Crossed products of C*-algebras with the weak expectation property
aa r X i v : . [ m a t h . OA ] J u l CROSSED PRODUCTS OF C ∗ -ALGEBRAS WITH THEWEAK EXPECTATION PROPERTY ANGSHUMAN BHATTACHARYA AND DOUGLAS FARENICK
Abstract. If α is an amenable action of a discrete group G on a unitalC ∗ -algebra A , then the crossed-product C ∗ -algebra A ⋊ α G has the weakexpectation property if and only if A has this property. Introduction
A weak expectation on a unital C ∗ -subalgebra B ⊂ B ( H ) is a unitalcompletely positive (ucp) linear map φ : B ( H ) → B ′′ (the double commutantof B ) such that φ ( b ) = b for every b ∈ B . A unital C ∗ -algebra A has theweak expectation property (WEP) if π ( A ) admits a weak expectation forevery faithful representation π of A on some Hilbert space H . Equivalently,if A ⊂ A ∗∗ ⊂ B ( H u ) denotes the universal representation of A , where A ∗∗ is the enveloping von Neumann algebra of A , then A has WEP if and onlyif there is a ucp map φ : B ( H u ) → A ∗∗ that fixes every element of A . Thenotion of weak expectation first arose in the work of C. Lance on nuclearC ∗ -algebras [4], where it was shown that every unital nuclear C ∗ -algebra hasWEP. Twenty years later E. Kirchberg established a number of importantproperties and characterisations of the weak expectation property in hispenetrating study of exactness [3].A C ∗ -algebra A has the quotient weak expectation property (QWEP) if A is a quotient of a C ∗ -algebra with WEP. The class of C ∗ -algebras with QWEPenjoys a number of permanence properties, many of which are enumeratedin [6, Proposition 4.1] and originate with Kirchberg [3]. For example, if A isa unital C ∗ -algebra with QWEP and if α is an amenable action of a discretegroup G on A , then the crossed product C ∗ -algebra A ⋊ α G has QWEP [6,Proposition 4.1(vi)].In contrast to QWEP, the weak expectation property appears to have fewpermanence properties. For example,
A ⊗ min B may fail to have WEP if A and B have WEP; one such example is furnished by A = B = B ( H ) [5]. Incomparison, if A and B are nuclear, then so is A ⊗ min B , and if A and B areexact, then so is A ⊗ min B [1, § Mathematics Subject Classification.
Primary 46L05; Secondary 46L06.
Key words and phrases. weak expectation property, amenable group, amenable action.The work of the second author is supported in part by the Natural Sciences and Engi-neering Research Council (NSERC) of Canada.
The purpose of this note is to establish the following permanence resultfor WEP (Theorem 2.1): if α is an amenable action of a discrete group G ona unital C ∗ -algebra A , then A ⋊ α G has the weak expectation property if andonly if A does. In this regard, the weak expectation property is consistentwith the analogous permanence results for nuclear and exact C ∗ -algebras [1,Theorem 4.3.4].Before turning to the proof, we note that Lance’s definition of WEP re-quires knowledge of all faithful representations of A . It is advantageous,therefore, to have alternate ways to characterise the weak expectation prop-erty. We mention two such ways below. Theorem 1.1. (Kirchberg’s Criterion [3])
A unital C ∗ -algebra A has theweak expectation property if and only if A ⊗ min C ∗ ( F ∞ ) = A ⊗ max C ∗ ( F ∞ ) . The second description is useful in cases where one desires to fix a par-ticular faithful representation of A . Theorem 1.2. (A Matrix Completion Criterion [2]) If A is a unital C ∗ -subalgebra of B ( H ) , then the following statements are equivalent: (1) A has the weak expectation property; (2) if, given p ∈ N and X , X ∈ M p ( A ) , there exist strongly positiveoperators A, B, C ∈ M p ( B ( H )) such that A + B + C = 1 and Y = A X X ∗ B X X ∗ C is strongly positive in M p ( B ( H )) , then there also exist ˜ A, ˜ B, ˜ C ∈M p ( A ) with the same property. By strongly positive one means a positive operator A for which there is areal δ > A ≥ δ The Main Result
Theorem 2.1. If α is an amenable action of a discrete group G on a unitalC ∗ -algebra A , then A ⋊ α G has the weak expectation property if and only if A does.Proof. We begin with two preliminary observations that are independent ofwhether A has WEP or not.The first observation is that, because α is an amenable action of G on A , the C ∗ -algebra A ⋊ α G coincides with the reduced crossed product C ∗ -algebra A ⋊ α, r G [1, Theorem 4.3.4(1)]. The second observation is that if ι : G → Aut( B ) denotes the trivial action of G on a unital C ∗ -algebra B ,then the action α ⊗ max ι of G on A ⊗ max B is amenable. (The action α ⊗ max ι ROSSED PRODUCTS OF C ∗ -ALGEBRAS WITH WEP 3 of G on A ⊗ max B satisfies α ⊗ max ι ( g )[ a ⊗ b ] = α g ( a ) ⊗ b for all g ∈ G, a ∈ A , b ∈ B [8, Remark 2.74].)To prove this second fact, using the properties that define α as an amenableaction [1, pp. 124-125], let { T i } i denote a net of finitely supported positive-valued functions T i : G → Z ( A ) (the centre of A ) such that P g ∈ G T i ( g ) = 1and lim i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X g ∈ G (cid:2) α g ( T i ( s − g )) − T i ( g ) (cid:3) ∗ (cid:2) α g ( T i ( s − g )) − T i ( g ) (cid:3)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) → s ∈ G. Define finitely supported positive-valued functions ˜ T i : G →Z ( A ⊗ max B ) by ˜ T i ( g ) = T i ( g ) ⊗ max B . Then P g ∈ G ˜ T i ( g ) = 1 A⊗ max B andthe limiting equation above holds with T i replaced with ˜ T i and α replacedwith α ⊗ max ι . Hence, the action α ⊗ max ι of G on A ⊗ max B is amenable.Assume now that A has the weak expectation property. By Kirchberg’sCriterion (Theorem 1.1), A ⊗ min C ∗ ( F ∞ ) = A ⊗ max C ∗ ( F ∞ ). Let ι : G → Aut (C ∗ ( F ∞ )) denote the trivial action of G on C ∗ ( F ∞ ). Thus, α ⊗ max ι isan amenable action. Hence,( A ⋊ α G) ⊗ min C ∗ ( F ∞ ) = ( A ⋊ α, r G) ⊗ min C ∗ ( F ∞ )= ( A ⊗ min C ∗ ( F ∞ )) ⋊ α ⊗ max ι, r G= (
A ⊗ max C ∗ ( F ∞ )) ⋊ α ⊗ max ι, r G= (
A ⊗ max C ∗ ( F ∞ )) ⋊ α ⊗ max ι G= ( A ⋊ α G) ⊗ max C ∗ ( F ∞ ) , where the final equality holds by [8, Lemma 2.75]. Another application ofKirchberg’s Criterion implies that A ⋊ α G has WEP.Conversely, assume that A ⋊ α G has the weak expectation property andthat A ⋊ α, r G is represented faithfully on a Hilbert space H . Thus, A ⊂ A ⋊ α, r G = A ⋊ α G ⊂ B ( H )also represents A faithfully on H . Let E : A ⋊ α, r G → A denote the canonicalconditional expectation of A ⋊ α, r G onto A [1, Proposition 4.1.9]. We nowuse the criterion of Theorem 1.2 for WEP.Suppose that p ∈ N , X , X ∈ M p ( A ), and A, B, C ∈ M p ( B ( H )) are suchthat A + B + C = 1 and the matrix Y = A X X ∗ B X X ∗ C ∈ M p ( B ( H )) A. BHATTACHARYA AND D. FARENICK is strongly positive. Because
A ⊂ A ⋊ α G and because A ⋊ α G has WEP,there are, by Theorem 1.2, ˜ A, ˜ B, ˜ C ∈ M p ( A ⋊ α G) such that˜ Y = ˜ A X X ∗ ˜ B X X ∗ ˜ C ∈ M p ( A ⋊ α G)is strongly positive and ˜ A + ˜ B + ˜ C = 1. Because ucp maps preserve strongpositivity, the matrix( E ⊗ id M )[ ˜ Y ] = E ( ˜ A ) X X ∗ E ( ˜ B ) X X ∗ E ( ˜ C ) ∈ M p ( A )is strongly positive and the diagonal elements sum to 1 ∈ M p ( A ). Thus, A ⊂ B ( H ) satisfies the criterion of Theorem 1.2 for WEP. (cid:3) A Direct Proof in the Case of Amenable Groups
The proof of Theorem 2.1 relies on the criteria for WEP given by The-orems 1.1 and 1.2, which seem far removed from the defining condition ofLance and thereby making the argument of Theorem 2.1 somewhat indi-rect. The purpose of this section is to present a more conceptual proofin the case of amenable discrete groups using Lance’s definition of WEPdirectly together with basic facts about amenable groups and C ∗ -algebras.In what follows, λ shall denote the left regular representation of G on theHilbert space ℓ (G) and e denotes the identity of G. Two properties thatan amenable group G is well known to have are:(i) A ⋊ α G = A ⋊ α, r G, for every unital C ∗ -algebra A , and(ii) G admits a Følner net—namely a net { F i } i ∈ Λ of finite subsets F i ⊂ Gsuch that, for every g ∈ G,lim i | F i ∩ gF i || F i | = 1 . (In fact the second property above characterises amenable groups.) Theorem 3.1. If α is an action of an amenable discrete group G on aunital C ∗ -algebra A , then A ⋊ α G has the weak expectation property if andonly if A does.Proof. Assume first that A ⋊ α G has the weak expectation property. Toshow that A has WEP, it is sufficient to show that if A is representedfaithfully as a unital C ∗ -subalgebra of B ( K ), for some Hilbert space K , andif π A u : A → B ( H A u ) is the universal representation of A , then there a ucpmap ω : B ( K ) → A ∗∗ such that ω ( a ) = π A u ( a ) for every a ∈ A .To this end, let A ⋊ α G ⊂ B ( H A ⋊ α G u ) be the universal representation of A ⋊ α G. Because A is unital, A is a unital C ∗ -subalgebra of A ⋊ α G. Hence,
A ⊂ A ⋊ α G ⊂ ( A ⋊ α G) ∗∗ ⊂ B ( H A ⋊ α G u ) ROSSED PRODUCTS OF C ∗ -ALGEBRAS WITH WEP 5 and we therefore, on the one hand, consider A as a unital C ∗ -subalgebra of B ( K ), where K = H A ⋊ α G u . On the other hand, A ⊂ A ⋊ α G = A ⋊ α, r G ⊂ B ( H A ⋊ α G u ) ⊗ min C ∗ r (G) ⊂ B ( H A ⋊ α G u ) ⊗ B (cid:0) ℓ (G) (cid:1) ⊂ B (cid:0) K ⊗ ℓ (G) (cid:1) , where ⊗ denotes the von Neumann algebra tensor product, yields anotherfaithful representation of A ⋊ α G—in this case, as a unital C ∗ -subalgebra of B (cid:0) K ⊗ ℓ (G) (cid:1) . Let ( A ⋊ α G) ′′ denote the double commutant of A ⋊ α G in B (cid:0) K ⊗ ℓ (G) (cid:1) .Using the vector state τ on B (cid:0) ℓ (G) (cid:1) defined by τ ( x ) = h xδ e , δ e i togetherwith the identity map id B ( K ) : B ( H A ⋊ α G u ) → B ( H A ⋊ α G u ), we obtain a normalucp map ψ = id B ( K ) ⊗ τ : B ( K ) ⊗ B (cid:0) ℓ (G) (cid:1) → B ( K ) . If E : A ⋊ α, r G → A denotes the conditional expectation of A ⋊ α, r G onto A whereby E (cid:16)P g a g λ g (cid:17) = a e , then, using the identification A ⋊ α G = A ⋊ α, r G,the restriction of ψ to ( A ⋊ α G) ′′ is a normal extension of ρ ◦ E , where ρ : A → B ( K ) is the faithful representation of A ⊂ B (cid:0)
K ⊗ ℓ (G) (cid:1) as aunital C ∗ -subalgebra of B ( K ). That is, we have the following commutativediagram: A ⋊ α G E −−−−→ A y y ρ ( A ⋊ α G) ′′ −−−−→ ψ B ( K ) . Because ψ is normal, the range of ψ | ( A ⋊ α G) ′′ is determined by ψ (cid:0) ( A ⋊ α G) ′′ (cid:1) = ( ψ ( A ⋊ α G))
SOT = ( ρ ( A )) SOT . In other words, the range of ψ | ( A ⋊ α G) ′′ is the strong-closure of the C ∗ -subalgebra A of A ⋊ α G in the enveloping von Neumann algebra ( A ⋊ α G) ∗∗ of A ⋊ α G. Therefore, by [7, Corollary 3.7.9], there is an isomorphism θ : ( ρ ( A )) SOT → A ∗∗ such that π A u = θ | ρ ( A ) .Now let π : ( A ⋊ α G) ∗∗ → ( A ⋊ α G) ′′ be the normal epimorphism thatextends the identity map of A ⋊ α G. Because A ⋊ α G has WEP, there is aucp map φ : B ( H A ⋊ α G u ) → ( A ⋊ α G) ∗∗ that fixes every element of A ⋊ α G.Hence, if ω = θ ◦ ψ | ( A ⋊ α G) ′′ ◦ π ◦ φ , then ω is a ucp map of B ( K ) → A ∗∗ for which ω ( a ) = π A u ( a ) for every a ∈ A . That is, A has WEP.Conversely, assume that A has the weak expectation property and that A is (represented faithfully as) a unital C ∗ -subalgebra of B ( H ) for some Hilbertspace H . Thus, we consider A and A ⋊ α G faithfully represented via
A ⊂ A ⋊ α G = A ⋊ α, r G ⊂ B (cid:0) H ⊗ ℓ (G) (cid:1) . A. BHATTACHARYA AND D. FARENICK
Note that u : G → B ( H A ⋊ α G u ) whereby u ( g ) = π A ⋊ α G u (1 ⊗ λ g ) is a uni-tary representation of G such that (1 ⊗ λ ) × π is the regular (covariant)representation associated with the dynamical system ( A , α, G).Let π A ⋊ α G u : A ⋊ α G → B ( H A ⋊ α G u ) be the universal representation of A ⋊ α G and define π : A → B ( H A u ) by π = π A ⋊ α G u |A ⋊ α G . Because π is afaithful representation of A and A has WEP, there is a ucp map φ : B ( H ) → π ( A ) ′′ ⊂ π A ⋊ α G u ( A ⋊ α G) ′′ such that φ ( π ( a )) = π ( a ) for every a ∈ A .As in [1, Proposition 4.5.1], if F ⊂ G is a finite nonempty subset andif p F ∈ B ( ℓ (G)) is the projection with range Span { δ f : f ∈ F } , then p F B ( ℓ (G)) p F is isomorphic to the matrix algebra M n for n = | F | , andso we obtain a ucp map φ F : B ( H ⊗ ℓ (G) → B ( H ) ⊗ M n defined by φ F ( x ) = (1 ⊗ p F ) x (1 ⊗ p F ). Next, let { e f,h } f,h ∈ F denote the matrix unitsof M n and define an action β of G on π ( A ) ′′ by β g ( y ) = u ( g ) y u ( g ) ∗ , for y ∈ π ( A ) ′′ . Observe that π ( A ) ′′ ⋊ β G ⊂ π A ⋊ α G u ( A ⋊ α G) ′′ .The linear map ψ F : π ( A ) ′′ ⊗ M n → A ⋊ β G for which ψ F ( y ⊗ e f,h ) = | F | − β f ( y ) u ( f h − ), for y ∈ π ( A ) ′′ , is a ucp map by the proof of [1, Lemma4.2.3]. Hence, θ F := ψ F ◦ ( φ ⊗ id M n ) ◦ φ F is a ucp map B (cid:0) H ⊗ ℓ (G) (cid:1) → π A ⋊ α G u ( A ⋊ α G) ′′ .Hence, if { F i } i is a Følner net in G and if θ i : B (cid:0) H ⊗ ℓ (G) (cid:1) → π A ⋊ α G u ( A ⋊ α G) ′′ is the ucp map constructed above, for each i , then the net { θ i } i admitsa cluster point θ relative to the point-ultraweak topology. Now, for every i ∈ Λ, aλ g ∈ A ⋊ α, r G, and ξ, η ∈ H A ⋊ α G , (cid:12)(cid:12) h (cid:0) θ ( aλ g ) − π A ⋊ α G u ( aλ g ) (cid:1) ξ, η i (cid:12)(cid:12) ≤ |h ( θ ( aλ g ) − θ F i ( aλ g )) ξ, η i| + (cid:12)(cid:12) h (cid:0) θ F i ( aλ g ) − π A ⋊ α G u ( aλ g ) (cid:1) ξ, η i (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) − | F i ∩ gF i || F i | (cid:19) h π A ⋊ α G u ( aλ g ) ξ, η i (cid:12)(cid:12)(cid:12)(cid:12) . Because θ is a cluster point of { θ i } i , we deduce that θ ( aλ g ) = π A ⋊ α G u ( aλ g ).Hence, by continuity, θ : B (cid:0) H ⊗ ℓ (G) (cid:1) → π A ⋊ α G u ( A ⋊ α G) ′′ is a ucp mapfor that extends the identity map on π A ⋊ α G u ( A ⋊ α G), which proves that A ⋊ α G has the weak expectation property. (cid:3) Remarks
The two proofs given in Theorems 2.1 and 3.1 of the implication A ⋊ α G has WEP ⇒ A has WEP depend only on the equality A ⋊ α G = A ⋊ α, r Grather than on the amenability of the action α or the group G itself.The arguments to establish Theorems 2.1 and 3.1 depend crucially on thefact that A is a unital C ∗ -algebra, and it would be of interest to know towhat extent such results remain true for non-unital C ∗ -algebras. ROSSED PRODUCTS OF C ∗ -ALGEBRAS WITH WEP 7 References [1] Nathanial P. Brown and Narutaka Ozawa, C ∗ -algebras and finite-dimensional approx-imations , Graduate Studies in Mathematics, vol. 88, American Mathematical Society,Providence, RI, 2008. MR 2391387 (2009h:46101)[2] Douglas Farenick, Ali S. Kavruk, and Vern I. Paulsen, C ∗ -algebras with the weakexpectation property and a multivariable analogue of Ando’s theorem on the numericalradius , J. Operator Theory (to appear).[3] Eberhard Kirchberg, On nonsemisplit extensions, tensor products and exactness ofgroup C ∗ -algebras , Invent. Math. (1993), no. 3, 449–489. MR 1218321 (94d:46058)[4] Christopher Lance, On nuclear C ∗ -algebras , J. Functional Analysis (1973), 157–176.MR 0344901 (49 An application of expanders to B ( l ) ⊗ B ( l ), J. Funct. Anal. (2003), no. 2, 499–510. MR 1964549 (2004d:46065)[6] , About the QWEP conjecture , Internat. J. Math. (2004), no. 5, 501–530.MR 2072092 (2005b:46124)[7] Gert K. Pedersen, C ∗ -algebras and their automorphism groups , London Mathemat-ical Society Monographs, vol. 14, Academic Press Inc. [Harcourt Brace JovanovichPublishers], London, 1979. MR MR548006 (81e:46037)[8] Dana P. Williams, Crossed products of C ∗ -algebras , Mathematical Surveys and Mono-graphs, vol. 134, American Mathematical Society, Providence, RI, 2007. MR 2288954(2007m:46003), Mathematical Surveys and Mono-graphs, vol. 134, American Mathematical Society, Providence, RI, 2007. MR 2288954(2007m:46003)