 6.1].
Note that condition (iv) above implies ϕ i (s, t) is a finite rank map on A for all s, t ∈ G, i ∈ I, but the latter is not equivalent to (iv).
Theorem 3.2. Let (A, G, α) be a C * -dynamical system, with G a discrete group. The following are equivalent: Each Φ i is a finite rank map since it is supported on finitely many diagonals, and the image of each diagonal is finite-dimensional; here we think of operators T on ℓ 2 (G) ⊗ H as block operator matrices (a p,q ) p,q∈G where for some C > 0. Since Φ i is finite rank by [14,Theorem 12.7] there exist a natural number n and completely bounded maps ν i : A ⋊ α,r G → M n and µ i : Since for ξ, η ∈ H, so conditions (i) and (iii) imply that Φ i (π(a)λ r ) converges to π(a)λ r in norm. Since sums of elements of this form are dense in A ⋊ α,r G and the S i , therefore the Φ i , are uniformly bounded it follows that Φ i (x) − x → 0 for all x ∈ A ⋊ α,r G. This implies that A ⋊ α,r G is exact by the discussion at the beginning of this section.
Given finite sets F ⊂ A, and R ⊂ G we can construct ϕ F,R,ǫ as above so that for all (s, t) ∈ ∆ R and a ∈ F for any a ∈ A and any finiteR ⊂ G. Finally, the condition on the diagonals is satisfied because θ is finite rank:  Proof. If A ⋊ α,r G is exact then the existence of a conditional expectation A⋊ α,r G → A immediately implies A is exact. Conversely, assume A is exact, with Φ j : A → B(H) an approximating net of u.c.p. maps, and T i : G → A a net implementing the amenable action. We may assume that A is faithfully embedded in H such that the action is implemented by unitary operators, so for each p ∈ G let u p be a unitary on H such that u p au * p = α p (a) (a ∈ A). Define which gives a net of Schur A-multipliers with uniformly bounded multiplier norm as in [10,Corollary 4.6].
Since the rth term of the sum is non-zero only if r ∈ s −1 F i ∩t −1 F i we have that ϕ i,j is supported on the strip ∆ where F i is the support of T i . As Φ j is finite rank it follows that ϕ i,j (s, t) is a finite rank map from A to B(H) for all s, t.
For each t ∈ G and each i, j the subspace of ℓ ∞ (G, B(H)) given by is finite-dimensional since Φ j is finite rank and T i has finite support. Finally Hence ϕ i,j(i) (s, t)(α −1 s (a))−α −1 s (a) → 0 uniformly on ∆ R for all a ∈ A.
Remark 3.5. By [2, Theorem 10.2.9] if G and A are both exact then so is A ⋊ α,r G. The converse is also well known; we observe that it can be quickly deduced from Theorem 3.2: if ϕ i satisfy Definition 3.1 then ϕ i (e, e) : A → B(H) implement exactness of A and, for any unit vector ξ ∈ H, the classical Schur multipliers (s, t) → ϕ i (s, t)(1 A )ξ, ξ implement exactness of G. Therefore the system (A, G, α) is exact in the sense defined above if and only if G is exact and A is exact.
Recall that a discrete group G is called exact if C * r (G) is an exact C *algebra. We now use our results to obtain Ozawa's characterisation of exact discrete groups [12]: G is exact if and only if the uniform Roe algebra C * u (G) = ℓ ∞ (G) ⋊ β,r G is nuclear (β denotes the translation action). Theorem 3.6. Let G be a discrete group. The following are equivalent: is an exact C * -algebra, so by Theorem 3.2 there exists a sequence (ϕ k ) k of (classical) Schur multipliers which are supported on a strip and converge to 1 uniformly on a strip; by Remark 3.3 each ϕ k may be chosen positive. Define We claim that each F k is a Herz-Schur (ℓ ∞ (G), G, β)-multiplier, and that the family (F k ) k satisfies [10, Definition 4.1] and therefore implements nuclearity of ℓ ∞ (G) ⋊ β,r G. For p, s, t ∈ G and f ∈ ℓ ∞ (G) calculate g. Theorem 2.1 with A = C for the existence of such maps V k ) and, for s ∈ G define where M f : ℓ 2 (G) → ℓ 2 (G) is the multiplication operator. Therefore F k is a Herz-Schur (ℓ ∞ (G), G, β)-multiplier by [11,Theorem 3.8]. It also follows that F k is a completely positive Herz-Schur (ℓ ∞ (G), G, β)-multiplier by [10,Theorem 2.8]. The support of F k is supp F k = {r ∈ G : (t, r −1 t) ∈ supp ϕ k for some t ∈ G}, which is finite by Definition 3.1(ii). We have F k (e) cb = sup t∈G |ϕ k (t, t)|.
By (1) and (2) we can choose ϕ k such that sup t∈G |ϕ k (t, t)| ≤ 1. To see that each F k (r) is finite rank observe that in the case A = C Definition 3.1(iv) reduces to the requirement that there are finitely many elements of ℓ ∞ (G) which span ran F k (r). Finally, when A = C Definition 3.1(iii) requires that |ϕ k (s, t) − 1| → 0 uniformly on ∆ R , for each finite R ⊆ G, so for r ∈ R → 0 for all φ ∈ ℓ ∞ (G) and all r ∈ G. This completes the proof by [10,Theorem 4.3].
Although (ii) =⇒ (i) is immediate from the definitions we note that if (F i ) is a sequence of Herz-Schur multipliers of (ℓ ∞ (G), G, β) implementing nuclearity of the uniform Roe algebra as in [10] then, for a unit vector ξ ∈ ℓ 2 (G), ϕ i (s, t) := N (F i )(s, t)(1)ξ, ξ defines a sequence of positive Schur multipliers implementing exactness of C * r (G) by Theorem 3.2. We note that the two conditions in Theorem 3.6 are known to be equivalent to property (A) of G, where G is viewed as a discrete metric space. We refer to the survey [16] for the necessary background on coarse spaces. Definition 3.1 may therefore be regarded as a generalisation of property (A).

The Operator Approximation Property
Recall that a C * -algebra A is said to have the operator approximation property (OAP) if there exists a net of finite rank continuous linear maps (Φ i ) i on A which converge to the identity in the stable point-norm topology, that is Φ i ⊗ id K(ℓ 2 ) → id A⊗K(ℓ 2 ) in point-norm, and A is said to have the strong operator approximation property (SOAP) if there exists a net of finite rank continuous linear maps (Φ i ) i on A which converge to the identity in the strong stable point-norm topology, that is Φ i ⊗ id B(ℓ 2 ) → id A⊗B(ℓ 2 ) in pointnorm. A locally compact group has the approximation property (AP) if there is a net (u i ) i of finitely supported functions on G which converge to the identity in the weak* topology of M cb A(G). Haagerup and Kraus [7] proved that a discrete group G has the AP if and only if C * r (G) has the SOAP if and only if C * r (G) has the OAP, and studied the behaviour of the AP and weak*OAP under the von Neumann algebra crossed product.
Dynamical systems involving the action of a group with the AP have recently been studied by Crann-Neufang [3] and Suzuki [15]. Suzuki shows that if a locally compact group G has the AP then, for any C * -dynamical system (A, G, α), A ⋊ α,r G has the (S)OAP if and only if A has the (S)OAP. In this section we give a Herz-Schur multiplier characterisation of the SOAP for A ⋊ α,r G, generalising the result of Haagerup-Kraus.
The weak* topology of M cb A(G) comes from the introduction of the space Q(G), which has M cb A(G) as its Banach space dual. Recall that Q(G) is defined as the completion of C c (G) in the norm given by the duality with M cb A(G): Following [7], for any C * -algebra C there is a collection of functionals on CB(C): take x ∈ C ⊗ K(ℓ 2 ) and ψ ∈ (C ′′ ⊗ B(ℓ 2 )) * , or x ∈ C ⊗ B(ℓ 2 ) and ψ ∈ (C ⊗ B(ℓ 2 )) * , and define ω x,ψ (T ) := ψ (T ⊗ id)x , T ∈ CB(C).
It is known [7, Proposition 1.5] that Moreover, G has AP if and only if there is a net (u i ) i of finitely supported functions on G such that ω x,ψ (u i ) → ω x,ψ (id) for any such x and ψ.  Let (A, G, α) be a C * -dynamical system, with G a discrete group. We say that (A, G, α) has the approximation property (AP) if there is a net (F i ) i∈I of Herz-Schur (A, G, α)-multipliers satisfying: i. each F i is finitely supported; ii. for each r ∈ G and i ∈ I, F i (r) is a finite rank map on A; iii. ω(F i ) → ω(id) for all ω ∈ Q(A, G, α).
In the arguments below we will use the following standard consequence of the Hahn-Banach theorem several times: if C is a convex set in a Banach space then the weak and norm closures of C coincide. Remarks 4.3. i. We note that we cannot deduce the AP for G from the AP for (A, G, α) in general. In fact, Lafforgue-de la Salle [8] have shown that there exist discrete exact groups without the AP, including SL(3, Z), and if G is such a group then (by [12]) ℓ ∞ (G) ⋊ β,r G is nuclear and so has the SOAP. By Theorem 4.4 below (ℓ ∞ (G), G, β) must then have the AP. ii. Condition (iii) above implies that A has the SOAP. In fact, for x ∈ A ⊗ B(ℓ 2 ) and ψ ∈ (π(A) ⊗ B(ℓ 2 )) * ,we have whereψ is an extension of ψ to a bounded linear functional on (A ⋊ α,r G)⊗B(ℓ 2 ). We have therefore F i (e) converges in the strong stable pointweak topology to the identity map. By the Hahn-Banach theorem, there exists a net in the convex hull of (F i (e)) i∈I that converges to id in the strong stable point-norm topology. Proof. (i) =⇒ (ii) Conditions (i) and (ii) of Definition 4.2 imply that the maps S F i on A ⋊ α,r G associated to the Herz-Schur (A, G, α)-multipliers F i are finite rank. By definition of Q(A, G, α) condition (iii) of Definition 4.2 implies that (S F i ⊗ id B(ℓ 2 ) )x → x weakly, for each x ∈ (A ⋊ α,r G) ⊗ B(ℓ 2 ). Hence, by the Hahn-Banach theorem, there is a convex combination of the S F i which converges to the identity in the strong stable point-norm topology.
(ii) =⇒ (i) Let (Φ i ) i be a net of completely bounded maps implementing the SOAP of A ⋊ α,r G. First, let us show that we may assume ran Φ i is contained in B := span{π(a)λ r : a ∈ A, r ∈ G}. Write Φ i = k i j=1 φ i j ⊗ T i j , where φ i j ∈ (A ⋊ α,r G) * and T i j ∈ A ⋊ α,r G. If the T i j are not elements of B then, for each n ∈ N find T i j,n ∈ B such that T i j −T i j,n < (nk i max j φ i j ) −1 . Then Φ i,n := k i j=1 φ i j ⊗T i j,n is a net which implements the SOAP of A⋊ α,r G and has ran Φ i,n ⊂ B.
Define F i : G → CB(A); F i (r)(a) := E(Φ i (π(a)λ r )λ r −1 ), where E is the canonical conditional expectation A ⋊ α,r G → A. Then each F i is a Herz-Schur (A, G, α)-multiplier by the same calculation as [10,Proposition 3.4], and the assumption on the range of Φ i implies that F i is finitely supported and finite rank.
To prove the required convergence let ∆ be the coaction of G on A⋊ α,r G, so ∆(π(a)λ r ) = λ r ⊗ π(a)λ r , and define V (δ r ⊗ ξ) := δ r ⊗ δ r ⊗ ξ (r ∈ G, ξ ∈ H); then V * ∆(x)V = x (x ∈ A ⋊ α,r G) and S F i = V * (id ⊗ Φ i )∆(·)V . For x ∈ (A ⋊ α,r G) ⊗ B(ℓ 2 ) and ψ ∈ ((A ⋊ α,r G) ⊗ B(ℓ 2 )) * calculate Remark 4.5. Crann-Neufang [3] also define a version of the AP for a C *dynamical system (A, G, α), based on a slice map property. They show [3,Corollary 4.11] that if G has the AP then any C * -dynamical system (A, G, α) has the AP in their sense. On the other hand, if (A, G, α) has the AP in the sense of Definition 4.2 then A must have the SOAP. Therefore these two notions of AP for a C * -dynamical system are different.