Amenable dynamical systems over locally compact groups
aa r X i v : . [ m a t h . OA ] A p r AMENABLE DYNAMICAL SYSTEMS OVER LOCALLY COMPACTGROUPS
ALEX BEARDEN AND JASON CRANN
Abstract.
We establish a Reiter property for amenable W ∗ -dynamical systems ( M, G, α ) over arbitrary locally compact groups. We also prove that a commutative C ∗ -dynamicalsystem ( C ( X ) , G, α ) is topologically amenable if and only if its universal W ∗ -dynamicalsystem is amenable. Our results answer three open questions from the literature; one ofAnantharaman–Delaroche from [3], and two from a recent preprint of Buss–Echterhoff–Willett [9]. Introduction
Amenability and its various manifestations have played an important role in the study ofdynamical systems and their associated operator algebras. Zimmer introduced a dynamicalversion of amenability [25] of an action of a locally compact group on a standard measurespace through a generalization of Day’s fixed point criterion, which has proven very usefulin ergodic theory and von Neumann algebras.Motivated by the structure of crossed products, Anantharaman-Delaroche generalizedZimmer’s notion of amenability to the level of W ∗ -dynamical systems ( M, G, α ) [1]. In[3] she characterized amenability of discrete W ∗ -dynamical systems through a Reiter typeproperty involving asymptotically G -invariant functions in C c ( G, M ) , generalizing Reiter’scondition for amenable groups. A related notion of topological amenability for commu-tative C ∗ -dynamical systems ( C ( X ) , G, α ) appeared in [19] in the context of topologicalgroupoids. This relation was made precise for discrete dynamical systems in [3] where itwas shown that ( C ( X ) , G, α ) is topologically amenable if and only if the associated bidualsystem ( C ( X ) ∗∗ , G, α ∗∗ ) is amenable. In this work we generalize the aforementioned tworesults from [3] to arbitrary locally compact groups (see Theorem 3.4 and Theorem 4.2). Thefirst result gives a Reiter property for amenable W ∗ -dynamical systems, while the secondshows that a commutative C ∗ -dynamical system ( C ( X ) , G, α ) is topologically amenable ifand only if the universal W ∗ -dynamical system of ( C ( X ) , G, α ) (in the sense of [11]) isamenable. These two results answer, in the affirmative, two questions recently posed byBuss–Echterhoff–Willett in [9].Although our approach parallels that of [3], there are significant technical hurdles in thelocally compact setting. One of primary importance is a continuous version of [3, Lemme3.1], whose validity was asked by Anantharaman–Delaroche in that paper. Using the theoryof vector-valued liftings we answer this question in the affirmative, and this forms the baseof our two main results. Another major hurdle is the passage from pointwise asymptotic G -invariance in Reiter’s property for amenable W ∗ -dynamical systems ( M, G, α ) to uniformasymptotic G -invariance on compacta. For this we establish a particular locally convex Mathematics Subject Classification.
Key words and phrases.
Dynamical systems, crossed products; locally compact groups; amenable actions. generalization of [17, Proposition 6.10] utilizing the Banach L ( G ) -module structure of theinjective tensor product L ( G ) ⊗ ε M c , where M c is the G -continuous part of M .Our results also shed insight into potential notions of (topologically) amenable co-actionsof arbitrary locally compact quantum groups, building on the existing notions in the discretecase [14, 23]. 2. Preliminaries
Dynamical Systems. A W ∗ -dynamical system ( M, G, α ) consists of a von Neumannalgebra M endowed with a homomorphism α : G → Aut( M ) of a locally compact group G such that for each x ∈ M , the map G ∋ s → α s ( x ) ∈ M is weak* continuous. Welet M c denote the unital C ∗ -subalgebra consisting of those x ∈ M for which s α s ( x ) isnorm continuous. We say that ( M, G, α ) is commutative if M is commutative. ( M, G, α ) is amenable if there exists a projection P : L ∞ ( G ) ⊗ M → M of norm one such that P ◦ ( λ s ⊗ α s ) = α s ◦ P , s ∈ G , where λ denotes the left translation action on L ∞ ( G ) . Forexample, ( L ∞ ( G ) , G, λ ) is always amenable, and G is amenable if and only if the trivialaction G y { x } is amenable, in which case P becomes a left invariant mean on L ∞ ( G ) .A C ∗ -dynamical system ( A, G, α ) consists of a C ∗ -algebra endowed with a homomorphism α : G → Aut( A ) of a locally compact group G such that for each a ∈ A , the map G ∋ s α s ( a ) ∈ A is norm continuous. In this paper we are mainly interested in commutative C ∗ -dynamical systems, where A = C ( X ) for a locally compact Hausdroff space X . Acommutative C ∗ -dynamical system ( C ( X ) , G, α ) is topologically amenable if there exists anet ( m i ) of continuous maps X → Prob( G ) (with respect to the weak* topology on Prob( G ) ),satisfying k δ s ∗ m i ( x ) − m i ( s · x ) k M ( G ) → uniformly on compacta of G × X . This notion coincides with amenability of the trans-formation groupoid G ⋊ X , as defined by Renault [19, Definition II.3.6]. Similar to the W ∗ -setting, ( C ( G ) , G, λ ) is always topologically amenable, and the trivial action G y { x } is topologically amenable if and only if G is amenable. By [7] and [4, Proposition 3.4], asecond countable locally compact group G is exact if and only if its action on its LUC-compactification β u G = spec(LUC( G )) is topologically amenable. Here, LUC( G ) is theunital C ∗ -algebra of left uniformly continuous functions on G .A function h : G × X → C is of positive type (with respect to G y X ) if for every x ∈ X ,every n ∈ N , s , ..., s n ∈ G and z , ...., z n ∈ C we have n X i,j =1 z i z j h ( s − i s j , s − i x ) ≥ . By [4, Proposition 2.5], ( C ( X ) , G, α ) is topologically amenable if and only if there exists anet ( h i ) of positive type functions in C c ( G × X ) tending to 1 uniformly on compact subsetsof G × X .2.2. Vector-Valued Integration.
Throughout this subsection S will be a locally compactHausdorff space with positive Radon measure µ .For a Banach space B , we let L ( S, B ) denote the space of (locally a.e. equivalence classesof) Bochner integrable functions f : S → B with the norm k f k = R S k f k dµ ( s ) . By thePettis Measurability Theorem and Bochner’s Theorem (see [20, Section 2.3]), for f : S → B supported on a σ -finite set, f ∈ L ( S, B ) if and only if f is weakly measurable, essentially separably valued, and satisfies R S k f ( s ) k dµ ( s ) < ∞ . In particular, there is a canonical map C c ( S, B ) → L ( S, B ) , where C c ( S, B ) denotes the continuous B -valued functions of compactsupport. It is well-known that L ( S, B ) ∼ = L ( S, µ ) ⊗ π B isometrically, where ⊗ π is theBanach space projective tensor product (see, e.g., [22, Proposition IV.7.14]).If M is a von Neumann algebra we have the following canonical identifications: ( L ∞ ( S, µ ) ⊗ M ) ∗ ∼ = L ( S, M ∗ ) ∼ = L ( S, µ ) ⊗ π M ∗ . (See [22, Proposition IV.7.14 and Theorem IV.7.17].) We remark that L ∞ ( S, µ ) ⊗ M does notnecessarily coincide with the space L ∞ ( S, M ) of essentially bounded w ∗ - locally measurablefunctions from S to M since we do not assume that M ∗ is separable (as one of our maintheorems concerns the case when M is the second dual of a commutative C ∗ -algebra). Onthe other hand, by [22, Theorem IV.7.17], for each F ∈ L ∞ ( S ) ⊗ M , there exists a weak*-measurable function ˜ F : S → M such that for every g ∈ L ( S, M ∗ ) , the function s F ( s ) , g ( s ) i is a measurable function on S , and h F, g i = Z S h ˜ F ( s ) , g ( s ) i dµ ( s ) for all g ∈ L ( S, M ∗ ) . In this case, we will say that ˜ F represents F , and usually abuse notation by omitting thetilde in the latter centered equation. There are some pitfalls that one must take care toavoid though—for example, if S = [0 , with Lebesgue measure, and M = ℓ ∞ [0 , is thespace of all bounded functions on [0 , , then the function f : S → M , f ( t ) = χ { t } , is nonzeroeverywhere, but f represents ∈ L ∞ ( S ) ⊗ M . Lemma 2.1. If M is a von Neumann algebra and ω ∈ M ∗ , there is a map ˜ ω : L ( S, M ) → L ( S, M ∗ ) determined by the formula h ˜ ω ( g )( s ) , x i = h ω, g ( s ) x i for g ∈ L ( S, M ) , s ∈ S , and x ∈ M . Moreover, k ˜ ω k ≤ k ω k .Proof. Using the canonical identifications, the map ˜ ω is just id ⊗ ω : L ( S ) ⊗ π M → L ( S ) ⊗ π M ∗ , where ω : M → M ∗ is the operator satisfying h ω ( y ) , x i = h ω, yx i for x, y ∈ M .The norm inequality is obvious. (cid:3) If A is a C ∗ -algebra, we let L ( S, A ) denote the Hilbert module completion of C c ( S, A ) under the A -valued inner product h ξ, η i = Z S ξ ( s ) η ( s ) ∗ dµ ( s ) , ξ, η ∈ C c ( S, A ) . Amenable W ∗ -dynamical systems In this section we establish a Reiter property for amenable W ∗ -dynamical systems, gen-eralizing [3, Théorème 3.3] from discrete groups to arbitrary locally compact groups. Werequire several preparations. The first is a continuous version of [3, Lemme 3.1].Given a locally compact Hausdorff space S with positive Radon measure µ , and a vonNeumann algebra M , we let K +1 ( S, Z ( M ) c ) = (cid:26) g ∈ C c ( S, Z ( M ) + c ) | Z S g ( s ) dµ ( s ) ≤ (cid:27) , where C c ( S, Z ( M ) + c ) is the space of norm continuous Z ( M ) + c -valued functions on S withcompact support. Let B M ( L ∞ ( S ) ⊗ M, M ) denote the Banach space of bounded M -bimodule ALEX BEARDEN AND JASON CRANN maps from L ∞ ( S ) ⊗ M to M , and let P denote the convex subset of B M ( L ∞ ( S ) ⊗ M, M ) given by the positive M -bimodule maps with norm ≤ . Every map P ∈ P is automaticallycompletely positive, so that k P k = k P (1) k .Each g ∈ K +1 ( S, Z ( M ) c ) gives rise to an element P g ∈ P by means of the formula h P g ( F ) , ω i = Z S h F ( s ) g ( s ) , ω i dµ ( s ) , F ∈ L ∞ ( S ) ⊗ M, ω ∈ M ∗ . The latter expression makes sense irrespective of the choice of representative of F since it isequal to h ˜ ω ( g ) , F i , viewing g ∈ L ( S, M ) . We will usually shorten the previously displayedformula by writing P g ( F ) = R S F ( s ) g ( s ) dµ ( s ) for F ∈ L ∞ ( S ) ⊗ M .Let P K := { P g | g ∈ K +1 ( S, Z ( M ) c ) } ⊆ P . Lemma 3.1.
Let S be a locally compact Hausdorff space with positive Radon measure µ andlet M be a commutative von Neumann algebra. Then P K is dense in P in the point-weak*topology of B ( L ∞ ( S ) ⊗ M, M ) .Proof. The majority of the proof follows that of [3, Lemme 3.1], but we include some detailsfor the convenience of the reader. First, B M ( L ∞ ( S ) ⊗ M, M ) = (( L ∞ ( S ) ⊗ M ⊗ πM M ∗ ) ∗ , where ⊗ πM is the M -bimodule Banach space projective tensor product. By definition ofthe projective tensor norm together with the Radon–Nikodym theorem, every element in ( L ∞ ( S ) ⊗ M ) ⊗ πM M ∗ is the equivalence class of an element of the form F ⊗ ϕ with F ∈ L ∞ ( S ) ⊗ M and ϕ ∈ M + ∗ , as shown in [3, Lemme 3.1]. By convexity it suffices to show that P is contained in the bipolar of P K . Let F ∈ L ∞ ( S ) ⊗ M and ϕ ∈ M + ∗ be such that Re h P g , F ⊗ ϕ i = Re ϕ (cid:18) Z S F ( s ) g ( s ) dµ ( s ) (cid:19) ≤ , g ∈ K +1 ( S, M c ) . If H = Re( F ) , then ϕ (cid:18) Z S H ( s ) g ( s ) dµ ( s ) (cid:19) ≤ , g ∈ K +1 ( S, M c ) . Let C denote the weak*-closure of { R S H ( s ) g ( s ) dµ ( s ) | g ∈ K +1 ( S, M c ) } in M . Given x , x ∈ C and a projection e ∈ M , we have x e + x (1 − e ) ∈ C . Indeed, pick nets ( g i ) , ( f j ) in K +1 ( S, M c ) such that x = w ∗ lim i Z S H ( s ) g i ( s ) dµ ( s ) , x = w ∗ lim j Z S H ( s ) f j ( s ) dµ ( s ) . Without loss of generality, we can assume the nets ( g i ) and ( f j ) have the same index set.Since M c is weakly dense in M ([16, Lemma 7.5.1]), by Kaplansky’s density theorem, picka net ( p k ) of positive operators in the unit ball of M c such that p k → e strongly (and henceweak*, by boundedness). Then x e + x (1 − e ) = w ∗ lim k w ∗ lim i Z S H ( s )( g i ( s ) p k + f i ( s )(1 − p k )) dµ ( s ) . Since g i (1 ⊗ p k ) + f i (1 ⊗ (1 − p k )) ∈ K +1 ( S, M c ) , combining the iterated limit into a single net,we see that x e + x (1 − e ) ∈ C . Then C is closed under finite suprema using the Stonianstructure of the spectrum of M , as in [3, Lemme 3.1]. Now, fix a ∗ -monomorphism ρ : L ∞ ( S, µ ) → ℓ ∞ ( S, µ ) , satisfying q ◦ ρ = id L ∞ ( S,µ ) , where ℓ ∞ ( S, µ ) is the C ∗ -algebra of bounded µ -measurable functions on S , and q : ℓ ∞ ( S, µ ) → L ∞ ( S, µ ) is the canonical quotient map. Such a lifting exists by [12, Corollary 2]. Fix s ∈ S . Then e s := ev s ◦ ρ ∈ L ∞ ( S, µ ) ∗ is a state on L ∞ ( S, µ ) . Let ( g si ) be a net of states in L ( S, µ ) approximating e s weak*. By a further approximation using norm density of C c ( S ) in L ( S, µ ) , we may take each g si ∈ C c ( S ) + with R S g si ( t ) dµ ( t ) ≤ . Viewing g si ∈ K +1 ( S, M c ) in the canonical way ( M c is unital), for every F ∈ L ∞ ( S ) ⊗ M , define a function F ρ : S → M by F ρ ( s ) = w ∗ lim i Z S g si ( t ) F ( t ) dµ ( t ) = w ∗ lim i P g si ( F ) . To see that this definition makes sense regardless of representative of F , note that for ω ∈ M ∗ , R S h g si ( t ) F ( t ) , ω i dµ ( t ) = h g si , (id ⊗ ω )( F ) i , where (id ⊗ ω )( F ) is the element in L ∞ ( S, µ ) defined h (id ⊗ ω )( F ) , g i = h F, g ⊗ ω i for g ∈ L ( S, µ ) .We claim that F ρ represents F . Indeed, to check measurability of ϕ g : s
7→ h F ρ ( s ) , g ( s ) i forall g ∈ L ( S, M ∗ ) , first take g to be a simple tensor in L ( S, M ∗ ) = L ( S ) ⊗ π M ∗ . In this case, ϕ g is a product of measurable functions, hence measurable. This implies the claim for allsimple functions g ∈ L ( S, M ∗ ) since these are sums of simple tensors. The claim for general g follows from this since a pointwise a.e.–limit of measurable functions is measurable. Theformula h F, g i = R S h F ( s ) , g ( s ) i dµ ( s ) for g ∈ L ( S, M ∗ ) is then readily checked for simpletensors g and improved to general g using the observation that F ρ is bounded.Since ( H ) ρ ( s ) ∈ C , we have ( H ) ρ ( s ) + = ( H ) ρ ( s ) ∨ ∈ C . Define m = sup s ∈ S ( H ) ρ ( s ) + ∈ M . Then by normality of ϕ , we have ϕ ( m ) ≤ . Since ( H ) ρ ( s ) ≤ m in M for all s , it followsthat H ≤ ⊗ m in L ∞ ( S ) ⊗ M . Indeed, if g ∈ L ( S, M ∗ ) + is a positive normal functionalon L ∞ ( S ) ⊗ M , then h H , g i = Z S h ( H ) ρ ( s ) , g ( s ) i dµ ( s ) ≤ Z S h m, g ( s ) i dµ ( s ) = h ⊗ m, g i . Thus, for every P ∈ P we have P ( H ) ≤ P (1 ⊗ m ) = mP (1) ≤ m, so that Re h P, F ⊗ ϕ i = ϕ ( P ( H )) ≤ ϕ ( m ) ≤ . Hence, P belongs to the bipolar of P K . (cid:3) Similar to [3], we consider the following two locally convex topologies on the Bochner space L ( S, M ) , where S and M are as in Lemma 3.1. The first, denoted τ n , is generated by thefamily of semi-norms { p ω | ω ∈ M + ∗ } , where p ω ( g ) = h ω, Z S | g ( s ) | dµ ( s ) i = Z S h| g ( s ) | , ω i dµ ( s ) . This is indeed well-defined since s
7→ | g ( s ) | is Bochner integrable whenever g is. The second,denoted τ F , is generated by the family of semi-norms { p F,ω | F ∈ L ∞ ( S ) ⊗ M, ω ∈ M + ∗ } , where p F,ω ( g ) = (cid:12)(cid:12)(cid:12)(cid:12) Z S h g ( s ) F ( s ) , ω i dµ ( s ) (cid:12)(cid:12)(cid:12)(cid:12) . To see that this is well-defined, define ˜ ω ( g ) : S → M ∗ by h x, ˜ ω ( g )( s ) i = h g ( s ) x, ω i for x ∈ M . Then by Lemma 2.1 ˜ ω ( g ) ∈ L ( S, M ∗ ) . A routine argument then shows that ALEX BEARDEN AND JASON CRANN s
7→ h F ( s ) , ˜ ω ( g )( s ) i = h g ( s ) F ( s ) , ω i is measurable, and integrability of this function is easyto check.Since p F,ω ( g ) ≤ k F k p ω ( g ) , it follows that τ n is stronger than τ F . Lemma 3.2.
Let V be a convex subset of L ( S, M ) such that every function in V is supportedon a σ -finite subset. Then V τ F = V τ n .Proof. Since τ n is stronger than τ F , it suffices to show that V τ F ⊆ V τ n . Let ( g i ) be a netin V converging to zero with respect to τ F . Then, by definition of τ F , ˜ ω ( g i ) → weakly in L ( S, M ∗ ) for all ω ∈ M + ∗ . By Mazur’s theorem, there exists a net ( g K,ε ) in V indexed byfinite subsets K of M + ∗ and ε > such that k ˜ ω ( g K,ε ) k L ( S,M ∗ ) < ε, ω ∈ K. For (an a.e.-representative of) g ∈ L ( S, M ) and s ∈ S , let g ( s ) = u s | g ( s ) | be the polardecomposition in M . Then, since h ω, | g ( s ) |i = |h ˜ ω ( g )( s ) , u ∗ s i| ≤ sup {|h ˜ ω ( g )( s ) , x i| : x ∈ M k·k≤ } = k ˜ ω ( g )( s ) k M ∗ for all ω ∈ M + ∗ and s ∈ S , we have p ω ( g ) = Z S h ω, u ∗ s g ( s ) i dµ ( s ) ≤ Z S k ˜ ω ( g )( s ) k M ∗ dµ ( s ) = k ˜ ω ( g ) k L ( S,M ∗ ) for all g ∈ L ( S, M ) . It follows that g K,ε → with respect to τ n . (cid:3) The next lemma will be used to upgrade pointwise asymptotic G -invariance in Reiter’sproperty to uniform asymptotic G -invariance on compacta. Such an upgrade in convergencecan be achieved through a modified version of [17, Proposition 6.10] using the Banach L ( G ) -module structure of the (Banach space) injective tensor product L ( S, µ ) ⊗ ε M c , where, asabove, M is commutative. In preparation, note that by combining the canonical maps C c ( S, M c ) → L ( S, µ ) ⊗ π M c → L ( S, µ ) ⊗ ε M c , we may view C c ( S, M c ) inside L ( S, µ ) ⊗ ε M c . In this case, k g k L ( S,µ ) ⊗ ε M c = k Z S | g ( s ) | dµ ( s ) k , g ∈ C c ( S, M c ) , which follows by identifying M c = C ( X ) for some compact Hausdorff space X and usingthe isometric identification L ( S, µ ) ⊗ ε C ( X ) = C ( X, L ( S, µ )) (see, e.g., [22, PropositionIV.7.3]). Thus, there is a canonical map K +1 ( S, M c ) → ( L ( S, µ ) ⊗ ε M c ) k·k ε ≤ . Lemma 3.3.
Let ( M, G, α ) be a commutative W ∗ -dynamical system. The following condi-tions are equivalent:(1) There exists a net ( g i ) in K +1 ( G, M c ) satisfying w ∗ lim i R G g i ( s ) ds = 1 and w ∗ lim i Z G | g i ( s ) − ( λ t ⊗ α t )( g i )( s ) | ds = 0 , for all t ∈ G. (2) There exists a net ( g i ) in K +1 ( G, M c ) satisfying w ∗ lim i R G g i ( s ) ds = 1 and w ∗ lim i Z G | g i ( s ) − ( λ t ⊗ α t )( g i )( s ) | ds = 0 , uniformly on compact subsets of G. Proof.
Since (2) clearly implies (1) , we only need to show (1) implies (2) . In preparation,note that for any g ∈ L ( G ) ⊗ ε M c , the function G ∋ t ( λ t ⊗ α t ) g ∈ L ( G ) ⊗ ε M c is bounded and continuous. This is clear if g ∈ L ( G ) ⊗ M c , and the general case followsfrom the fact that λ t ⊗ α t is an isometry for each t ∈ G . Hence, L ( G ) ⊗ ε M c becomes aBanach M ( G ) -module in the canonical fashion: µ ⋆ g = Z G ( λ t ⊗ α t ) g dµ ( t ) , where the integral is norm convergent (see, e.g., [6, IV.4, no.7]). In what follows the estimate Z G h ω, | f ⋆ g ( s ) |i ds ≤ k ω kk f ⋆ g k L ( G ) ⊗ ε M c ≤ k ω kk f k L ( G ) k g k L ( G ) ⊗ ε M c , will be used several times without comment, where ω ∈ M + ∗ , f ∈ L ( G ) , and g ∈ C c ( G, M c ) .Let C ⊆ G be compact, K ⊆ ( M ∗ ) + k·k =1 be finite, and ε > . Put C = C ∪ { e } , and δ = 10 − ε . Let f ∈ C c ( G ) be a state in L ( G ) . There exists a neighbourhood U of e suchthat k λ y f − f k L ( G ) < δ , whenever y ∈ U . Since C is compact, there exists a compactneighbourhood V of e such that t − V t ⊆ U for every t ∈ C . Hence for every r ∈ V and t ∈ C , k λ t − rt f − f k L ( G ) < δ . Let h = | V | − χ V . Then for t ∈ C we have k h ∗ ( λ t f ) − λ t f k L ( G ) = Z G (cid:12)(cid:12)(cid:12)(cid:12) Z G h ( r ) f ( t − r − s ) dr − Z G h ( r ) f ( t − s ) dr (cid:12)(cid:12)(cid:12)(cid:12) ds ≤ Z G h ( r ) (cid:18) Z G | f ( t − r − ts ) − f ( s ) | ds (cid:19) dr = Z V h ( r ) k λ t − rt f − f k L ( G ) dr < δ. Take a compact set C ′ ⊂ G such that R G \ C ′ f ( s ) ds < δ . Then C ′′ = C − C ′ is compact suchthat tC ′′ C ′ ⊇ C ′ for every t ∈ C and(1) Z G \ C ′′ f ( t − s ) ds = Z G \ tC ′′ C ′ f ( s ) ds ≤ Z G \ C ′ f ( s ) ds < δ. Pick a compact neighbourhood W of e such that k h ∗ δ t − h k L ( G ) < δ for every t ∈ W . Thenthere is an open neighbourhood W ′ of e for which W ′ W ′− ⊆ W . As C ′′ is compact, thereare c , ..., c m ∈ C ′′ such that C ′′ ⊆ ∪ mi =1 W ′ c i . Since each W ′ c i satisfies W ′ c i ( W ′ c i ) − ⊆ W ,there exists a finite partition { B j | j = 1 , ..., n } for C ′′ consisting of non-empty Borel setssuch that B j B − j ⊆ W for all j . For every j = 1 , ..., n choose b j ∈ B j . Then for all t ∈ B j (2) k h ∗ δ t − h ∗ δ b j k L ( G ) = k h ∗ δ tb − j − h k L ( G ) < δ. Now, the norm continuity of the action G y M ∗ implies that { ( α t ) ∗ ( ω ) | t ∈ V, ω ∈ K } is norm compact in M ∗ . Since on bounded subsets of M the weak* topology coincideswith uniform convergence on compact subsets, condition (1) implies the existence of g ∈ K +1 ( G, M c ) satisfying h ( α t ) ∗ ( ω ) , Z G | g ( s ) − ( δ b j ⋆ g )( s ) | ds i < δ, ALEX BEARDEN AND JASON CRANN for every j = 1 , ..., n , ω ∈ K and t ∈ V . Hence, for every t ∈ V , ω ∈ K , and j = 1 , ..., n , h ω, Z G | δ t ⋆ ( δ b j ⋆ g − g )( s ) | ds i = Z G h ω, | δ t ⋆ ( δ b j ⋆ g − g )( s ) |i ds = Z G h ω, δ t ⋆ ( | δ b j ⋆ g − g | )( s ) i ds = Z G h ω, α t ( | δ b j ⋆ g − g | )( t − s ) i ds = Z G h ω, α t ( | δ b j ⋆ g − g | )( s ) i ds = Z G h ( α t ) ∗ ( ω ) , | δ b j ⋆ g − g | ( s ) i ds< δ. It follows that h ω, Z G | h ⋆ δ b j ⋆ g ( s ) − h ⋆ g ( s ) | ds i = Z G h ω, | h ⋆ δ b j ⋆ g ( s ) − h ⋆ g ( s ) |i ds ≤ Z G h ω, Z G h ( t ) | δ t ⋆ ( δ b j ⋆ g − g )( s ) | dt i ds = Z G Z G h ( t ) h ω, | δ t ⋆ ( δ b j ⋆ g − g )( s ) |i dt ds = Z G h ( t ) (cid:18) Z G h ω, | δ t ⋆ ( δ b j ⋆ g − g )( s ) |i ds (cid:19) dt< δ. (3)Now, for t ∈ C and ω ∈ K , Z G h ω, | δ t ⋆ ( f ⋆ g )( s ) − h ⋆ δ t ⋆ ( f ⋆ g )( s ) |i ds = Z G h ω, | (( λ t f ) ⋆ g )( s ) − h ⋆ (( λ t f ) ⋆ g )( s ) |i ds = Z G h ω, | ( λ t f − h ∗ ( λ t f )) ⋆ g ( s ) |i ds = h ω, Z G | ( λ t f − h ∗ ( λ t f )) ⋆ g ( s ) | ds i≤ k ω kk ( λ t f − h ∗ ( λ t f )) ⋆ g k L ( G ) ⊗ ε M c ≤ k λ t f − h ∗ ( λ t f ) k L ( G ) k g k L ( G ) ⊗ ε M c < δ. (4)Also, if f ′ = λ t f , then f ′ is a state and applying inequalities (1), (2) and (3) we see that Z G h ω, | ( h ⋆ f ′ ⋆ g − h ⋆ g )( s ) |i ds ≤ Z G Z G f ′ ( r ) h ω, | ( h ⋆ δ r ⋆ g − h ⋆ g )( s ) |i dr ds = Z G (cid:18) Z G \ C ′′ f ′ ( r ) h ω, | ( h ⋆ δ r ⋆ g − h ⋆ g )( s ) |i dr + Z C ′′ f ′ ( r ) h ω, | ( h ⋆ δ r ⋆ g − h ⋆ g )( s ) |i dr (cid:19) ds< δ + Z G (cid:18) n X j =1 Z B j f ′ ( r ) h ω, | (( h ∗ δ r − h ∗ δ b j ) ⋆ g )( s ) |i dr + n X j =1 Z B j f ′ ( r ) h ω, | ( h ⋆ δ b j ⋆ g − h ⋆ g )( s ) |i dr (cid:19) ds = 2 δ + n X j =1 Z B j f ′ ( r ) Z G h ω, | (( h ∗ δ r − h ∗ δ b j ) ⋆ g )( s ) |i ds dr + n X j =1 Z B j f ′ ( r ) Z G h ω, | ( h ⋆ δ b j ⋆ g − h ⋆ g )( s ) |i ds dr< δ. (5)Finally, let t ∈ C . Since t ∈ C and e ∈ C , by (4) and (5) we have Z G h ω, | ( δ t ⋆ ( f ⋆ g ) − f ⋆ g )( s ) |i ds ≤ Z G h ω, | ( f ′ ⋆ g − h ⋆ ( f ′ ⋆ g ))( s ) |i ds + Z G h ω, | ( h ⋆ ( f ′ ⋆ g )) − h ⋆ g )( s ) |i ds + Z G h ω, | ( h ⋆ g − h ⋆ f ⋆ g )( s ) |i ds + Z G h ω, | ( h ⋆ f ⋆ g − f ⋆ g )( s ) |i ds< δ + 4 δ + 4 δ + δ = ε. It follows that the net ( f ⋆ g i ) lies in K +1 ( G, M c ) and satisfies w ∗ lim i Z G | f ⋆ g i ( s ) − ( λ t ⊗ α t )( f ⋆ g i )( s ) | ds = 0 , uniformly for t in compact subsets of G . In addition, since w ∗ lim i Z G g i ( s ) ds = 1 , and { ( α t ) ∗ ( ω ) | t ∈ supp( f ) , ω ∈ K } is norm compact in M ∗ , we have Z G h ω, f ⋆ g i ( s ) i ds = Z G Z G f ( t ) h ω, δ t ⋆ g i ( s ) i dt ds = Z G Z G f ( t ) h ( α t ) ∗ ( ω ) , g i ( s ) i dt ds = Z G f ( t ) Z G h ( α t ) ∗ ( ω ) , g i ( s ) i ds dt → Z G f ( t ) dt = 1 = h ω, i for all ω ∈ K . Hence, the net ( f ⋆ g i ) satisfies condition (2). (cid:3) We are now in position to generalize [3, Théorème 3.3] to locally compact groups. Theequivalences in the next theorem were independently obtained for exact locally compactgroups using different techniques by Buss–Echterhoff–Willett in the recent preprint [9]. Theauthors of that paper ask whether the equivalence holds for arbitrary locally compact groups[9, Question 9.1]. We therefore answer this question in the affirmative.
Theorem 3.4.
Let ( M, G, α ) be a W ∗ -dynamical system. The following conditions areequivalent:(1) There exists a net ( h i ) of positive type functions in C c ( G, Z ( M ) c ) such that(a) h i ( e ) ≤ for all i ;(b) lim i h i ( t ) = 1 weak*, uniformly on compact subsets.(2) There exists a net ( ξ i ) in C c ( G, Z ( M ) c ) such that(a) h ξ i , ξ i i ≤ for all i ;(b) h ( λ t ⊗ α t ) ξ i , ξ i i → weak*, uniformly on compact subsets.(3) There exists a net ( g i ) in K +1 ( G, Z ( M ) c ) such that(a) R G g i ( s ) ds → weak*;(b) R G | ( λ t ⊗ α t ) g i ( s ) − g i ( s ) | ds → weak*, uniformly on compact subsets.(4) There exists a G -equivariant projection of norm one from L ∞ ( G ) ⊗ M onto M .(5) There exists a G -equivariant projection of norm one from L ∞ ( G ) ⊗ Z ( M ) onto Z ( M ) .Proof. (1) implies (2) by [3, Proposition 2.5] and density of C c ( G, Z ( M ) c ) in L ( G, Z ( M ) c ) ,and (2) implies (1) is obvious by taking h i ( t ) = h ( λ t ⊗ α t ) ξ i , ξ i i (noting that the compactsupport of ξ i implies the range of h i indeed lies in the norm closed subalgebra Z ( M ) c ) . (2) ⇔ (3) follows more or less immediately from [3, Lemme 3.2] applied to the commutative C ∗ -dynamical system ( Z ( M ) c , G, α ) . (3) ⇒ (4) : Suppose there exists a net ( g i ) in K +1 ( G, Z ( M ) c ) satisfying conditions a ) and b ) above. By the properties of ( g i ) , each P g i is a positive contraction. Passing to asubnet we may assume that ( P g i ) converges weak* to some P in B ( L ∞ ( G ) ⊗ M, M ) , which isnecessarily a projection of norm one by (a).Fix t ∈ G , F ∈ ( L ∞ ( G ) ⊗ M ) + and ω ∈ M + ∗ . Then h P ( λ t ⊗ α t ( F )) , ω i = lim i Z G h g i ( s )( λ t ⊗ α t )( F )( s ) , ω i ds = lim i Z G h α t ( α t − ( g i ( s )) F ( t − s )) , ω i ds = lim i Z G h α t − ( g i ( s )) F ( t − s ) , ( α t ) ∗ ( ω ) i ds = lim i Z G h α t − ( g i ( ts )) F ( s ) , ( α t ) ∗ ( ω ) i ds = lim i Z G h (( λ t − ⊗ α t − ) g i )( s ) F ( s ) , ( α t ) ∗ ( ω ) i ds. Since ( λ t ⊗ α t ) g i ( s ) − g i ( s ) ∈ Z ( M ) c is self-adjoint for each s ∈ G , we have h (( λ t − ⊗ α t − ) g i − g i )( s ) F ( s ) , ( α t ) ∗ ( ω ) i = h p F ( s )(( λ t − ⊗ α t − ) g i − g i )( s ) p F ( s ) , ( α t ) ∗ ( ω ) i≤ h p F ( s ) | (( λ t − ⊗ α t − ) g i − g i )( s ) | p F ( s ) , ( α t ) ∗ ( ω ) i = h| (( λ t − ⊗ α t − ) g i − g i )( s ) | F ( s ) , ( α t ) ∗ ( ω ) i≤ k F kh| (( λ t − ⊗ α t − ) g i − g i )( s ) | , ( α t ) ∗ ( ω ) i , for every s, t ∈ G . Property ( b ) of ( g i ) then implies that h P ( λ t ⊗ α t ( F )) , ω i = lim i Z G h g i ( s ) F ( s ) , ( α t ) ∗ ( ω ) i = h α t ( P ( F )) , ω i , which yields (4). (4) ⇒ (5) is obvious by restriction, using the ⊗ M − M -bimodule property of projectionsof norm one L ∞ ( G ) ⊗ M → M .It remains to show that (5) implies (3) . Let P : L ∞ ( G ) ⊗ Z ( M ) → Z ( M ) be a G -equivariant projection of norm one. By Lemma 3.1 applied to the commutative von Neumannalgebra Z ( M ) , P lies in the point-weak* closure of P K . Hence, there is a net ( g i ) of functionsin K +1 ( G, Z ( M ) c ) satisfying P ( F ) = w ∗ lim i P g i ( F ) = w ∗ lim i Z G g i ( s ) F ( s ) ds, F ∈ L ∞ ( G, Z ( M )) . In particular, w ∗ lim i R G g i ( s ) ds . The G -equivariance of P implies that lim i Z G h g i ( s )( λ t ⊗ α t )( F )( s ) , ω i ds = lim i Z G h g i ( s ) F ( s ) , ( α t ) ∗ ( ω ) i ds for all F ∈ L ∞ ( G, Z ( M )) , ω ∈ Z ( M ) ∗ and t ∈ G . But Z G h g i ( s )( λ t ⊗ α t )( F )( s ) , ω i ds = Z G h ( λ t − ⊗ α t − )( g i )( s ) F ( s ) , ( α t ) ∗ ( ω ) i ds, as shown above, so it follows that (( λ t ⊗ α t )( g i ) − g i ) → with respect to τ F (on L ( G, Z ( M )) )for all t ∈ G . Just as in [3, pg. 307], one can use Lemma 3.2 applied to V = K +1 ( G, Z ( M ) c ) and an argument involving direct sums of Z ( M ) with copies of L ( G, Z ( M )) to show theexistence of a net ( g j ) in K +1 ( G, Z ( M ) c ) such that (( λ t ⊗ α t )( g j ) − g j ) → with respect to τ n for all t ∈ G , which implies a pointwise version of property 3(b), and R G g j ( s ) ds w ∗ −→ in Z ( M ) , which is property 3(a). Property 3(b) then follows from Lemma 3.3. (cid:3) As a corollary to Theorem 3.4 (and its proof), we obtain a different proof of the fact thata W ∗ -dynamical system ( M, G, α ) over an arbitrary locally compact group G is amenable ifand only if the restricted action ( Z ( M ) , G, α ) is amenable [2, Corollaire 3.6]4. Amenable commutative C ∗ -dynamical systems A discrete commutative C ∗ -dynamical system ( C ( X ) , G, α ) is topologically amenable ifand only if the corresponding bidual system ( C ( X ) ∗∗ , G, α ∗∗ ) is amenable [3, Théorème 4.9](see also [3, Remarque 4.10]). In the locally compact setting, the bidual action need not gen-erate a W ∗ -dynamical system, as the pertinent weak* continuity condition can fail. However,as shown in [11], there is a unique universal W ∗ -dynamical system associated ( C ( X ) , G, α ) ,which coincides with the bidual system in the discrete case. It is therefore natural to exam-ine the relationship between topological amenability of ( C ( X ) , G, α ) and amenability of itsuniversal W ∗ -dynamical system. In this section we establish the equivalence between thesetwo properties. This generalizes [3, Théorème 4.9] from discrete groups to locally compactgroups, and answers the recently posed [9, Question 9.4] in the affirmative. We begin with an overview of the universal W ∗ -dynamical system ( M, G, α ) associated to ( C ( X ) , G, α ) from [11], taking an L ( G ) -module perspective. In [9], they study ( M, G, α ) (for general ( A, G, α ) ) from a different, equivalent perspective.First, C ( X ) becomes a right L ( G ) -module in the canonical fashion by slicing the corre-sponding representation α : C ( X ) ∋ h ( s α s − ( h )) ∈ C b ( G, C ( X )) ⊆ L ∞ ( G ) ⊗ C ( X ) ∗∗ . By duality we obtain a left L ( G ) -module structure on M ( X ) via α ∗ | L ( G ) ⊗ π M ( X ) : L ( G ) ⊗ π M ( X ) → M ( X ) . Then G acts in a norm-continuous fashion on the essential submodule M ( X ) c := h L ( G ) ∗ M ( X ) i , where h·i denotes closed linear span. The same argument in [16, Lemma 7.5.1] shows that M ( X ) c coincides with the norm-continuous part of M ( X ) , hence the notation. This factwas also noted by Hamana in [10, Proposition 3.4(i)]. We therefore obtain a point-weak*continuous action of G on the dual space M ( X ) ∗ c by positive surjective isometries. Clearly(6) M ( X ) ∗ c ∼ = C ( X ) ∗∗ /M ( X ) ⊥ c isometrically and weak*-weak* homeomorphically as right L ( G ) -modules, where the canon-ical L ( G ) -module structure on C ( X ) ∗∗ is obtained by slicing the normal cover of α , whichis the normal ∗ -homomorphism e α = ( α ∗ | L ( G ) ⊗ π M ( X ) ) ∗ : C ( X ) ∗∗ → L ∞ ( G ) ⊗ C ( X ) ∗∗ . Note that e α | C b ( X ) is the unique strict extension of α to C b ( X ) = M ( C ( X )) , and is thereforeinjective [13, Proposition 2.1]. However, on C ( X ) ∗∗ , e α can have a large kernel. On the onehand, its kernel is of the form (1 − z ) C ( X ) ∗∗ for some projection z ∈ C ( X ) ∗∗ . On theother hand, by definition of the L ( G ) -action on C ( X ) ∗∗ , Ker( e α ) = M ( X ) ⊥ c . It follows that M ( X ) ∗ c is isometrically weak*-weak* order isomorphic to zC ( X ) ∗∗ , where we equip M ( X ) ∗ c with the quotient operator system structure from C ( X ) ∗∗ . We can therefore transport thepoint-weak* continuous G action on M ( X ) ∗ c to M := zC ( X ) ∗∗ , yielding a W ∗ -dynamicalsystem ( M, G, α ) , where α : G → Aut( zC ( X ) ∗∗ ) is given by α t ( zF ) = z (( α t ) ∗∗ ( F )) , F ∈ C ( X ) ∗∗ , t ∈ G. The associated normal ∗ -homomorphism α : M → L ∞ ( G ) ⊗ M is ( id ⊗ Ad( z )) ◦ e α | M . Hence, the L ( G ) -action on M satisfies ( zF ) ∗ f = ( f ⊗ id ) α ( F ) = Ad( z )(( f ⊗ id ) e α ( F )) = z ( F ∗ f ) , for f ∈ L ( G ) , and F ∈ C ( X ) ∗∗ . We emphasize that with this structure M is not necessarilyan L ( G ) -submodule of C ( X ) ∗∗ , rather Ad( z ) : C ( X ) ∗∗ → M is an L ( G ) -quotient map.There is, however, a G -equivariant isometry j z : M → C ( X ) ∗∗ such that q ◦ j z = id M ,where q = i ∗ is the adjoint of the inclusion i : M ( X ) c ⊆ M ( X ) : First consider the positive L ( G ) -morphism Φ : C ( X ) ∗∗ ∋ F ( E ⊗ id ) e α ( F ) ∈ C ( X ) ∗∗ , where E ∈ L ∞ ( G ) ∗ is a weak* cluster point of a fixed bounded approximate identity ( f k ) for L ( G ) consisting of states. Since h Φ( F ) , µ i = lim k h F, f k ∗ µ i , F ∈ C ( X ) ∗∗ , µ ∈ M ( X ) , it follows that M ( X ) ⊥ c = (1 − z ) C ( X ) ∗∗ ⊆ Ker(Φ) , so we obtain an induced positivecontractive L ( G ) -morphism j z : M → C ( X ) ∗∗ satisfying h q ( j z ( zF )) , f ∗ µ i = h j z ( zF ) , f ∗ µ i = lim k h F, f k ∗ f ∗ µ i = h zF, f ∗ µ i , for each F ∈ C ( X ) ∗∗ , f ∈ L ( G ) and µ ∈ M ( X ) . Whence, j z is an isometry. Also, since Φ ◦ ( α t ) ∗∗ = ( α t ) ∗∗ ◦ Φ , as is easily checked, it follows that j z is G -equivariant.Finally, as e α | C b ( X ) is an injective ∗ -homomorphism, for all F ∈ C b ( X ) we have k F k = k e α ( F ) k = k e α ( zF ) k = k zF k . It follows that
Ad( z ) : C b ( X ) ֒ → M is a G -equivariant isometry, and that the following G -equivariant diagram commutes(7) MC b ( X ) C ( X ) ∗∗ . j z Ad( z ) In the proof of the main result, we require a slight generalization of the fact that a linearfunctional on C b ( X ) is strictly continuous if and only if it is σ ( C b ( X ) , M ( X )) -continuous.If S is a compact Hausdorff space, we let τ st,uc and τ w ∗ ,uc be the topologies on C ( S, C b ( X )) induced by the family of semi-norms p f ( h ) = sup s ∈ S k h ( s ) f k C ( X ) , f ∈ C ( X ) , and p µ ( h ) = sup s ∈ S |h h ( s ) , µ i| , µ ∈ M ( X ) , respectively. Lemma 4.1.
Let S and X be locally compact Hausdorff spaces, with S compact. A linearfunctional on C ( S, C b ( X )) is τ st,uc -continuous if and only if it is τ w ∗ ,uc -continuous.Proof. Given h ∈ C ( S, C b ( X )) and f ∈ C ( X ) , by compactness of S and continuity of h ,there exists s ∈ S such that p f ( h ) = k h ( s ) f k . Pick µ ∈ M ( X ) k·k =1 for which k h ( s ) f k = |h h ( s ) f, µ i| = |h h ( s ) , f · µ i| . Since p f · µ ( h ) = sup s ∈ S |h h ( s ) , f · µ i| ≤ sup s ∈ S k h ( s ) f k = |h h ( s ) , f · µ i| , it follows that p f ( h ) = p f · µ ( h ) .Conversely, given h ∈ C ( S, C b ( X )) and µ ∈ M ( X ) , by Cohen’s factorization theorem forBanach modules there exists f ∈ C ( X ) and ν ∈ M ( X ) for which µ = f · ν . Then p µ ( h ) = sup s ∈ S |h h ( s ) f, ν i| ≤ k ν k p f ( h ) . The claim follows. (cid:3)
Theorem 4.2.
A commutative C ∗ -dynamical system ( C ( X ) , G, α ) is topologically amenableif and only its universal W ∗ -dynamical system ( M, G, α ) is amenable.Proof. Suppose ( C ( X ) , G, α ) is topologically amenable. By [4, Proposition 2.2(3)], thereexists a net ( g i ) in C c ( G × X ) + such that(8) lim i Z G g i ( t, x ) dt = 1 uniformly on compact subsets of X , and(9) lim i Z G | g i ( st, sx ) − g i ( t, x ) | dt = 0 uniformly on compact subsets of G × X . By an appropriate scaling, we may assume theinduced functions g i ∈ C c ( G, C ( X )) satisfy R G g i ( s ) ds ≤ . An elementary measure theoryargument shows that for a bounded net ( f i ) in C b ( X ) and f ∈ C b ( X ) , if f i → f uniformlyon compact subsets of X , then f i w ∗ −→ f in C ( X ) ∗∗ (where we view C b ( X ) ⊆ C ( X ) ∗∗ in thecanonical way). It follows here that the net ((1 ⊗ z )˜ g i ) ∈ K +1 ( G, M ) satisfies Theorem 3.4(3). Thus, ( M, G, α ) is amenable.Conversely, assume that ( M, G, α ) is amenable. By Theorem 3.4 (2), there exists a net ( ξ i ) in C c ( G, M c ) such that h ξ i , ξ i i ≤ for all i and w ∗ lim h ( λ t ⊗ α t ) ξ i , ξ i i = 1 , for t ∈ G uniformly on compact subsets. Then ξ ′ i = ( id ⊗ j z ) ξ i defines a net in C c ( G, C ( X ) ∗∗ ) .By [24, Proposition 2.2, Lemma 2.4], C c ( G, C ( X ) ∗∗ ) injects contractively into the self-dualcompletion L ( G, C ( X )) ′′ := B C ( X ) ( L ( G, C ( X )) , C ( X ) ∗∗ ) of the Hilbert C ( X ) -module L ( G, C ( X )) via ξ (cid:18) η
7→ h η, ξ i C ( X ) ∗∗ = Z G ξ ( s ) ∗ η ( s ) ds (cid:19) . Thus, we may view the net ( ξ ′ i ) inside the unit ball of L ( G, C ( X )) ′′ . By the Kaplanksydensity theorem for Hilbert C ∗ -modules [24, Corollary 2.7], for each i , there exists a net ( ξ i,j ) in C c ( G, C ( X )) k·k L G,C X )) ≤ such that h µ, h ξ ′ i − ξ i,j , ξ ′ i − ξ i,j ii / → , µ ∈ M ( X ) + . In particular, when µ ∈ M ( X ) + c = zM ( X ) + , the commutative diagram (7) implies that h µ, h ξ i − zξ i,j , ξ i − zξ i,j ii / → , µ ∈ M ( X ) + c , where zξ i,j is shorthand for (1 ⊗ z ) ξ i,j . Then, as in [3, Théorème 4.9], |h µ, h ( λ t ⊗ α t ) zξ i,j , zξ i,j ii − h µ, h ( λ t ⊗ α t ) ξ i , ξ i ii|≤ |h µ, h ( λ ⊗ α t ) zξ i,j , zξ i,j − ξ i ii| + |h µ, h ( λ t ⊗ α t )( zξ i,j − ξ i ) , ξ i ii|≤ h µ, h ( λ t ⊗ α t ) zξ i,j , zξ i,j − ξ i i ∗ h ( λ t ⊗ α t ) zξ i,j , zξ i,j − ξ i ii / + h µ, h ( λ t ⊗ α t )( zξ i,j − ξ i ) , ξ i i ∗ h ( λ t ⊗ α t )( zξ i,j − ξ i ) , ξ i ii / ≤ kh zξ i,j , zξ i,j ik / h µ, h zξ i,j − ξ i , zξ i,j − ξ i ii / + hkh ξ i , ξ i ik / h ( α t ) ∗ ( µ ) , h zξ i,j − ξ i , zξ i,j − ξ i ii / ≤ h µ, h zξ i,j − ξ i , zξ i,j − ξ i ii / + h ( α t ) ∗ ( µ ) , h zξ i,j − ξ i , zξ i,j − ξ i ii / . Once again using the norm continuity of the predual action G y M ∗ and the equivalencebetween weak* convergence on bounded subsets of M and uniform convergence on compactsets, the above estimates imply that h µ, h ( λ t ⊗ α t ) zξ i,j , zξ i,j ii → , µ ∈ M ( X ) c , uniformly for t in compact subsets of G .Let h i,j ( t ) = h ( λ t ⊗ α t ) ξ i,j , ξ i,j i , t ∈ G . Then h i,j is a continuous compactly supported C ( X ) + -valued function of positive type satisfying h i,j ( e ) ≤ and h zµ, h i,j ( t ) i = h zµ, h ( λ t ⊗ zα t ) ξ i,j , zξ i,j ii = h zµ, h ( λ t ⊗ α t ) zξ i,j , zξ i,j ii → h zµ, i , for every µ ∈ M ( X ) , uniformly for t in compact subsets of G . That is, the net zh i,j ( t ) → z weak* in C ( X ) ∗∗ , uniformly on compacta. Fix a non-negative function g ∈ C c ( G ) whichintegrates to 1. Then ( h i,j ∗ (1 ⊗ g )) is also net of compactly supported C ( X ) + -valuedfunctions of positive type. Moreover, ( h i,j ∗ (1 ⊗ g ))( e ) = Z G g ( s ) α s − ( h i,j ( e )) ds ≤ Z G g ( s ) ds = 1 and h µ, ( h i,j ∗ (1 ⊗ g ))( t ) i = h g ∗ µ, h i,j ( t ) i = h g ∗ µ, zh i,j ( t ) i → h g ∗ µ, z i = h µ, i , for every µ ∈ M ( X ) , uniformly on compacta. Applying Lemma 4.1 to the restriction h i,j | K ∈ C ( K, C b ( X )) for every compact subset K ⊆ G , the standard convexity argument yields anet ( h k ) of continuous compactly supported C ( X ) -valued functions of positive type on G for which h i ( e ) ≤ and h i ( t ) → strictly, uniformly on compacta. Since the strict topologyand the topology of uniform convergence on compacta agree on bounded subsets of C b ( X ) [8, Theorem 1], ( C ( X ) , G, α ) is topologically amenable by [4, Proposition 2.5(3)]. (cid:3) Remark 4.3.
Results of Suzuki [21] imply that Theorem 4.2 does not extend to actions oflocally compact (even discrete) groups on non-commutative C ∗ -algebras, with topologicalamenability replaced with strong amenability (see [9, Definition 3.4]). Remark 4.4.
It would be natural if topological amenability of ( C ( X ) , G, α ) were equivalentto L ( G ) -injectivity of C ( X ) ∗∗ . By [5] (which generalizes work of Monod [15]), amenabilityof ( M, G, α ) is equivalent to L ( G ) -injectivity of M ∼ = M ( X ) ∗ c = h L ( G ) ∗ M ( X ) i ∗ . Bygeneral Banach module theory (see, e.g., [18, Proposition 2.1.11]), this in turn is equivalentto L ( G ) -injectivity of C ( X ) ∗∗ in the bounded sense , where morphisms are not necessarilycontractive. One can show that the corresponding injectivity constant is ≤ , however, it isnot clear how, if possible, to push this to 1. Acknowledgements
The authors would like to thank Mehrdad Kalantar and Rufus Willett for helpful discus-sions at various points during this project. The second author was partially supported bythe NSERC Discovery Grant RGPIN-2017-06275.
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E-mail address : [email protected] Department of Mathematics, University of Texas at Tyler, Tyler, TX 75799
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