Amenability and weak containment for actions of locally compact groups on $C^*$-algebras

Abstract

In this paper we introduce and study new notions of amenability for actions of locally compact groups on C ˚-algebras. Our definition extends the definition of amenability for actions of discrete groups due to Claire Anantharaman-Delaroche. We show that our definition has several characterizations and consequences analogous to those known in the discrete case.One of our main results generalises a theorem of Matsumura: we show that for an action of an exact locally compact group G on a locally compact space X the full and reduced crossed products C 0 pXq¸max G and C 0 pXq¸r ed G coincide if and only if the action of G on X is measurewise amenable. We also show that the analogue of this theorem does not hold for actions on noncommutative C ˚-algebras.

Amenable dynamical systems over locally compact groups

“…The first result gives a Reiter property for amenable W * -dynamical systems, while the second shows that a commutative C * -dynamical system (C 0 (X), G, α) is topologically amenable if and only if the universal W * -dynamical system of (C 0 (X), G, α) (in the sense of [11]) is amenable. These two results answer, in the affirmative, two questions recently posed by Buss-Echterhoff-Willett in [9].…”

“…We also prove that a commutative C * -dynamical system (C 0 (X), G, α) is topologically amenable if and only if its universal W * -dynamical system is amenable. Our results answer three open questions from the literature; one of Anantharaman-Delaroche from [3], and two from a recent preprint of Buss-Echterhoff-Willett [9].…”

Topological boundaries of covariant representations

“…We also have the following characterization of strong amenability of the G-action on B π,u , which extends [25, Theorem 1.1] and follows immediately by [15,Theorem 7.5].…”

“…An action of G on a unital C*-algebra B is (strongly) topologically amenable if the action of G on the Gelfand spectrum of the center Z(B) of B is topologically amenable (c.f., [2]). A related notion is defined in [15,Definition 3.4]: an action of G on a C*-algebra B is strongly amenable if there is a net of norm-continuous, compactly supported, positive type functions θ i : G → ZM (A) such that θ i (e) ≤ 1 for all i (where e is the identity of G), and θ i (g) → 1, strictly and uniformly for g in compact subsets of G. The two notions are equivalent when B is commutative (and unital) [15,Theorem 5.13]. Note that in [15] the authors also define the notion of amenable action [15,Definition 3.4], which is in general weaker than strong amenability.…”

“…One could add a rather methodological comment to these by reminding that recently an "injective" crossed product is introduced (c.f. [14], [15], [16]) which could eventually be related to the boundary in this more general sense.…”