A weak expectation property for operator modules, injectivity and amenable actions
aa r X i v : . [ m a t h . OA ] S e p A WEAK EXPECTATION PROPERTY FOR OPERATOR MODULES,INJECTIVITY AND AMENABLE ACTIONS
ALEX BEARDEN AND JASON CRANN
Abstract.
We introduce an equivariant version of the weak expectation property(WEP) at the level of operator modules over completely contractive Banach algebras A .We prove a number of general results—for example, a characterization of the A -WEPin terms of an appropriate A -injective envelope, and also a characterization of those A for which A -WEP implies WEP. In the case of A = L ( G ) , we recover the G -WEPfor G - C ∗ -algebras in recent work of Buss–Echterhoff–Willett [8]. When A = A ( G ) , weobtain a dual notion for operator modules over the Fourier algebra. These dual notionsare related in the setting of dynamical systems, where we show that a W ∗ -dynamicalsystem ( M, G, α ) with M injective is amenable if and only if M is L ( G ) -injective ifand only if the crossed product G ¯ ⋉ M is A ( G ) -injective. Analogously, we show thata C ∗ -dynamical system ( A, G, α ) with A nuclear and G exact is amenable if and onlyif A has the L ( G ) -WEP if and only if the reduced crossed product G ⋉ A has the A ( G ) -WEP. Introduction
A connection of central importance in abstract harmonic analysis is that betweenproperties of locally compact groups and properties of their associated operator algebras.For instance, it is well-known that amenability of a locally compact group G impliesnuclearity of its reduced C ∗ -algebra C ∗ λ ( G ) and injectivity of its von Neumann algebra V N ( G ) [18], and that the converse is true in the setting of inner amenable groups [3, 32].While the equivalence does not hold for connected groups [10], it was recently shownthat amenability of a locally compact group G is equivalent to injectivity of V N ( G ) asan operator module over the Fourier algebra A ( G ) [12]. Thus, to fully capture propertiesof G one should not only consider the operator algebra structure of V N ( G ) , but also itsoperator A ( G ) -module structure.This perspective suggests the following question at the C ∗ -level: can one captureamenability of a locally compact group G through the A ( G ) -module structure of itsreduced group C ∗ -algebra C ∗ λ ( G ) ? Motivated by this question, we introduce an equi-variant weak expectation property (WEP) at the level of operator modules over a com-pletely contractive Banach algebra. We answer the above question by showing that G is amenable if and only if C ∗ λ ( G ) has the A ( G ) -WEP, thereby giving a module versionof a well-known result of Lance [29].From a dynamical systems perspective, V N ( G ) = C ¯ ⋉ G and C ∗ λ ( G ) = C ⋉ G , where G acts trivially on C . If G acts non-trivially on a von Neumann algebra or C ∗ -algebra, thenthe dual co-action induces a canonical operator A ( G ) -module structure on the respective Mathematics Subject Classification.
Key words and phrases.
Weak expectation property; operator modules; injectivity; amenable actions. crossed products, and it is natural to investigate whether amenability of the action canbe recovered from this additional structure. In this paper we establish this for a largeclass of W ∗ - and C ∗ -dynamical systems over locally compact groups. Specifically, weprove that a W ∗ -dynamical system ( M, G, α ) with M injective is amenable if and onlyif the crossed product G ¯ ⋉ M is A ( G ) -injective (Theorem 5.2), and that a C ∗ -dynamicalsystem ( A, G, α ) with A nuclear and G exact is amenable (in the sense of [9]) if and onlyif G ⋉ A has the A ( G ) -WEP (see Theorem 5.5).A notion of G -WEP for C ∗ -dynamical systems was defined in recent work of Buss,Echterhoff and Willett [8]. We show that this notion coincides with our L ( G ) -WEP, andwe perform a detailed study of the resulting Γ -WEP for C ∗ -dynamical systems ( A, Γ , α ) over a discrete group Γ . Among other things, we show that A has the Γ -WEP if andonly if A ∗∗ contains a Γ -equivariant copy of the Γ -injective envelope I Γ ( A ) , and that theequivariant analogue of the QWEP conjecture fails: there is a Γ - C ∗ -algebra B that isnot a Γ -quotient of any A which has the Γ -WEP. We also introduce a notion of WEPfor actions Γ y A , which, together with Lance’s WEP of A entails the Γ -WEP of A .As applications of our techniques, we show that the continuous G -WEP as defined in[9] coincides with the G -WEP (Proposition 4.5), we generalize a homological character-ization of amenable commutative W ∗ -dynamical systems [36] to the non-commutativecontext (Proposition 5.1), and we generalize a hereditary property of amenability [1]from discrete to arbitrary W ∗ -dynamical systems (Corollary 5.3).The structure of the paper is as follows. After a preliminary section on operatormodules and dynamical systems, we begin in section 3 with the definition of the A -WEP for operator modules over completely contractive Banach algebras. We exploresome basic examples and show, among other things, that a C ∗ -algebra A has the A -WEP if and only if it has Lance’s WEP. In section 4 we study the Γ -WEP for discrete C ∗ -dynamical systems, and in section 5 we pursue examples of the A ( G ) -WEP in thesetting of W ∗ - and C ∗ -dynamical systems.2. Preliminaries
Operator Modules.
Let Op denote the category of operator spaces and com-pletely contractive maps. Given a completely contractive Banach algebra A , an object X ∈ Op is a right operator A -module if it is a right Banach A -module for which themodule action extends to a complete contraction X b ⊗ A → X , where b ⊗ denotes theoperator space projective tensor product. We let modA denote the category of rightoperator A -modules with completely contractive module homomorphisms. Left modulesare defined analogously, and the resulting category is denoted Amod . We let A denotethe unitization of A . Note that any X ∈ modA is naturally a completely contractive A -module via x · ( a, λ ) := x · a + λx, x ∈ X, a ∈ A, λ ∈ C . Given X ∈ Op , the space CB ( A , X ) is a completely contractive right A -module via ϕ · a ( b ) = ϕ ( ab ) , a ∈ A, b ∈ A , ϕ ∈ CB ( A , X ) . If, in addition, X ∈ modA , there is a canonical completely isometric A -morphism j X : X ֒ → CB ( A , X ) given by j X ( x )( a ) = x · a, x ∈ X, a ∈ A . We also write j X for the same map taking values in CB ( A, X ) .We say that X ∈ modA is faithful if for every non-zero x ∈ X , there is a ∈ A suchthat x · a = 0 , and we say that X is essential if h X · A i = X , where h·i denotes theclosed linear span. Given Z ∈ Amod , the the module tensor product X b ⊗ A Z is definedby X b ⊗ A Z = X b ⊗ Z/N, N = h x · a ⊗ z − x ⊗ a · z | x ∈ X, a ∈ A, z ∈ Z i . An operator module X ∈ modA is said to be relatively injective if there exists amorphism ϕ : CB ( A , X ) → X such that ϕ ◦ j X = id X . When X is faithful, this isequivalent to the existence a morphism ϕ : CB ( A, X ) → X such that ϕ ◦ j X = id X bythe operator analogue of [13, Proposition 1.7].We say that X is A -injective if for every Y, Z ∈ modA , every completely isometricmorphism κ : Y ֒ → Z , and every morphism ϕ : Y → X , there exists a morphism e ϕ : Z → X such that e ϕ ◦ κ = ϕ , that is, the following diagram commutes: ZY X e ϕκ ϕ Theorem 2.1.
Every X ∈ modA admits an injective envelope in modA , denoted I A ( X ) , called the A -injective envelope of X . The following properties hold:(1) If Y ∈ modA and ψ : I A ( X ) → Y is an A -module morphism, then ψ is com-pletely isometric if its restriction to X is completely isometric.(2) If ψ : I A ( X ) → I A ( X ) is an A -module morphism whose restriction to X is theidentity map, then ψ is the identity map.Proof. Since X is an operator space we may view it inside B ( H ) for some Hilbert space H . Then the canonical embedding j X : X ֒ → CB ( A , X ) extends to a completelyisometric A -morphism X ֒ → CB ( A , B ( H )) . Since B ( H ) is an injective operator space, CB ( A , B ( H )) is injective in modA (see, e.g., the proof of [12, Proposition 2.3], which ismodelled off of [20, Lemma 1]). It follows that the category modA admits sufficientlymany injectives, and properties (1) and (2) follow similarly from the work of Hamana[20, 21]. (cid:3) A key point in the previous proof is the fact that CB ( A , X ) is A -injective if X is aninjective operator space. The following refinement of this fact in the special case that A has a contractive approximate identity and X is a dual operator space will be usedseveral times later. Lemma 2.2. If A has a contractive approximate identity and M is an injective dualoperator space, then CB ( A, M ) is A -injective.Proof. Let κ : X ֒ → Y be an inclusion in modA , and let ϕ : X → CB ( A, M ) be an A -morphism. Let ( e λ ) be a cai for A . Define ϕ λ : X → M , ϕ λ ( x ) = ϕ ( x )( e λ ) . Then ( ϕ λ ) ALEX BEARDEN AND JASON CRANN is a net of complete contractions in CB ( X, M ) . Let ϕ be a limit point of ( ϕ λ ) in theweak*-topology in CB ( X, M ) . Since M is injective, there exists a complete contraction ˜ ϕ : Y → M such that ˜ ϕ ◦ κ = ϕ . Define ˜ ϕ : Y → CB ( A, M ) , ˜ ϕ ( y )( a ) = ˜ ϕ ( ya ) . It isstraightforward to check that ˜ ϕ is an A -morphism such that ˜ ϕ ◦ κ = ϕ . (cid:3) Let A be a C ∗ -algebra. A non-degenerate representation π : A → B ( H ) has the weakexpectation property (WEP) if there exists a unital completely positive map ϕ : B ( H ) → π ( A ) ′′ such that ϕ ( π ( a )) = π ( a ) , a ∈ A . The C ∗ -algebra A has Lance’s WEP if everyfaithful non-degenerate representation has the WEP [29, Definition 2.8].An object X ∈ Op has the operator space weak expectation property if for anyinclusion X ֒ → Y in Op there exists a completely contractive map ϕ : Y → X ∗∗ suchthat ϕ ( x ) = x for all x ∈ X . It was shown implicitly in [28, pg. 459] that a C ∗ -algebra A has the operator space WEP if and only if it has Lance’s WEP. The underlying reasonis that weak expectations are automatically completely positive: Lemma 2.3.
Let A ⊆ B be an inclusion of C ∗ -algebras. Any contraction E : B → A ∗∗ satisfying E ( a ) = a for all a ∈ A is a completely positive A -bimodule contraction. If, inaddition, M ( A ) ⊆ B , then E is an M ( A ) -bimodule map.Proof. We follow [28, pg. 459]. The map ( E ∗ | A ∗ ) ∗ : B ∗∗ → A ∗∗ satisfies ( E ∗ | A ∗ ) ∗ ◦ i ∗∗ = id A ∗∗ , where i is the inclusion A ⊆ B . Then i ∗∗ ◦ ( E ∗ | A ∗ ) ∗ : B ∗∗ → B ∗∗ is a projection ofnorm one onto i ∗∗ ( A ∗∗ ) , and therefore a completely positive i ∗∗ ( A ∗∗ ) -bimodule map byTomiyama’s theorem [43]. Since i is an injective ∗ -homomorphism one sees that ( E ∗ | A ∗ ) ∗ is a i ∗∗ ( A ∗∗ ) − A ∗∗ -bimodule map. It follows that E = ( E ∗ | A ∗ ) ∗ | B is an A -bimodule map,and a M ( A ) -bimodule map in the case when M ( A ) ⊆ B . (cid:3) Locally compact groups and dynamical systems.
Let G be a locally compactgroup. The set of coefficient functions of the left regular representation, A ( G ) = { u : G → C : u ( s ) = h λ ( s ) ξ, η i , ξ, η ∈ L ( G ) , s ∈ G } , is called the Fourier algebra of G . It was shown by Eymard that, endowed with thenorm k u k A ( G ) = inf {k ξ k L ( G ) k η k L ( G ) : u ( · ) = h λ ( · ) ξ, η i} ,A ( G ) is a Banach algebra under pointwise multiplication [15, Proposition 3.4]. Further-more, it is the predual of the group von Neumann algebra V N ( G ) , where the duality isgiven by h u, λ ( s ) i = u ( s ) , u ∈ A ( G ) , s ∈ G. Eymard also showed that the space of functions ϕ : G → C for which there exists astrongly continuous unitary representation π : G → B ( H π ) and ξ, η ∈ H π such that ϕ ( s ) = h π ( s ) ξ, η i , s ∈ G , is a unital Banach algebra (with pointwise multiplication)under the norm k ϕ k B ( G ) = inf {k ξ k H π k η k H π : ϕ ( · ) = h π ( · ) ξ, η i} , called the Fourier-Stieltjes algebra of G [15, Proposition 2.16], denoted by B ( G ) . It isknown that B ( G ) is isometrically isomorphic to the dual of the full group C ∗ -algebra C ∗ ( G ) . Under this identification, states on C ∗ ( G ) correspond to positive definite func-tions of norm one on G . A W ∗ -dynamical system ( M, G, α ) consists of a von Neumann algebra M endowedwith an action α : G → Aut( M ) of a locally compact group G such that for each x ∈ M , the map G ∋ s α s ( x ) is weak* continuous. Every action induces a normal G -equivariant injective ∗ -homomorphism α : M → L ∞ ( G ) ⊗ M via h α ( x ) , F i = Z G h α s − ( x ) , F ( s ) i ds, F ∈ L ( G, M ∗ ) = ( L ∞ ( G ) ⊗ M ) ∗ and a corresponding right L ( G ) -module structure on M [42, 18.6]. Note that thepredual M ∗ becomes a left operator L ( G ) -module via α ∗ : L ( G ) b ⊗ M ∗ → M ∗ .The crossed product of M by G , denoted G ¯ ⋉ M , is the von Neumann subalgebra of B ( L ( G )) ⊗ M generated by α ( M ) and V N ( G ) ⊗ .A C ∗ -dynamical system ( A, G, α ) consists of a C ∗ -algebra endowed with a continuousgroup action α : G → Aut( A ) such that for each a ∈ A , the map G ∋ s α s ( a ) ∈ A isnorm continuous.A covariant representation ( π, σ ) of ( A, G, α ) consists of a representation π : A →B ( H ) and a unitary representation σ : G → B ( H ) such that π ( α s ( a )) = σ ( s ) π ( a ) σ ( s ) − for all s ∈ G . Given a covariant representation ( π, σ ) , we let ( π × σ )( f ) = Z G π ( f ( t )) σ ( t ) dt, f ∈ C c ( G, A ) . The full crossed product G ⋉ f A is the completion of C c ( G, A ) in the norm k f k = sup ( π,σ ) k ( π × σ )( f ) k , where sup is taken over all covariant representations ( π, σ ) of ( A, G, α ) .Let A ⊆ B ( H ) be a faithful non-degenerate representation of A . Then ( α, λ ⊗ is acovariant representation on L ( G, H ) , where α ( a ) ξ ( t ) = α t − ( a ) ξ ( t ) , ( λ ⊗ s ) ξ ( t ) = ξ ( s − t ) , ξ ∈ L ( G, H ) . The reduced crossed product G ⋉ A is defined to be the norm closure of ( α × ( λ ⊗ C c ( G, A )) . This definition is independent of the faithful non-degenerate representa-tion A ⊆ B ( H ) . We often abbreviate α × ( λ ⊗ as α × λ .Analogous to the group setting, dual spaces of crossed products can be identified withcertain A ∗ -valued functions on G . We review aspects of this theory below and refer thereader to [39, Chapters 7.6, 7.7] for details.For each C ∗ -dynamical system ( A, G, α ) there is a universal covariant representation ( π, σ ) such that G ⋉ f A ⊆ C ∗ ( π ( A ) ∪ σ ( G )) ⊆ M ( G ⋉ f A ) . Each functional ϕ ∈ ( G ⋉ f A ) ∗ then defines a function Φ : G → A ∗ by h Φ( s ) , a i = ϕ ( π ( a ) σ ( s )) , a ∈ A, s ∈ G. Let B ( G ⋉ f A ) denote the resulting space of A ∗ -valued functions on G . An element Φ ∈ B ( G ⋉ f A ) is positive definite if it arises from a positive linear functional ϕ as above.We let A ( G ⋉ f A ) denote the subspace of B ( G ⋉ f A ) whose associated functionals ϕ are ALEX BEARDEN AND JASON CRANN of the form ϕ ( x ) = ∞ X n =1 h ξ n , α × λ ( x ) η n i , x ∈ G ⋉ f A, for sequences ( ξ n ) and ( η n ) in L ( G, H ) with P ∞ n =1 k ξ n k < ∞ and P ∞ n =1 k η n k < ∞ .Then A ( G ⋉ f A ) is a norm closed subspace of ( G ⋉ f A ) ∗ which can be identified with (( G ⋉ A ) ′′ ) ∗ .A function h : G → A is of positive type (with respect to α ) if for every n ∈ N , and s , ..., s n ∈ G , the matrix [ α s i ( h ( s − i s j )] ∈ M n ( A ) + . Every C ∗ -dynamical system ( A, G, α ) admits a unique universal W ∗ -dynamical system ( A ′′ α , G, α ) [23]. We review this construction taking an L ( G ) -module perspective. In[9], they study ( A ′′ α , G, α ) from a different, equivalent perspective.First, A becomes a right operator L ( G ) -module in the canonical fashion by slicingthe corresponding non-degenerate representation α : A ∋ a ( s α s − ( a )) ∈ C b ( G, A ) ⊆ L ∞ ( G ) ⊗ A ∗∗ . Explicitly, this action is given by(1) a ∗ f = Z G f ( s ) α s − ( a ) ds for a ∈ A, f ∈ L ( G ) , where the integral is norm convergent. By duality we obtain aleft operator L ( G ) -module structure on A ∗ via α ∗ | L ( G ) b ⊗ A ∗ : L ( G ) b ⊗ A ∗ → A ∗ . Then G acts in a norm-continuous fashion on the essential submodule A ∗ c := h L ( G ) ∗ A ∗ i . The same argument in [39, Lemma 7.5.1] shows that A ∗ c coincides with the norm-continuous part of A ∗ , hence the notation. This fact was also noted by Hamana in[22, Proposition 3.4(i)]. We therefore obtain a point-weak* continuous action of G onthe dual space ( A ∗ c ) ∗ by surjective isometries. Clearly(2) ( A ∗ c ) ∗ ∼ = A ∗∗ / ( A ∗ c ) ⊥ completely isometrically and weak*-weak* homeomorphically as right L ( G ) -modules,where the canonical L ( G ) -module structure on A ∗∗ is obtained by slicing the normalcover of α , which is the normal ∗ -homomorphism e α = ( α ∗ | L ( G ) b ⊗ A ∗ ) ∗ : A ∗∗ → L ∞ ( G ) ⊗ A ∗∗ . Note that e α | M ( A )) is the unique strict extension of α , and is therefore injective [30,Proposition 2.1]. However, on A ∗∗ , e α can have a large kernel. On the one hand, itskernel is of the form (1 − z ) A ∗∗ for some projection z ∈ Z ( A ∗∗ ) . On the other hand,by definition of the L ( G ) -action on A ∗∗ , Ker( e α ) = ( A ∗ c ) ⊥ . It follows that ( A ∗ c ) ∗ iscompletely isometrically weak*-weak* order isomorphic to zA ∗∗ , where we equip ( A ∗ c ) ∗ with the quotient operator system structure from A ∗∗ . We can therefore transport thepoint-weak* continuous G action on ( A ∗ c ) ∗ to A ′′ α := zA ∗∗ , yielding a W ∗ -dynamicalsystem ( A ′′ α , G, α ) , where α : G → Aut( A ′′ α ) is given by α t ( zx ) = z (( α t ) ∗∗ ( x )) , x ∈ A ∗∗ , t ∈ G. We emphasize that with this structure A ′′ α is not necessarily an L ( G ) -submodule of A ∗∗ ,rather Ad( z ) : A ∗∗ → A ′′ α is an L ( G ) -quotient map.3. The weak expectation property for operator modules
Throughout this section, unless otherwise stated, A denotes a fixed completely con-tractive Banach algebra. For a Banach (or operator) space X , we let i X : X ֒ → X ∗∗ denote the canonical inclusion. Definition 3.1.
An object X ∈ modA has the weak expectation property ( A -WEP) iffor any completely isometric morphism κ : X ֒ → Y there exists a morphism ψ : Y → X ∗∗ such that ψ ◦ κ = i X . Examples 3.2. (1) Clearly, we recover the operator space WEP when A = C .(2) Any A -injective module has the A -WEP. If, in addition, X ∈ modA is a dualmodule in the sense that there exists a Y ∈ Amod with X = Y ∗ and h xa, y i = h x, ay i for all x ∈ X , y ∈ Y , and a ∈ A (equivalently, X has an operator spacepredual with respect to which X ∋ x x · a ∈ X is weak*-weak* continuous foreach a ∈ A ), then X has the A -WEP if and only if X is A -injective. This followsquickly from the fact that the adjoint of the inclusion Y ֒ → X ∗ is an A -moduleprojection X ∗∗ → X . Remark 3.3.
Notions of WEP for a C ∗ -algebra A relative to another C ∗ -algebra B appeared in the unpublished manuscript [33]. They are defined in terms of relative weakinjectivity of inclusions of the type A ⊆ L ( E B ) , where E B is a Hilbert B -module and L ( E B ) is the C ∗ -algebra of adjointable operators on E B . Although similar in nature,these notions differ from ours as there need not be a canonical B -module structure on A , in general. Even when A = B = C ∗ λ ( F ) , by [33, Example 5.4], A has the AW EP inthe sense of [33, Definition 3.1], but it follows from Proposition 3.10 and [29] that A doesnot have the A -WEP in the sense of Definition 3.1. Besides, in this paper we are mainlyinterested in the case where the underlying Banach algebra A is not a C ∗ -algebra.We record a number of equivalent conditions for later use. Theorem 3.4.
The following are equivalent for X ∈ modA :(1) X has the A -WEP.(2) For any inclusion κ : X ֒ → Y there exists a morphism ϕ : X ∗ → Y ∗ such that κ ∗ ◦ ϕ = id X ∗ .(3) For any inclusion κ : X ֒ → Z ∗ into a dual A -module Z ∗ , there exists a morphism ψ : Z ∗ → X ∗∗ such that ψ ◦ κ = i X .(4) For any inclusion κ : X ֒ → Z ∗ into a dual A -module Z ∗ , there exists a morphism ˜ ψ : Z ∗ → κ ( X ) weak* such that ˜ ψ ◦ κ = id X .(5) There is an A -module embedding i : I A ( X ) ֒ → X ∗∗ such that i | X = i X .(6) The inclusion i X : X ֒ → X ∗∗ factors through an injective module in modA .(7) For every inclusion X ֒ → Y in modA and Z ∈ Amod the canonical map X b ⊗ A Z ֒ → Y b ⊗ A Z is a complete isometry. ALEX BEARDEN AND JASON CRANN
Proof. (1) = ⇒ (2) : Suppose there exists a morphism ψ : Y → X ∗∗ such that ψ ◦ κ = i X .Then ϕ := ψ ∗ | X ∗ : X ∗ → Y ∗ is a morphism that satisfies h κ ∗ ◦ ϕ ( x ∗ ) , x ∗∗ i = h x ∗ , ψ ∗∗ ◦ κ ∗∗ ( x ∗∗ ) i = h x ∗ , ( ψ ◦ κ ) ∗∗ ( x ∗∗ ) i = h x ∗ , x ∗∗ i , which shows κ ∗ ◦ ϕ = id X ∗ . (2) = ⇒ (1) : Let κ : X ֒ → Y be an inclusion. Suppose there exists a morphism ϕ : X ∗ → Y ∗ such that κ ∗ ◦ ϕ = id X ∗ . Then ψ := ϕ ∗ | Y : Y → X ∗∗ is a morphism andfor each x ∈ X and x ∗ ∈ X ∗ we have h x ∗ , ψ ( κ ( x )) i = h κ ∗ ◦ ϕ ( x ∗ ) , x i = h x ∗ , x i , whichshows ψ ◦ κ = i X . (1) = ⇒ (3) : Obvious. (3) = ⇒ (1) : Follows from the fact that Y embeds into Y ∗∗ as a submodule. (1) = ⇒ (4) : First, since i ∗ Z ◦ κ ∗∗ | X = κ it follows that i ∗ Z ◦ κ ∗∗ ( X ∗∗ ) ⊆ κ ( X ) weak* . Nowlet ψ : Z ∗ → X ∗∗ be a morphism with ψ ◦ κ = i X . Then the composition ˜ ψ := i ∗ Z ◦ κ ∗∗ ◦ ψ is the desired morphism. (4) = ⇒ (1) : Let κ : X ֒ → Y be an inclusion. Let ˜ κ := κ ∗∗ ◦ i X : X → Y ∗∗ . Notethat ˜ κ ( X ) weak* = κ ∗∗ ( X ∗∗ ) . Now by (4) there is a morphism ˜ ψ : Y ∗∗ → ˜ κ ( X ) weak* suchthat ˜ ψ ◦ ˜ κ = id X . Thus the restriction of ( κ ∗∗ ) − ◦ ˜ ψ : Y ∗∗ → X ∗∗ to Y is the desiredmorphism. (1) = ⇒ (5) : Follows from the definition. (5) = ⇒ (1) : Let κ : X ֒ → Y be an inclusion. By injectivity, the map κ − : κ ( X ) → X extends to a morphism ϕ : Y → I A ( X ) . Then the composition i ◦ ϕ : Y → X ∗∗ is amorphism that satisfies ψ ◦ κ = i X . (5) = ⇒ (6) : Obvious. (6) = ⇒ (1) : If there exists an injective module I and a morphisms ϕ : X → I and ψ : I → X ∗∗ satisfying ψ ◦ ϕ = i X , then for any inclusion κ : X → Y there is a morphism e ϕ : Y → I with e ϕ ◦ κ = ϕ . Then ψ ◦ e ϕ : Y → X ∗∗ is the desired weak expectation. (2) ⇐⇒ (7) : Let κ : X ֒ → Y be an inclusion. Then ( κ ⊗ id Z ) : X b ⊗ A Z → Y b ⊗ A Z isa complete isometry for every Z ∈ Amod if and only if κ is a weak retract, meaningthere exists a morphism ϕ : X ∗ → Y ∗ satisfying κ ∗ ◦ ϕ = id X ∗ . This follows verbatimfrom (the operator space analogue) of [14, 1.9]. (cid:3) An inclusion
X ֒ → Y in modA for which X b ⊗ A Z ֒ → Y b ⊗ A Z is a complete isometryfor every Z ∈ Amod is said to be flat . Remark 3.5.
The equivalence of (1) and (7) in Theorem 3.4 is the operator moduleanalogue of Lance’s characterization of the WEP for a C ∗ -algebra A by means of theso-called extension property for ⊗ max [29, Theorem 3.3], meaning that for any inclusion A ⊆ B of C ∗ -algebras, and any C ∗ -algebra C , we have the inclusion A ⊗ max C ⊆ B ⊗ max C .3.1. A -WEP vs. WEP. It is natural to wonder whether the A -WEP implies theoperator space WEP. This is false, in general, as the following example shows. Example 3.6.
Let Γ be a non-amenable discrete group, A = B (Γ) , the Fourier-Stieltjesalgebra of Γ , and X = W ∗ (Γ) = C ∗ (Γ) ∗∗ the universal von Neumann algebra of Γ . Since B (Γ) is unital it follows that is 1-projective over itself in the sense of [12, Section 2],so that (by the module version of [5, Theorem 3.5]) W ∗ (Γ) = B (Γ) ∗ is B (Γ) -injective,thus has the B (Γ) -WEP. However, if W ∗ (Γ) had the operator space WEP, then, as itis a dual space, it would necessarily be injective by Example 3.2 (2). This would entailnuclearity of C ∗ (Γ) and hence the amenability of Γ , hence we have a contradiction. The following theorem characterizes the A for which the implication A -WEP ⇒ WEPalways holds. If A is unital, then A ∗ has the A -WEP. So the equivalence of (1) and (5)of this theorem says that, in the unital case, as long as A ∗ is not a counterexample to“ A -WEP implies WEP,” then there are no counterexamples. Theorem 3.7.
For a completely contractive Banach algebra A , the following are equiv-alent: (1) Every operator module over A that is A -injective is injective. (2) Every operator module over A that has the A -WEP has the WEP.If A has a cai, then the following are also equivalent to the above: (3) For any Hilbert space H , CB ( A, B ( H )) is injective. (4) The operator space dual A ∗ is injective. (5) The operator space dual A ∗ has the WEP.Proof. (1) = ⇒ (2) : Suppose X has the A -WEP, so that I A ( X ) embeds in X ∗∗ via anembedding that restricts to the identity on X . By (1), there is also an embedding I ( X ) ⊆ I A ( X ) that restricts to the identity on X . By composing these embeddings, wesee that X has the WEP. (2) = ⇒ (1) : If X is A -injective, then X has the A -WEP. By (2), X has the WEP,and so I ( X ) ⊆ X ∗∗ via an embedding restricting to the identity on X . Since X is A -injective, there is an A -morphism Φ : X ∗∗ → X extending the identity. Restricting Φ to a copy of I ( X ) yields a complete contraction I ( X ) → X restricting to the identityon X . It follows that X is injective.Now assume that A is unital. (1) = ⇒ (3) : Follows from 2.2. (3) = ⇒ (1) : If X is A -injective, then there exists an A -morphism Φ : CB ( A, B ( H )) → X that restricts to the identity on the canonical copy of X in CB ( A, B ( H )) via j X . It isthen clear that if (3) holds, X is injective. (3) = ⇒ (4) : Obvious (take H = C ). (4) = ⇒ (3) : If H is a Hilbert space with dimension I , then we may canonicallyidentify CB ( A, B ( H )) with the space M I ( A ∗ ) of matrices [ ϕ ij ] indexed by a set withcardinality I with entries in A ∗ such that the finitely supported submatrices of [ ϕ ij ] are uniformly bounded in norm. If A ∗ is injective, then there is a Hilbert space K ,completely isometric representation A ∗ ⊆ B ( K ) , and completely contractive projection Φ : B ( K ) → A ∗ that restricts to the identity on A ∗ . The canonical amplification Φ I : B ( H ⊗ K ) = M I ( B ( K )) → M I ( A ∗ ) = CB ( A, B ( H )) is then a completely contractiveprojection, which implies that CB ( A, B ( H )) is injective since B ( H ⊗ K ) is. (4) ⇐⇒ (5) : This is a simple consequence of the fact that the adjoint of the inclusion A ֒ → A ∗∗ is a conditional expectation onto A ∗ . (cid:3) Say that a left A -module X is an h -module over A if the module action extends toa complete contraction A ⊗ h X → X , where ⊗ h is the Haagerup tensor product. Sincethere is a canonical complete contraction A b ⊗ X → A ⊗ h X , it follows that every h -moduleover A is an operator A -module.The following result, and hence also the corollary below it, actually holds in generalfor any nondegenerate h -module over an approximately unital Banach algebra with an operator space structure [7, comment above Theorem 2.2]. For convenience, we providean elementary proof in the case that the algebra is a C ∗ -algebra. Proposition 3.8. If X is a non-degenerate h -module over a C ∗ -algebra A , then thereis an A -module structure on I ( X ) extending that on X , and I ( X ) is an h -module over A with this structure.Proof. First assume A is unital. Let m : A ⊗ h X → X be a complete contractionextending the module action. Since ⊗ h is injective, A ⊗ h X ⊆ A ⊗ h I ( X ) . Thus m extends to a complete contraction ˜ m : A ⊗ h I ( X ) → I ( X ) . Define an action of A on I ( X ) by a · η = ˜ m ( a ⊗ η ) for a ∈ A , η ∈ I ( X ) .The only nontrivial property to check in order to prove that this is a module actionis the identity ( ab ) · x = a · ( b · x ) . Fix a unitary a ∈ A , and let ϕ : I ( X ) → I ( X ) be themap η a − · ( a · η ) . Since ϕ is a complete contraction restricting to the identity on X ,we have ϕ = id by rigidity. It follows similarly from rigidity that for unitaries a, b ∈ A , η = ( ab ) − · ( a · ( b · η )) . So ( ab ) · η = ( ab ) · (( ab ) − · ( a · ( b · η ))) = a · ( b · η ) for all unitaries a, b ∈ A and η ∈ I ( X ) . Since the unitaries in A span A , the desiredidentity holds for general a, b ∈ A . Thus I ( X ) admits an A -module structure extendingthat on X for which I ( X ) becomes an h -module over A .If A is non-unital, then X is canonically an h -module over the unitization A [6, 3.1.11].So by the first part of the present proof, there is some A -module structure on I ( X ) extending that on X for which I ( X ) is a h -module over A . Restricting to A and usingfunctoriality of the Haagerup tensor product gives the desired A -module structure on I ( X ) . (cid:3) Corollary 3.9. If X is a nondegenerate h -module over a C ∗ -algebra A , and X has the A -WEP, then X has the WEP. In particular, every C ∗ -algebra A which has the A -WEPhas the WEP.Proof. By Proposition 3.8, I ( X ) is an h -module over A . Hence I ( X ) ∈ Amod by thecomment above Proposition 3.8. So if X has the A -WEP, then there is a completelycontractive ( A -module) map ψ : I ( X ) → X ∗∗ that restricts to the inclusion on X . (cid:3) Corollary 3.10. A C ∗ -algebra A has the A -WEP if and only if it has the WEP.Proof. One direction follows immediately from Corollary 3.9. Suppose A has the WEP.Then the inclusion A ֒ → A ∗∗ factors through the injective envelope I ( A ) , say through acompletely positive contraction E : I ( A ) → A ∗∗ . Since E is a weak expectation, it is an A -bimodule map by Lemma 2.3. Moreover, I ( A ) an injective operator A -module (see[16, pg. 60]). This follows from Wittstock’s bimodule extension theorem [45, Theorem4.1] and Tomiyama’s theorem [43] on conditional expectations: any faithful inclusion A ⊆ B ( H ) lifts to a complete contraction ϕ : I ( A ) → B ( H ) , which is automaticallya complete isometry by rigidity. Thus, injectivity of I ( A ) and Tomiyama’s theoremyield a completely contractive A -bimodule projection P : B ( H ) → I ( A ) . Since B ( H ) is A -injective by [45, Theorem 4.1], it follows that I ( A ) is A -injective. Thus, the inclusion A ֒ → A ∗∗ factors through an A -injective module, implying A has the A -WEP by Theorem3.4 (6). (cid:3) Remark 3.11.
Corollary 3.10 is the WEP analogue of [16, Theorem 3.2] which states(in particular) that a unital C ∗ -algebra A is injective if and only if it is A -injective.3.2. The A -module C . Since a completely contractive A -module structure on an oper-ator space X is equivalent to a complete contraction A → CB ( X ) , there is a one-to-onecorrespondence between characters (i.e., multiplicative linear functionals) on A and com-pletely contractive A -module structures on C . For a character ϕ on A , denote by C ϕ the space C with the corresponding A -module structure, i.e., a · z = ϕ ( a ) z for a ∈ A , z ∈ C . It is natural to ask for conditions on ϕ that are equivalent to the A -WEP of C ϕ ,which by Example 3.2 (2) is equivalent to A -injectivity of C ϕ .The characterization below uses the notion of ϕ -amenability due to Kaniuth, Lau, andPym [26]. For a character ϕ on A , A is said to be ϕ -amenable if there exists a boundedlinear functional m on A ∗ such that h m, ϕ i = 1 and h m, f · a i = ϕ ( a ) h m, f i for all a ∈ A and f ∈ A ∗ . Such a functional m is called a ϕ -mean . Proposition 3.12.
For a character ϕ on A , if C ϕ is A -injective, then A is ϕ -amenablewith a ϕ -mean of norm one. The converse holds if A has a cai.Proof. Suppose that C ϕ is A -injective. Define a map i : C ϕ → A ∗ by i ( z ) = zϕ . Then i is completely contractive since ϕ is contractive, and h i ( a · z ) , b i = ϕ ( a ) zϕ ( b ) = z h ϕ, ba i = h ai ( z ) , b i for all a, b ∈ A , z ∈ C . So i is an A -morphism. By assumption, there is an A -morphism m : A ∗ → C such that m ◦ i = id C . Then k m k ≤ , h m, ϕ i = h m, i (1) i = 1 , and for any f ∈ A ∗ and a ∈ A , h m, f a i = a h m, f i = ϕ ( a ) h m, f i .For the converse, assume that A has a cai and that A is ϕ -amenable with a ϕ -meanof norm one. Let i : C ϕ → A ∗ be as above. By similar calculations to those above, anynorm-one ϕ -mean m : A ∗ → C ϕ is an A -morphism such that m ◦ i = id C . Then C ϕ is A -injective since, by Lemma 2.2, A ∗ is A -injective. (cid:3) The G -WEP In [8], Buss, Echterhoff and Willett introduced a notion of G -WEP for C ∗ -dynamicalsystems over locally compact groups G : a G - C ∗ -algebra A has the G -WEP if for any G -equivariant inclusion A ⊆ B into another G - C ∗ -algebra B , there exists a G -equivariantcompletely positive contraction E : B → A ∗∗ which restricts to the identity on A . Inthis subsection, we show that this notion coincides with the L ( G ) -WEP. Useful toolsin this regard are the following two lemmas. Lemma 4.1.
Let A be a completely contractive Banach algebra with a (two-sided) con-tractive approximate identity, and let X ∈ modA . Then there exists a completely con-tractive A -module map ϕ : X → h X · A i ∗∗ such that ϕ | h X · A i = i h X · A i . Hence h X · A i ֒ → X is a flat inclusion.Proof. Let ( e α ) be a (two-sided) contractive approximate identity for A . Then for each α , R e α : X ∋ x x · e α ∈ h X · A i is completely contractive, so the composition ϕ α := i h X · A i ◦ R e α : X → h X · A i ∗∗ is completely contractive. Let ϕ be a limit point of ( ϕ α ) in the weak*-topology of CB ( X, h X · A i ∗∗ ) . Then ϕ is completely contractive, and it is straightforward to checkthat ϕ is an A -module map with ϕ | h X · A i = i h X · A i .The last statement follows as in the proof of (2) ⇔ (7) in Theorem 3.4. (cid:3) Lemma 4.2.
Let A be a completely contractive Banach algebra with a (two-sided) con-tractive approximate identity, and let X ∈ modA be essential (i.e., h X · A i = X ). Then X has the A -WEP if and only if for every completely isometric morphism κ : X ֒ → Y into an essential A -module Y , there exists a morphism ϕ : Y → X ∗∗ such that ϕ ◦ κ = i X .Proof. The forward direction is obvious. For the converse, let κ : X ֒ → Y be an inclusionof A -modules. Then κ ( X ) ⊆ h Y · A i since X is essential. So there exists a morphism ψ : h Y · A i → X ∗∗ such that ϕ ◦ κ = i X , and ψ extends to a morphism h Y · A i ∗∗ → X ∗∗ .Composing this map with the morphism Y → h Y · A i ∗∗ guaranteed by Lemma 4.1 yieldsthe desired map Y → X ∗∗ . (cid:3) Let ( M, G, α ) be a W ∗ -dynamical system. Then M is canonically a module over M ( G ) via the following weak*-convergent integral:(3) x ∗ µ = Z G α s − ( x ) dµ ( s ) , x ∈ M, µ ∈ M ( G ) . The continuous part of M , M c := { x ∗ f : f ∈ L ( G ) , x ∈ M } , coincides with the set { x ∈ M : t α t ( x ) is norm-continuous } , and is a G - C ∗ -algebra.Moreover, for each x ∈ M c , the integral (3) is norm convergent. M c is a module over M ( G ) and an essential module in mod L ( G ) . These facts are well-known (see [39] or[22], for example). Proposition 4.3.
Let ( A, G, α ) be a C ∗ -dynamical system. Then A has the G -WEP ifand only if it has the L ( G ) -WEP.Proof. Suppose A has the L ( G ) -WEP and that A ⊆ B is an inclusion of G - C ∗ -algebras.Then there exists an L ( G ) -morphism E : B → A ∗∗ restricting to the identity on A . Forany f ∈ L ( G ) , s ∈ G , and b ∈ B we have E ( α s ( b ∗ f )) = E (( b ∗ f ) ∗ δ s ) = E ( b ∗ ( f ∗ δ s )) = E ( b ) ∗ ( f ∗ δ s ) = α s ( E ( b ) ∗ f ) = α s ( E ( b ∗ f )) . Since B satisfies h B ∗ L ( G ) i = B , it follows that E is a G -equivariant complete con-traction. By Lemma 2.3 E is necessarily a completely positive contraction, whence A has the G -WEP.Conversely, suppose that A has the G -WEP. Let Y ∈ mod L ( G ) and κ : A ֒ → Y be a completely isometric L ( G ) -morphism. We will show that there exists an L ( G ) -morphism ϕ : Y → A ∗∗ such that ϕ ◦ κ = i A . By Lemma 4.2, we may assume Y is essential. Let A ⊆ B ( H ) be a faithful ∗ -representation. Since A ∈ mod L ( G ) isessential and L ( G ) has a contractive approximate identity, it follows that the canonical ∗ -homomorphic L ( G ) -morphism j : A → CB ( L ( G ) , B ( H )) is completely isometric. ByLemma 2.2, CB ( L ( G ) , B ( H )) is L ( G ) -injective. So there exists an L ( G ) -morphism ψ : Y → CB ( L ( G ) , B ( H )) such that ψ ◦ κ = j . Since Y is essential, ψ ( Y ) ⊆ B ,where B = hCB ( L ( G ) , B ( H )) · L ( G ) i . Since CB ( L ( G ) , B ( H )) = B ( H ) ⊗ L ∞ ( G ) is a G - W ∗ -algebra, B is a G - C ∗ -algebra containing a ∗ -homomorphic copy of A . So there is a G -equivariant complete contraction, which must also be an L ( G ) -morphism, ρ : B → A ∗∗ such that ρ ◦ j = i A . Then ϕ := ρ ◦ ψ : Y → A ∗∗ is an L ( G ) -morphism, and ϕ ◦ κ = ρ ◦ ψ ◦ κ = ρ ◦ j = i A . (cid:3) In a similar fashion, we have the analogous result for injectivity. Here, we say that G - C ∗ -algebra is G -injective if it is injective in the category of G - C ∗ -algebras and completelypositive G -equivariant contractions, denoted G - C ∗ - alg . Proposition 4.4.
Let ( A, G, α ) be a C ∗ -dynamical system. Then A is G -injective if andonly if it is L ( G ) -injective.Proof. Suppose A has is injective in mod L ( G ) and that B ⊆ C is an inclusion of G - C ∗ -algebras. If ϕ : B → A is a morphism in G - C ∗ - alg , then by norm convergence inthe C ∗ -analogue of (3), ϕ is an L ( G ) -module map. By L ( G ) -injectivity there exists an L ( G ) -morphism e ϕ : C → A extending ϕ . By Lemma 2.3 e ϕ is necessarily a completelypositive contraction, and as above it is G -equivariant. Whence A is G -injective.Conversely, if A is G -injective, and ι : X ֒ → Y is an inclusion in mod L ( G ) , and ϕ : X → A is an L ( G ) -morphism. Let A ⊆ B ( H ) be a faithful inclusion and j A : A ֒ → CB ( L ( G ) , B ( H )) the canonical embedding. By [12, Proposition 2.3] (which isinspired by [21, Lemma 2.2]) CB ( L ( G ) , B ( H )) is L ( G ) -injective. Thus, there existsan L ( G ) -morphism e ϕ : Y → CB ( L ( G ) , B ( H )) extending j A ◦ ϕ . Since CB ( L ( G ) , B ( H )) = ( L ( G ) b ⊗T ( H )) ∗ = ( L ∞ ( G ) ⊕ C ⊗B ( H ) is a G - W ∗ -algebra, its continuous part CB ( L ( G ) , B ( H )) c is a G - C ∗ -algebra. Moreover, j A ( A ) ⊆ CB ( L ( G ) , B ( H )) c . By G -injectivity there is a G -equivariant completely pos-itive contraction Φ : CB ( L ( G ) , B ( H )) c → A satisfying Φ ◦ j A = id A . As above, Φ is necessarily an L ( G ) -morphism, and the composition Φ ◦ e ϕ : Y → A is the desired L ( G ) -morphism extending ϕ . (cid:3) In [9, Definition 8.1], Buss, Echterhoff and Willett introduced a notion of continuous G -WEP for a C ∗ -dynamical system ( A, G, α ) : for any G -equivariant inclusion A ⊆ B into a G - C ∗ -algebra B , there exists a completely positive G -equivariant contraction E : B → A ′′ α for which E | A is the canonical inclusion A ⊆ A ′′ α . By [9, Lemma 8.1], the G -WEP implies the continuous G -WEP. We now establish the converse. Proposition 4.5.
Let ( A, G, α ) be a C ∗ -dynamical system. Then A has the G -WEP ifand only if it has the continuous G -WEP.Proof. First, we construct a unital completely positive L ( G ) -module map A ′′ α → A ∗∗ which restricts to the identity on A . Let e α : A ∗∗ → L ∞ ( G ) ⊗ A ∗∗ be the normal cover ofthe non-degenerate representation α : A → M ( C ( G ) ⊗ ∨ A ) ⊆ L ∞ ( G ) ⊗ A ∗∗ , let ( f i ) bea contractive approximate identity for L ( G ) consisting of states, and let m ∈ L ∞ ( G ) ∗ be a weak* limit of a subnet ( f i j ) . The map Φ : A ∗∗ ∋ a ( m ⊗ id ) e α ( a ) ∈ A ∗∗ , is a unital completely positive L ( G ) -morphism, the latter property following from theasymptotic centrality of ( f i j ) : for every a ∈ A ∗∗ , f ∈ L ( G ) and µ ∈ A ∗ , h Φ( a ∗ f ) , µ i = lim j h a ∗ f, f i j ∗ µ i = lim j h a, f ∗ f i j ∗ µ i = lim j h a, f i j ∗ f ∗ µ i = lim j h a ∗ f i j , f ∗ µ i = h Φ( a ) , f ∗ µ i = h Φ( a ) ∗ f, µ i . By definition of Φ , it follows that ( A ∗ c ) ⊥ = (1 − z ) A ∗∗ ⊆ Ker(Φ) , so we obtain an inducedunital completely positive L ( G ) -morphism (still denoted) Φ : A ′′ α → A ∗∗ . Then for every a ∈ A and µ ∈ A ∗ , h Φ( za ) , µ i = h ( m ⊗ id ) e α ( za ) , µ i = h ( m ⊗ id ) α ( a ) , µ i = lim j h a ∗ f i j , µ i = h a, µ i , where the last equality follows from continuity of s α s ( a ) .Now, suppose A has the continuous G -WEP and B is a G - C ∗ -algebra for which A ⊆ B .Then there exists a completely positive G -equivariant contraction E : B → A ′′ α whichrestricts to the inclusion A ⊆ A ′′ α . By norm convergence of (1) it follows that E is L ( G ) -equivariant. Then Φ ◦ E : B → A ∗∗ is a completely positive L ( G ) -morphismwhich restricts to the identity on A . By L ( G ) -essentiality of B , the same argumentfrom the proof of Proposition 4.3 shows that Φ ◦ E is G -equivariant. Hence, A has the G -WEP. (cid:3) A C ∗ -algebra is said to have the QWEP if it is a quotient of a C ∗ -algebra withthe WEP. In [28], Kirchberg conjectured that every C ∗ -algebra has the QWEP. Untilrecently, this conjecture stood as one of the most important open problems in operatoralgebra theory, where it was shown to be false [24]. In the setting of G - C ∗ -algebras, wecan simply show that the G -equivariant analogue of the QWEP conjecture is false. Say a G - C ∗ -algebra has the G -QWEP if it is the image under a G -equivariant ∗ -homomorphismof a G -C ∗ -algebra with the G -WEP. Proposition 4.6.
Let G be a non-amenable locally compact group. Then no C ∗ -algebrawith a G -invariant state has the G -QWEP. In particular, C ∗ ( G ) does not have the G -QWEP.Proof. Let A be a C ∗ -algebra admitting a G -invariant state ϕ , and suppose for con-tradiction that q : B → A is a G -equivariant surjective ∗ -homomorphism, where B isa G - C ∗ -algebra with the G -WEP. Then there is a G -equivariant copy of I L ( G ) ( B ) in B ∗∗ such that the inclusion I L ( G ) ( B ) ֒ → B ∗∗ restricts to the canonical inclusion of B .Note that B ∗∗ ∈ I L ( G ) ( B ) . (Indeed, if Φ : B ∗∗ → I L ( G ) ( B ) is a complete contrac-tion such that Φ B = id B , then each b in B is in the multiplicative domain of Φ , so Φ(1 B ∗∗ ) b = Φ( b ) = b . Thus Φ(1 B ∗∗ ) = 1 B ∗∗ .) So C ֒ → I L ( G ) ( B ) via z z B ∗∗ , whichimplies the existence of an L ( G ) -embedding I L ( G ) ( C ) ֒ → I L ( G ) ( B ) ⊆ B ∗∗ . Composingthis inclusion with ϕ ∗∗ ◦ q ∗∗ : B ∗∗ → C yields an L ( G ) -module map I L ( G ) ( C ) → C restricting to the identity on C . By L ( G ) -rigidity, I L ( G ) ( C ) = C , so that C is L ( G ) -injective. However, this would give the existence of a G -invariant state L ∞ ( G ) → C ,thus contradicting non-amenability of G . (cid:3) The amenable radical of a locally compact group G is the unique amenable normalsubgroup of G containing all other amenable normal subgroups of G . By [27], C ∗ λ ( G ) admits a tracial state if and only if the amenable radical of G is open (which of courseholds in particular for discrete groups). Since a tracial state on C ∗ λ ( G ) is necessarily G -invariant, we get the following as a corollary. Corollary 4.7. If G is a non-amenable locally compact group with open amenable rad-ical, then C ∗ λ ( G ) does not have the G -QWEP. The discrete group case.
Throughout this section Γ is a discrete group, and bya Γ - C ∗ -algebra we mean a unital C ∗ -algebra A on which Γ acts by ∗ -automorphisms.By a Γ -map we always mean a Γ -equivariant unital completely positive (ucp) map.If A is a Γ - C ∗ -algebra, then as noted above there is a canonical action of ℓ (Γ) on A such that A with this action belongs to ℓ (Γ) mod . Clearly, in this setting, Γ -mapsand unital ℓ (Γ) -morphisms coincide. It follows verbatim from Proposition 4.4 that a Γ - C ∗ -algebra A is injective in ℓ (Γ) mod if and only if it is injective in the categoryof Γ - C ∗ -algebras with Γ -maps, the only difference being we consider unital morphisms.Similarly, by Proposition 4.3 the ℓ (Γ) -WEP for A coincides with the analogous notionin the category of Γ - C ∗ -algebras with Γ -maps. Thus, we say that a Γ - C ∗ -algebra A hasthe Γ -WEP if it has ℓ (Γ) -WEP.Let A be a C ∗ -algebra. The C ∗ -algebra ℓ ∞ (Γ , A ) of bounded A -valued functions on Γ turns into a Γ - C ∗ -algebra (or a Γ -von Neumann algebra if A is a von Neumann algebra)with the action ( t · f )( s ) = f ( t − s ) ( s, t ∈ Γ and f ∈ ℓ ∞ (Γ , A )) . The map ι : ℓ ∞ (Γ) → ℓ ∞ (Γ , A ) defined by(4) ι ( f )( s ) = f ( s ) A is a Γ -equivariant C ∗ - (or von Neumann-)embedding. In this setup, if ϕ : A → A is aucp map between C ∗ -algebras, then e ϕ : ℓ ∞ (Γ , A ) → ℓ ∞ (Γ , A ) , f ϕ ◦ f is a Γ -map.If A is injective, then ℓ ∞ (Γ , A ) is injective in ℓ (Γ) mod . The latter claim follows from aproof similar to that of Hamana’s in [21, Lemma 2.2], where this statement is proved forslightly different categories. We won’t need the following, but we note that the aboveworks in more general setup. If X ⊆ B ( H ) is an operator space, then ℓ ∞ (Γ , X ) is an ℓ (Γ) -submodule of ℓ ∞ (Γ , B ( H )) . So in particular, ℓ ∞ (Γ , X ) is in ℓ (Γ) mod . If X isinjective, then a proof similar to the one mentioned above in [21, Lemma 2.2] shows that ℓ ∞ (Γ , X ) is injective in ℓ (Γ) mod .If A is a Γ - C ∗ -algebra and π : A → B ( H ) is a faithful nondegenerate ∗ -representation,then the map(5) j π : A → ℓ ∞ (Γ , B ( H )) , j π ( a )( t ) = π ( t − a ) , for a ∈ A and t ∈ Γ , is a Γ -embedding. Note that j π ( A ) weak* ⊆ ℓ ∞ (cid:16) Γ , π ( A ) weak* (cid:17) , wherethe weak*-closures are taken in ℓ ∞ (Γ , B ( H )) , and B ( H ) , respectively.In the following, we use I Γ ( A ) to denote Hamana’s Γ -injective envelope of a Γ - C ∗ -algebra A (see [21]). This is the injective envelope of A in the category of Γ -operatorsystems with Γ -equivariant u.c.p. maps. Some of the proofs below also use the injective envelope I ( A ) of a C ∗ -algebra A ,and in particular, the fact that A has the WEP if and only if there is a completelyisometric embedding I ( A ) ֒ → A ∗∗ fixing elements in A (this is attributed to Blackadarin [38, Introduction]). Theorem 4.8. If A is a Γ - C ∗ -algebra, the following are equivalent:(1) A has the Γ -WEP.(2) For every Γ - C ∗ -algebra B and completely isometric Γ -map κ : A ֒ → B , there isa Γ -map ψ : B → A ∗∗ such that ψ ◦ κ = ι A .(3) For every faithful nondegenerate representation π : A → B ( H ) , there is a Γ -map Φ : ℓ ∞ (Γ , B ( H )) → j π ( A ) weak* ⊆ ℓ ∞ (Γ , B ( H )) such that Φ( a ) = a for all a ∈ j π ( A ) .(4) Letting π u : A → B ( H u ) denote the universal representation, there is a Γ -map Φ : ℓ ∞ (Γ , B ( H u )) → j π u ( A ) ′′ ∼ = A ∗∗ such that Φ( a ) = a for all a ∈ j π u ( A ) .(5) For every faithful covariant representation π : (Γ y A ) → B ( H ) , there is a Γ -map Φ : B ( H ) → π ( A ) ′′ such that Φ( a ) = a for all a ∈ π ( A ) .(6) Letting (1 ⊗ λ, π ) : (Γ y A ) → B ( H u ⊗ ℓ (Γ)) denote the regular representationinduced from the universal representation of A , there is a Γ -map Φ : B ( H u ⊗ ℓ (Γ)) → π ( A ) ′′ ∼ = A ∗∗ such that Φ( a ) = a for all a ∈ π ( A ) .(7) There is a Γ -embedding I Γ ( A ) ֒ → A ∗∗ .Proof. (1) ⇐⇒ (2): This follows immediately from Proposition 4.3.(2) = ⇒ (3): Assume (2), and let π : A → B ( H ) be a faithful representation.Since j π : A → ℓ ∞ (Γ , B ( H )) is a completely isometric Γ -map, there is by (2) a Γ -map ψ : ℓ ∞ (Γ , B ( H )) → A ∗∗ such that ψ ◦ j π = ι A . By the universal property of A ∗∗ , there isa normal ∗ -homomorphism ϕ : A ∗∗ → j π ( A ) weak* such that ϕ ( a ) = j π ( a ) for all a ∈ A .By normality, ϕ is a Γ -map. Thus ϕ ◦ ψ satisfies the conclusion of (3).(3) = ⇒ (4): The only thing not completely trivial about this implication is theassertion that j π u ( A ) ′′ ∼ = A ∗∗ . This follows from the universal property of A ∗∗ togetherwith the fact that the canonical inclusion ˜ π u : A ∗∗ ֒ → B ( H u ) induces a normal injective ∗ -homomorphism j ˜ π u : A ∗∗ ֒ → ℓ ∞ (Γ , B ( H u )) that restricts to j π u on A .(4) = ⇒ (7): This direction follows since ℓ ∞ (Γ , B ( H u )) is injective in the category of Γ - C ∗ -algebras with Γ -maps (see [21, Lemma 2.2]). Thus there is a Γ -map ϕ : I Γ ( A ) → ℓ ∞ (Γ , B ( H u )) such that ϕ ( a ) = j π u ( a ) for all a ∈ A . With Φ from (4), the composition Φ ◦ ϕ : I Γ ( A ) → A ∗∗ gives a Γ -embedding by Γ -essentiality (see [21, Section 2]).(7) = ⇒ (5): Assume that there is a Γ -embedding κ : I Γ ( A ) ֒ → A ∗∗ , and let π : (Γ y A ) → B ( H ) be a covariant representation. By Γ -injectivity, the inclusion A ֒ → I Γ ( A ) extends to a Γ -map ˜ κ : B ( H ) → I Γ ( A ) . By the universal property of A ∗∗ , there isa Γ -map ψ : A ∗∗ → π ( A ) ′′ such that ψ ( a ) = π ( a ) for all a ∈ A . The composition ψ ◦ ˜ κ : B ( H ) → π ( A ) ′′ is the desired Γ -map.(5) = ⇒ (6): This follows similarly to the implication (3) = ⇒ (4).(6) = ⇒ (4): Since ℓ ∞ (Γ , B ( H u )) ⊆ B ( H u ⊗ ℓ (Γ)) canonically as a Γ -subspace, therestriction of the map guaranteed by (6) satisfies the claim in (4).(7) = ⇒ (2): This follows from a routine use of Γ -injectivity. (cid:3) Theorem 4.9. If Γ ⋉ A has the WEP then A has the Γ -WEP. Proof.
Denote by E : Γ ⋉ A → A the canonical conditional expectation, which is a Γ -map, and let E ∗∗ : (Γ ⋉ A ) ∗∗ → A ∗∗ be its second adjoint. So, using [21, Theorem3.4], we have the following Γ -maps I Γ ( A ) → I (Γ ⋉ A ) → (Γ ⋉ A ) ∗∗ → A ∗∗ whose composition I Γ ( A ) → A ∗∗ restricts to identity on A , hence is a Γ -embedding by Γ -essentiality. Thus, by Theorem 4.8, A has the Γ -WEP. (cid:3) The WEP for actions.
A discrete group Γ is said to act amenably on a Γ - C ∗ -algebra A if the bidual action Γ y A ∗∗ is amenable. It is known that a group canact amenably on a non-nuclear C ∗ -algebra (or a non-injective von Neumann algebra).Motivated by the above characterization, and in order to include actions on C ∗ -algebraswithout the WEP, we make the following definition. Definition 4.10.
We say an action Γ y A has the WEP if there is a Γ -map Φ : ℓ ∞ (Γ , A ∗∗ ) → A ∗∗ such that (Φ ◦ j )( a ) = a for all a ∈ A .Obviously, from the definitions, if A has the Γ -WEP then Γ y A has the WEP. Proposition 4.11. If Γ y A has the WEP and A has the WEP, then A has the Γ -WEP.Proof. Since A has the WEP, there is a Γ -embedding ι : ℓ ∞ (Γ , I ( A )) ֒ → ℓ ∞ (Γ , A ∗∗ ) thatfixes elements in ℓ ∞ (Γ , A ) . With j : A → ℓ ∞ (Γ , I ( A )) and j π u : A → ℓ ∞ (Γ , A ∗∗ ) the Γ -embeddings described as in Equation 5, we have ι ◦ j ( a ) = j π u ( a ) . Since Γ y A hasthe WEP, we can compose to get a Γ -map ℓ ∞ (Γ , I ( A )) → A ∗∗ restricting to the identityon the canonical copies of A . Since ℓ ∞ (Γ , I ( A )) is Γ -injective, the result follows fromTheorem 3.4 (6). (cid:3) Proposition 4.12. If Γ y A is amenable, then Γ y A has the WEP.Proof. Consider the automorphism κ : ℓ ∞ (Γ , A ∗∗ ) → ℓ ∞ (Γ , A ∗∗ ) defined by κ ( f )( t ) = t ( f ( t )) , t ∈ Γ . For s ∈ Γ and f ∈ ℓ ∞ (Γ , A ∗∗ ) we have κ ( sf )( t ) = t ( sf ( t )) = t ( f ( s − t )) = s ( s − t ( f ( s − t ))) = s ( κ ( f )( s − t )) = s ( sκ ( f )( t )) , which shows κ ( sf ) = α s ( κ ( f )) , where α : Γ y ℓ ∞ (Γ , A ∗∗ ) is the diagonal action.Moreover, for every a ∈ A , κ ( j π u ( a ))( t ) = t ( j π u ( a )( t )) = t ( π u ( t − a )) = π u ( a ) , which shows κ ( j π u ( a )) = 1 ⊗ a .The result is now clear by the definition of the WEP for the action and the fact that anaction Γ y A is amenable if and only if there is an α -equivariant conditional expectation E : ℓ ∞ (Γ , A ∗∗ ) ∼ = ℓ ∞ (Γ) ⊗ A ∗∗ → ⊗ A ∗∗ ∼ = A ∗∗ . (cid:3) Combining the previous two propositions, we get:
Corollary 4.13. If Γ y A is amenable and A has the WEP, then A has the Γ -WEP. Example 4.14.
Let ∂ F Γ denote the Furstenberg boundary of Γ . Then C ( ∂ F Γ) hasthe Γ -WEP. This follows by Theorem 4.8 and the fact that I Γ ( C ( ∂ F Γ)) = C ( ∂ F Γ) .In particular, if Γ is not exact, then C ( ∂ F Γ) has the Γ -WEP but Γ y C ( ∂ F Γ) is notamenable. Theorem 4.15.
The following are equivalent for a discrete group Γ .(1) Γ is amenable.(2) C ∗ λ (Γ) has the WEP.(3) C ∗ λ (Γ) has the Γ -WEP.Proof. The equivalence of (1) and (2) is Lance’s theorem [29]. Since ℓ (Γ) ∗ = ℓ ∞ (Γ) isinjective, the implication (3) = ⇒ (2) follows from Theorem 3.7. Now suppose Γ isamenable and C ∗ λ (Γ) has the WEP. Then the action Γ y C ∗ λ (Γ) is amenable. Hence C ∗ λ (Γ) has the Γ -WEP by Proposition 4.13. (cid:3) We will prove in the next section that the above conditions are also equivalent to C ∗ λ (Γ) having the A (Γ) -WEP.In the case of a general locally compact group G , the implication (2) ⇒ (1) above isknown to not hold in general, e.g., for G = SL (2 , R ) .4.3. On the kernel of actions with the WEP.
Let R (Γ) denote the amenable rad-ical of Γ , that is, the unique amenable normal subgroup of Γ that contains all otheramenable normal subgroups of Γ . By [17, Proposition 7] and [25, Theorem 3.11], R (Γ) = ker (Γ y I Γ ( C )) . So ker (Γ y I Γ ( A )) may be considered as a generalizedamenable radical of Γ corresponding to the action Γ y A . Proposition 4.16.
For any Γ - C ∗ -algebra A , ker (Γ y I Γ ( A )) ⊆ R (Γ) . In particular, ker (Γ y I Γ ( A )) is amenable.Proof. Since I Γ ( C ) ⊆ I Γ ( A ) via a Γ -embedding, the result follows from the fact men-tioned above that R (Γ) is the kernel of the action Γ y I Γ ( C ) . (cid:3) Evidently, the inclusion ker (Γ y I Γ ( A )) ⊆ ker (Γ y A ) always holds. The converseinclusion also clearly holds if A is Γ -injective. We show below that it holds under thegenerally weaker assumption that A has the Γ -WEP. Theorem 4.17. If A has the Γ -WEP, then ker (Γ y I Γ ( A )) = ker (Γ y A ) .Proof. As noted above, ker (Γ y I Γ ( A )) ⊆ ker (Γ y A ) always holds.For the other inclusion, suppose A has the Γ -WEP. Then there is a Γ -embedding ϕ : I Γ ( A ) ֒ → A ∗∗ such that ϕ restricts to the canonical inclusion on A . If s ∈ ker (Γ y A ) ,then by weak*-continuity, s is also in the kernel of the action Γ y A ∗∗ . Thus s also actstrivially on everything in I Γ ( A ) , i.e., s ∈ ker (Γ y I Γ ( A )) . (cid:3) The converse of Theorem 4.17 is not true: if Γ is trivial, then ker (Γ y I Γ ( A )) =ker (Γ y A ) = Γ for all A . Example 4.18. (1) Consider the trivial action Γ y C . Then ker (Γ y I Γ ( C )) = R (Γ) and ker (Γ y C ) = Γ . So for Γ non-amenable, this gives an example when ker (Γ y I Γ ( A )) = ker (Γ y A ) .(2) Consider the canonical action Γ y ℓ ∞ (Γ) . Since ℓ ∞ (Γ) is Γ -injective, it follows ker (Γ y I Γ ( ℓ ∞ (Γ))) = ker (Γ y ℓ ∞ (Γ)) , and it is easy to see that these areboth the trivial subgroup. So for Γ non-amenable, this gives an example when ker (Γ y I Γ ( A )) = R (Γ) . (3) Consider the canonical inner action Γ y C ∗ λ (Γ) . It is straightforward to checkthat ker (Γ y C ∗ λ (Γ)) is the center Z (Γ) . If Γ is amenable, then C ∗ λ (Γ) hasthe Γ -WEP by Theorem 4.15, so we get ker (Γ y I Γ ( C ∗ λ (Γ))) = Z (Γ) by Theo-rem 4.17. Hence for Γ amenable and non-abelian, this gives an example when ker (Γ y I Γ ( A )) is nontrivial and not equal to R (Γ) . Corollary 4.19. If Γ is a discrete group with trivial amenable radical and A is a Γ - C ∗ -algebra with the Γ -WEP, then the action Γ y A is faithful.Proof. If R (Γ) is trivial, then so is ker (Γ y I Γ ( A )) by Proposition 4.16. The result thusfollows from Theorem 4.17. (cid:3) Amenability of stabilizers.
The theorem below can be considered as a general-ization of Lance’s result [29] that C ∗ λ (Γ) has the WEP iff Γ is amenable. Theorem 4.20.
Let Γ y X be a minimal action on a compact Hausdorff space X suchthat Γ y C ( X ) has the WEP. Then the stabilizer of any point x ∈ X is amenable.In particular, if Γ ⋉ C ( X ) has the WEP then the stabilizer of any point x ∈ X isamenable.Proof. Let x ∈ X , and let Λ = { g ∈ Γ : gx = x } be the stabilizer subgroup of x . Denoteby P x : C ( X ) → ℓ ∞ (Γ) the injective *-homomorphism P x ( f )( g ) := f ( gx ) for g ∈ Γ . Forany g ∈ Γ , h ∈ Λ , and f ∈ C ( X ) we have P x ( f )( gh ) = f ( ghx ) = f ( gx ) = P x ( f )( g ) , which shows P x ( C ( X )) ⊂ ℓ ∞ (Γ / Λ) . Hence this yields a faithful representation π : C ( X ) → B ( ℓ (Γ / Λ)) , and we have j π ( C ( X )) weak* ⊂ j id ( ℓ ∞ (Γ / Λ)) , where id : ℓ ∞ (Γ / Λ) → B ( ℓ (Γ / Λ)) is the canonical inclusion. Now suppose the ac-tion Γ y C ( X ) has the WEP, that is, there exists a Γ -map Φ : ℓ ∞ (Γ , π ( C ( X )) ′′ ) → j π ( C ( X )) weak* . Then ϕ = δ Λ ◦ j − id ◦ Φ ◦ ι is a Λ -invariant state on ℓ ∞ (Γ) , where ι : ℓ ∞ (Γ) → ℓ ∞ (Γ , B ( ℓ (Γ / Λ))) is the Γ -embedding defined in (4). Hence, Λ isamenable. (cid:3) Amenable actions and the A ( G ) -WEP In this section we characterize amenable dynamical systems through the A ( G ) -modulestructure of their associated crossed products. Our techniques are streamlined using theperspective of co-actions, so we begin with a quick overview of the relevant definitions.This perspective also hints at potential quantum group generalizations of our results.We postpone this investigation for future work.Let G be a locally compact group. The adjoint of convolution L ( G ) b ⊗ L ( G ) → L ( G ) is a co-associative co-multiplication ∆ : L ∞ ( G ) → L ∞ ( G ) ⊗ L ∞ ( G ) satisfying ∆( f )( s, t ) = f ( st ) , for all f ∈ L ∞ ( G ) . There are left and right fundamental unitaries W, V ∈ B ( L ( G × G )) which implement ∆ in the sense that ∆( f ) = W ∗ (1 ⊗ M f ) W = V ( M f ⊗ V ∗ , f ∈ L ∞ ( G ) . They are given respectively by
W ξ ( s, t ) = ξ ( s, s − t ) , V ξ ( s, t ) = ξ ( st, t ) δ G ( t ) / , s, t ∈ G, ξ ∈ L ( G × G ) , where δ G is the modular function. These fundamental unitaries are intimately relatedto the left and right regular representations λ, ρ : G → B ( L ( G )) given by λ ( s ) ξ ( t ) = ξ ( s − t ) , ρ ( s ) ξ ( t ) = ξ ( ts ) δ G ( s ) / , s, t ∈ G, ξ ∈ L ( G ) . It follows that W ∈ L ∞ ( G ) ⊗ V N ( G ) and V ∈ V N ( G ) ′ ⊗ L ∞ ( G ) .The adjoint of pointwise multiplication A ( G ) b ⊗ A ( G ) → A ( G ) defines a co-associativeco-multiplication b ∆ : V N ( G ) → V N ( G ) ⊗ V N ( G ) satisfying b ∆( λ ( s )) = λ ( s ) ⊗ λ ( s ) , s ∈ G . There are left and right fundamental unitaries c W , b V ∈ B ( L ( G × G )) whichimplement the co-product via b ∆( x ) = c W ∗ (1 ⊗ x ) c W = b V ( x ⊗ b V ∗ , x ∈ V N ( G ) . They are given specifically by c W ξ ( s, t ) = ξ ( ts, t ) , b V ξ ( s, t ) = W ξ ( s, t ) = ξ ( s, s − t ) , s, t ∈ G, ξ ∈ L ( G × G ) . Note that c W ∈ V N ( G ) ⊗ L ∞ ( G ) .Both co-products ∆ and b ∆ admit left and right extensions to B ( L ( G )) via the fun-damental unitaries. For instance, ∆ l : B ( L ( G )) ∋ T W ∗ (1 ⊗ T ) W ∈ L ∞ ( G ) ⊗B ( L ( G )) and(6) ∆ r : B ( L ( G )) ∋ T V ( T ⊗ V ∗ ∈ B ( L ( G )) ⊗ L ∞ ( G ) . These maps, in turn, yield operator L ( G ) -module structures on B ( L ( G )) given by T · f = ( f ⊗ id )∆ l ( T ) = Z G f ( s ) λ ( s ) ∗ T λ ( s ) ds and f · T = ( id ⊗ f )∆ r ( T ) = Z G f ( s ) ρ ( s ) T ρ ( s ) ∗ ds, for f ∈ L ( G ) and T ∈ B ( L ( G )) . Analogously, the lifted co-products b ∆ l and b ∆ r yield an operator A ( G ) -bimodule structure on B ( L ( G )) . In this case the left and rightmodule structures coincide since b V = W = σ c W ∗ σ . Indeed, for every u ∈ A ( G ) and T ∈ B ( L ( G )) we have T · u = ( u ⊗ id ) b ∆ l ( T ) = ( u ⊗ id ) c W ∗ (1 ⊗ T ) c W = ( id ⊗ u ) W ( T ⊗ W ∗ = ( id ⊗ u ) b V ( T ⊗ b V = u · T When G is discrete, the A ( G ) -module action is nothing but Schur product with thematrix [ u ( st − )] s,t ∈ G , where u ∈ A ( G ) .Viewing the extended co-multiplication ∆ r as a map B ( L ( G )) → B ( L ( G )) ⊗B ( L ( G )) ,its pre-adjoint defines a completely contractive multiplication on the space of trace-class operators T ( L ( G )) via ⊲ : T ( L ( G )) b ⊗T ( L ( G )) ∋ ω ⊗ τ ω ⊲ τ = ∆ r ∗ ( ω ⊗ τ ) ∈ T ( L ( G )) . The resulting bimodule structure on B ( L ( G )) satisfies T ⊲ ρ = ( ρ ⊗ id ) V ( T ⊗ V ∗ , ρ ⊲ T = ( id ⊗ ρ ) V ( T ⊗ V ∗ = π ( ρ ) · T for T ∈ B ( L ( G )) , ρ ∈ T ( L ( G )) , where π : T ( L ( G )) ։ L ( G ) is the canonical quotientmap given by restriction to L ∞ ( G ) . Hence, the left ⊲ -module structure degenerates tothe left L ( G ) -module structure defined above. The analogous construction exists for b ∆ r ,yielding a dual product b ⊲ on T ( L ( G )) and a corresponding b ⊲ -bimodule structure on B ( L ( G )) . Dually to ⊲ , the left b ⊲ -module structure degenerates to the left A ( G ) -action.If ( M, G, α ) is a W ∗ -dynamical system, the induced normal ∗ -homomorphism α : M → L ∞ ( G ) ⊗ M , α ( x )( s ) = α s − ( x ) , x ∈ M , s ∈ G , is co-associative in the sense that (∆ ⊗ id ) ◦ α = ( id ⊗ α ) ◦ α. The corresponding L ( G ) -module structure is determined by x ⋆ f = ( f ⊗ id ) α ( x ) = Z G f ( s ) α s − ( x ) , f ∈ L ( G ) , x ∈ M. It follows from the fixed point description of G ¯ ⋉ M (see, e.g., [40, Theorem 16.1.15])that G ¯ ⋉ M = { T ∈ B ( L ( G )) ⊗ M | ( id ⊗ α s − )( T ) = (Ad( ρ ( s )) ⊗ id )( T ) ∀ s ∈ G } . By point-weak* continuity of W ∗ -dynamical systems, the condition ( id ⊗ α s − )( T ) =(Ad( ρ ( s )) ⊗ id )( T ) for all s ∈ G is equivalent to ( id ⊗ f ⊗ id )( id ⊗ α )( T ) = ( id ⊗ f ⊗ id )(∆ r ⊗ id )( T ) , f ∈ L ( G ) (by approximating point masses by suitable nets in L ( G ) ). Hence, G ¯ ⋉ M = { X ∈ B ( L ( G )) ⊗ M | ( id ⊗ α )( X ) = (∆ r ⊗ id )( X ) } The system ( M, G, α ) admits a dual co-action b α : G ¯ ⋉ M → V N ( G ) ⊗ ( G ¯ ⋉ M ) of V N ( G ) on the crossed product, given by(7) b α ( X ) = ( c W ∗ ⊗ ⊗ X )( c W ⊗ , X ∈ G ¯ ⋉ M. On the generators we have b α (ˆ x ⊗
1) = ( c W ∗ (1 ⊗ ˆ x ) c W ) ⊗ , ˆ x ∈ V N ( G ) and b α ( α ( x )) =1 ⊗ α ( x ) , x ∈ M . Moreover, by [44, Theorem 2.7] ( G ¯ ⋉ M ) b α = { X ∈ G ¯ ⋉ M | b α ( X ) = 1 ⊗ X } = α ( M ) . This co-action yields a canonical right operator A ( G ) -module structure on the crossedproduct G ¯ ⋉ M via X · u = ( u ⊗ id ) b α ( X ) , X ∈ G ¯ ⋉ M, u ∈ A ( G ) . Assuming M is standardly represented on H , there exists a strongly continuous unitaryrepresentation u : G → B ( H ) and corresponding generator U ∈ L ∞ ( G ) ⊗B ( H ) such that α ( x ) = U ∗ (1 ⊗ x ) U , x ∈ M [19, Corollary 3.11]. At the level of vectors ξ ∈ C c ( G, H ) ⊆ L ( G ) ⊗ H we have(8) α ( x ) ξ ( s ) = U ∗ (1 ⊗ x ) U ξ ( s ) = u ∗ s xu s ξ ( s ) , s ∈ G, x ∈ M. A W ∗ -dynamical system ( M, G, α ) is amenable if there exists a projection of normone P : L ∞ ( G ) ⊗ M → M such that P ◦ ( λ s ⊗ α s ) = α s ◦ P , s ∈ G , where λ also denotesthe left translation action on L ∞ ( G ) . For example, ( L ∞ ( G ) , G, λ ) is always amenable,and G is amenable if and only if the trivial action G y { x } is amenable, in whichcase P becomes a left invariant mean on L ∞ ( G ) . When G is second countable and M = L ∞ ( X, µ ) for a regular G -space ( X, µ ) (see [36, Definition 2.1.1] for a definition),it is known that that ( M, G, α ) is amenable if and only if M is relatively G -injective[36, Theorem 5.7.1]. At the level on L ( G ) -modules, we now show that this equivalenceholds in general. Proposition 5.1. A W ∗ -dynamical system ( M, G, α ) is amenable if and only if M isrelatively L ( G ) -injective.Proof. Assume M is standardly represented in B ( H ) , and let U ∈ L ∞ ( G ) ⊗B ( H ) be theunitary implementation of α . By commutativity of L ∞ ( G ) and formula (8) it followsthat both Ad( U ) and Ad( U ∗ ) leave L ∞ ( G ) ⊗ M invariant. Moreover, for every ξ, η ∈ C c ( G, H ) , h ∈ L ∞ ( G ) , x ∈ M , we have h U ∗ ( λ s ⊗ α s ( h ⊗ x )) U ξ, η i = Z G h U ∗ ( λ s ⊗ α s ( h ⊗ x )) U ξ ( t ) , η ( t ) i dt = Z G h ( s − t t ) h u t − α s ( x ) u t ξ ( t ) , η ( t ) i dt = Z G h ( s − t t ) h u t − s xu s − t ξ ( t ) , η ( t ) i dt = Z G h ( t ) h u t − xu t ξ ( st ) , η ( st ) i dt = h U ∗ ( h ⊗ x ) U ( λ s − ⊗ ξ, ( λ s − ⊗ η i = h ( λ s ⊗ id )( U ∗ ( h ⊗ x ) U ) ξ, η i . It follows that
Ad( U ∗ ) ◦ ( λ s ⊗ α s ) = ( λ s ⊗ id ) ◦ Ad( U ∗ ) .Now, suppose ( M, G, α ) is amenable. Then there exists a projection P : L ∞ ( G ) ⊗ M → M of norm one such that P ◦ ( λ s ⊗ α s ) = α s ◦ P , s ∈ G . Then Q := P ◦ Ad( U ) satisfies Q ◦ ( λ s ⊗ id ) = P ◦ Ad( U ) ◦ ( λ s ⊗ id ) = P ◦ ( λ s ⊗ α s ) ◦ Ad( U ) = α s ◦ Q, s ∈ G, and Q ( α ( x )) = P (1 ⊗ x ) = x . That is, Q : L ∞ ( G ) ⊗ M → M is a completely con-tractive G -equivariant left inverse to α . The composition α ◦ Q is then a G -equivariantprojection of norm one from L ∞ ( G ) ⊗ M onto the G -invariant von Neumann subalge-bra α ( M ) . By [2, Lemme 2.1], there exists an L ( G ) -equivariant projection of normone Ψ : L ∞ ( G ) ⊗ M → α ( M ) . The map α − ◦ Ψ : L ∞ ( G ) ⊗ M → M is then an L ( G ) -module left inverse to α . Noting that under the canonical identification CB ( L ( G ) , M ) =( L ( G ) b ⊗ M ∗ ) ∗ = L ∞ ( G ) ⊗ M , the action α is the canonical embedding j M : M →CB ( L ( G ) , M ) . Since M is a faithful L ( G ) -module, it follows that M is relativelyinjective over L ( G ) .Conversely, relative L ( G ) -injectivity of M entails the existence of a completely con-tractive L ( G ) -morphism Φ : L ∞ ( G ) ⊗ M → M such that Φ ◦ α = id M . It follows ina similar manner to the previous paragraph that there is a G -equivariant completely contractive left inverse Ψ to α . Define P : L ∞ ( G ) ⊗ M ∋ X Ψ( U ∗ XU ) ∈ M . Then P ◦ ( λ s ⊗ α s ) = Ψ ◦ Ad( U ∗ ) ◦ ( λ s ⊗ α s ) = Ψ( λ s ⊗ id ) ◦ Ad( U ∗ ) = α s ◦ P, and P (1 ⊗ x ) = Ψ( α ( x )) = x , so that P is a projection of norm witnessing the amenabilityof ( M, G, α ) . (cid:3) We now establish a perfect duality between amenability of W ∗ -dynamical systems ( M, G, α ) with M injective and A ( G ) -injectivity of the associated crossed product G ¯ ⋉ M .This generalizes [12, Corollary 5.3], which corresponds to M = C , in which case G ¯ ⋉ M = V N ( G ) . By virtually the same argument, the equivalence of (2) and (3) below persists tothe level of co-actions of co-amenable locally compact quantum groups on von Neumannalgebras. This will appear in future work. A similar result for actions of discrete quantumgroups on von Neumann algebras was obtained independently in [35]. Theorem 5.2.
Let ( M, G, α ) be a W ∗ -dynamical system with M injective. The followingconditions are equivalent:(1) ( M, G, α ) is amenable;(2) G ¯ ⋉ M is A ( G ) -injective;(3) M is L ( G ) -injective.Proof. (1) = ⇒ (2) By Proposition 5.1, there exists a completely contractive right L ( G ) -module map P : L ∞ ( G ) ⊗ M → M such that P ◦ α = id M . Define Θ( P ) : B ( L ( G )) ⊗ M →B ( L ( G )) ⊗ M by Θ( P ) = ( id ⊗ P )(∆ r ⊗ id ) , where ∆ r is the right extension of the co-product on L ∞ ( G ) (see (6)). It follows that h Θ( P )( T ) , ρ ⊗ ω i = h P ( T ⊲ ρ ) , ω i , T ∈ B ( L ( G )) ⊗ M, ρ ∈ T ( L ( G )) , ω ∈ M ∗ , where we abuse notation by letting ⊲ also denote the right action of T ( L ( G )) on the firstleg of B ( L ( G )) ⊗ M , that is, T ⊲ ρ = ( ρ ⊗ id ⊗ id )(∆ r ⊗ id )( T ) . With this representationit is clear that Θ( P ) is a right ⊲ -module map. The argument from [12, Proposition 4.2](applied to commutative quantum groups) amplifies to B ( L ( G )) ⊗ M and shows that Θ( P ) is a left b ⊲ -module map, where ρ b ⊲ T = ( id ⊗ ρ ⊗ id )( b ∆ r ⊗ id )( T ) . Hence, Θ( P ) isa left (equivalently, right) A ( G ) -module map where the A ( G ) -action is on the first legof B ( L ( G )) ⊗ M .The module property of P is equivalent to α ◦ P = ( id ⊗ P )(∆ ⊗ id ) , so for every X ∈ B ( L ( G )) ⊗ M , ( id ⊗ α )(Θ( P )( X )) = ( id ⊗ α )( id ⊗ P )(∆ r ⊗ id )( X )= ( id ⊗ id ⊗ P )( id ⊗ ∆ ⊗ id )(∆ r ⊗ id )( X )= ( id ⊗ id ⊗ P )( id ⊗ ∆ r ⊗ id )(∆ r ⊗ id )( X )= ( id ⊗ id ⊗ P )(∆ r ⊗ id ⊗ id )(∆ r ⊗ id )( X )= (∆ r ⊗ id )( id ⊗ P )(∆ r ⊗ id )( X )= (∆ r ⊗ id )(Θ( P )( X )) . On the one hand, this implies Θ( P )( X ) = ( id ⊗ P )( id ⊗ α )(Θ( P )( X )) = ( id ⊗ P )(∆ r ⊗ id )(Θ( P )( X )) = Θ( P )(Θ( P )( X ) , so that Θ( P ) is a projection of norm one. On the other hand, since G ¯ ⋉ M = { X ∈ B ( L ( G )) ⊗ M | ( id ⊗ α )( X ) = (∆ r ⊗ id )( X ) } , the chain of equalities above entails Θ( P )( B ( L ( G )) ⊗ M ) ⊆ G ¯ ⋉ M . Moreover, if X ∈ G ¯ ⋉ M then Θ( P )( X ) = ( id ⊗ P )(∆ r ⊗ id )( X ) = ( id ⊗ P )( id ⊗ α )( X ) = X, and we see that Θ( P ) is an A ( G ) -equivariant projection of norm one onto G ¯ ⋉ M .Since V N ( G ) admits an A ( G ) -invariant state [41, Theorem 4], it follows from (theproof of) [11, Theorem 5.5] that B ( L ( G )) is injective as a left, and hence right, op-erator A ( G ) -module. Since M is an injective von Neumann algebra, it follows that B ( L ( G )) ⊗ M is injective in mod A ( G ) . Hence, G ¯ ⋉ M is injective in mod A ( G ) via Θ( P ) . (2) = ⇒ (3) If G ¯ ⋉ M is injective in mod A ( G ) , then there exists an A ( G ) -equivariantprojection of norm one E : B ( L ( G )) ⊗ M → G ¯ ⋉ M . By the A ( G ) -module property wehave (see [12, Corollary 4.3]) E ( L ∞ ( G ) ⊗ M ) ⊆ L ∞ ( G ) ⊗ M ∩ G ¯ ⋉ M = α ( M ) . Hence P = α − ◦ E : L ∞ ( G ) ⊗ M → M is a completely contractive left inverse to α .Moreover, since E is an G ¯ ⋉ M -bimodule map, it is a V N ( G ) ⊗ -bimodule map, and itfollows from [11, Theorem 4.9] that E is a right L ( G ) -module map. Hence, P is a right L ( G ) -module left inverse to α , implying M is relatively injective in mod L ( G ) . Since M is also an injective von Neumann algebra, (3) follows from [12, Proposition 2.3]. (3) = ⇒ (1) If M is L ( G ) -injective then it is relatively L ( G ) -injective. Whence, ( M, G, α ) is amenable by Proposition 5.1. (cid:3) As an application of the above results, we obtain a generalized version of [1, Propo-sition 4.1] to the locally compact setting (when the cocycle is trivial). Note that it isprecisely the A ( G ) -module structure in condition (2) which allows for the generalizedversion: as remarked in [1, Remarque 4.4 (b)], the verbatim generalization of [1, Propo-sition 4.1] to the locally compact setting does not hold. Corollary 5.3.
Let ( M, G, α ) be a W ∗ -dynamical system. The following conditions areequivalent:(1) ( M, G, α ) is amenable;(2) There exists an A ( G ) -module projection of norm one from B ( L ( G )) ⊗ M onto G ¯ ⋉ M ;(3) For every extension ( N, G, β ) of ( M, G, α ) , there exists a G -equivariant projectionof norm one from N onto M .Proof. The equivalence of (1) and (2) follows from the proof of Theorem 5.2. The onlydifference, here, is that M is not necessarily injective. In this case, condition (2) impliesthe relative L ( G ) -injectivity of M , which implies (1) by Proposition 5.1. (1) = ⇒ (3) If ( N, G, β ) is any extension of ( M, G, α ) , then by definition (see [1,Définition 3.3] there is a linear projection of norm one from N onto M . By Proposition5.1, M is relatively L ( G ) -injective, so there is an L ( G ) -module projection P of norm one from N onto M . Since the continuous parts N c and M c of N and M coincide with h N ∗ L ( G ) i and h M ∗ L ( G ) i , respectively (see [39, Lemma 7.5.1]), it follows that P induces a G -equivariant projection of norm one from N c onto M c . Hence, (3) followsfrom [2, Lemme 2.1]. (3) = ⇒ (1) Simply apply (3) to the extension ( L ∞ ( G ) ⊗ M, G, λ ⊗ α ) of ( M, G, α ) . (cid:3) Buss, Echterhoff and Willett recently introduced the following notion of amenabil-ity for C ∗ -dynamical systems [9]: ( A, G, α ) is amenable if there exists a net of norm-continuous, compactly supported, positive type functions h i : G → Z ( A ′′ α ) such that k h i ( e ) k ≤ for all i , and h i ( s ) → weak* in A ′′ α , uniformly for s in compact subsets of G . It was subsequently shown by the authors of this paper that this notion coincideswith amenability of the universal enveloping system ( A ′′ α , G, α ) [4, Theorem 4.2], and,for commutative systems ( C ( X ) , G, α ) , coincides with topological amenability of thetransformation group ( G, X ) [4, Corollary 4.12].In [9, Proposition 7.10], it was shown that for C ∗ -dynamical systems ( A, G, α ) with A nuclear, amenability implies that A has the G -WEP (equivalently, the L ( G ) -WEPby Propositions 4.3 and 4.5), and that both conditions are equivalent if G is exact. Wenow complement this result by establishing a C ∗ -analogue of Theorem 5.2. Theorem 5.4.
Let ( A, G, α ) be a C ∗ -dynamical system with A nuclear. Consider thefollowing conditions:(1) ( A, G, α ) is amenable;(2) G ⋉ A has the A ( G ) -WEP;(3) A has the L ( G ) -WEP.Then (1) ⇒ (2) ⇒ (3) , and if G is exact, the conditions are equivalent.Proof. (1) = ⇒ (2) : If ( A, G, α ) is amenable, then by [4, Theorem 4.2] there exists a net ( h i ) of continuous compactly supported functions h i : G → CB ( A ) , whose correspondingHerz-Schur multipliers Θ( h i ) : G ⋉ A → G ⋉ A (see [34] and [4]) satisfy Θ( h i )( α × λ ( f )) = Z G α ( h i ( s )( f ( s )))( λ s ⊗ ds, f ∈ C c ( G, A ) , k Θ( h i ) k cb ≤ and Θ( h i ) → id G ⋉ A in the point norm topology. Note that each Θ( h i ) isan A ( G ) -morphism.As G ⋉ f A ∼ = G ⋉ A via the regular representation [9, Proposition 5.9], we have B ( G ⋉ f A ) = ( G ⋉ f A ) ∗ = ( G ⋉ A ) ∗ . Since each h i is compactly supported, and compactly supported elements of B ( G ⋉ A ) + liein A ( G ⋉ A ) + [39, Lemma 7.7.6], it follows that Θ( h i ) ∗ maps ( G ⋉ A ) ∗ into A ( G ⋉ A ) =( G ⋉ A ) ′′∗ . Representing A faithfully inside A ′′ α , it follows that ( G ⋉ A ) ′′ = G ¯ ⋉ A ′′ α .Hence, the adjoint of the co-restriction of Θ( h i ) ∗ defines a completely contractive A ( G ) -morphism Φ i : G ¯ ⋉ A ′′ α → ( G ⋉ A ) ∗∗ . Clustering the resulting net (Φ i ) to an A ( G ) -morphism Φ ∈ CB ( G ¯ ⋉ A ′′ α , ( G ⋉ A ) ∗∗ ) , andappealing to the point norm convergence Θ( h i ) → id G ⋉ A , it follows that the diagram G ¯ ⋉ A ′′ α G ⋉ A ( G ⋉ A ) ∗∗ Φ commutes. Since A is nuclear, A ′′ α is injective, and since ( A ′′ α , G, α ) is amenable, Theorem5.2 guarantees the A ( G ) -injectivity of G ¯ ⋉ A ′′ α . Hence, by Theorem 3.4, G ⋉ A has the A ( G ) -WEP. (2) = ⇒ (3) : If G ⋉ A has the A ( G ) -WEP then the canonical inclusion i : G ⋉ A ֒ → B ( L ( G )) ⊗ A ′′ α is A ( G ) -flat, implying the existence of a A ( G ) -equivariant weakexpectation B ( L ( G )) ⊗ A ′′ α → ( G ⋉ A ) ∗∗ , which, by Lemma 2.3 is a completely positive M ( G ⋉ A ) -bimodule map. Composing with the canonical A ( G ) -morphism ( G ⋉ A ) ∗∗ ։ G ¯ ⋉ A ′′ α we obtain a completely positive A ( G ) -morphism E : B ( L ( G )) ⊗ A ′′ α → G ¯ ⋉ A ′′ α (which is also an M ( G ⋉ A ) -bimodule map) making the following diagram commute: B ( L ( G )) ⊗ A ′′ α G ⋉ A G ¯ ⋉ A ′′ αE The left fundamental unitary of L ∞ ( G ) satisfies W ⊗ ∈ M ( C ( G ) ⊗ ∨ ( G ⋉ A )) by the“left handed version” of [31, Lemma 3.3]. Thus, for any f ∈ L ( G ) , ( f ⊗ id ⊗ id )( id ⊗ E )( W ⊗
1) = E (( f ⊗ id ⊗ id )( W ⊗ f ⊗ id ⊗ id )( W ⊗ , as ( f ⊗ id ⊗ id )( W ⊗ ∈ M ( G ⋉ A ) . Then ( id ⊗ E )( W ⊗
1) = W ⊗ , so that W ⊗ liesin the multiplicative domain of the unital completely positive map ( id ⊗ E ) . Then, asin [37, Theorem 3.2], the module property of unital completely positive maps over theirmultiplicative domains implies that E ( T · f ) = E (( f ⊗ id ⊗ id )( W ⊗ ∗ (1 ⊗ T )( W ⊗ f ⊗ id ⊗ id )( id ⊗ E )(( W ⊗ ∗ (1 ⊗ T )( W ⊗ f ⊗ id ⊗ id )(( W ⊗ ∗ (1 ⊗ E ( T ))( W ⊗ E ( T ) · f, where T · f is the action of L ( G ) on the left leg of B ( L ( G )) ⊗ A ′′ α .Now, the dual co-action b α : G ¯ ⋉ A ′′ α → V N ( G ) ⊗ ( G ¯ ⋉ A ′′ α ) on the crossed productsatisfies ( G ¯ ⋉ A ′′ α ) b α = α ( A ′′ α ) . Since A ( G ) acts trivially on L ∞ ( G ) , the A ( G ) -moduleproperty of E implies E | L ∞ ( G ) ⊗ A ′′ α : L ∞ ( G ) ⊗ A ′′ α → ( G ¯ ⋉ A ′′ α ) b α = α ( A ′′ α ) , so we obtain a right L ( G ) -module map Ψ : L ∞ ( G ) ⊗ A ′′ α → A ∗∗ via Φ ◦ α − ◦ E | L ∞ ( G ) ⊗ A ′′ α ,where Φ : A ′′ α → A ∗∗ is the L ( G ) -morphism constructed in the proof of Proposition 4.5.Since α ( A ) = α ( A ) ⊆ M ( G ⋉ A ) , E is an M ( G ⋉ A ) -bimodule map, and Φ | A = i A (seethe proof of Proposition 4.5) the following diagram commutes L ∞ ( G ) ⊗ A ′′ α A A ∗∗ . Ψ α | A i A Since A ′′ α is an injective von Neumann algebra and L ∞ ( G ) is L ( G ) -injective, it followsthat L ∞ ( G ) ⊗ A ′′ α is L ( G ) -injective. Hence, the canonical inclusion i A : A ֒ → A ∗∗ factorsthrough an injective L ( G ) -module, implying that A has the L ( G ) -WEP.Finally, if G is exact, the equivalence between the G -WEP and the L ( G ) -WEP(Proposition 4.3) together with [9, Proposition 7.10] shows that (3) ⇒ (1) . (cid:3) As a special case, we now establish the promised A ( G ) -equivariant analogue of Lance’stheorem for discrete groups [29]. Corollary 5.5.
Let G be a locally compact group. Then G is amenable if and only if C ∗ λ ( G ) has the A ( G ) -WEP.Proof. The forward direction follows immediately from (1) ⇒ (2) in Theorem 5.5 appliedto ( C , G, trivial) . Conversely, if C ∗ λ ( G ) has the A ( G ) -WEP, then there is an A ( G ) -weakexpectation Ψ : B ( L ( G )) → C ∗ λ ( G ) ∗∗ that restricts to the identity map on C ∗ λ ( G ) . ByLemma 2.3 Ψ is M ( C ∗ λ ( G )) − , hence G -equivariant. Composing with the canonical map C ∗ λ ( G ) ∗∗ → V N ( G ) , we get a G -equivariant A ( G ) -morphism Φ : B ( L ( G )) → V N ( G ) .Since A ( G ) acts trivially on L ∞ ( G ) , we have Φ( L ∞ ( G )) ⊂ V N ( G ) b ∆ = C . Hence Φ | L ∞ ( G ) defines an invariant mean, and G is amenable. (cid:3) Acknowledgements
The authors would like to thank Mehrdad Kalantar for his involvement in the earlystages of the project. The second author was partially supported by the NSERC Dis-covery Grant RGPIN-2017-06275.
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