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

Alcides Buss, Siegfried Echterhoff+1 more|arXiv (Cornell University)|Mar 6, 2020

Advanced Operator Algebra Research54 references8 citations

TL;DR

This paper introduces a new notion of amenability for actions of locally compact groups on C*-algebras, generalizing Anantharaman-Delaroche's definition from discrete groups. It establishes that for actions on commutative C*-algebras, this amenability is equivalent to measurewise amenability, resolving a long-standing open problem. The key result generalizes Matsumura's theorem: for an exact locally compact group acting on a locally compact space X, the full and reduced crossed products of C₀(X) agree if and only if the action is amenable.

ABSTRACT

In this work we introduce and study a new notion 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 permanence properties analogous to those known in the discrete case. For example, for actions on commutative $C^*$-algebras, we show that our notion of amenability is equivalent to measurewise amenability. Combined with a recent result of Alex Bearden and Jason Crann, this also settles a long standing open problem about the equivalence of topological amenability and measurewise amenability for a second countable $G$-space $X$. We use our new notion of amenability to study when the maximal and reduced crossed products agree. One of our main results generalizes 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 $C_0(X) times_\max G$ and $C_0(X) times_{\operatorname{red}} G$ coincide if and only if the action of $G$ on $X$ is amenable. We also show that the analogue of this theorem does not hold for actions on noncommutative $C^*$-algebras. Finally, we study amenability as it relates to more detailed structure in the case of $C^*$-algebras that fibre over an appropriate $G$-space $X$, and the interaction of amenability with various regularity properties such as nuclearity, exactness, and the (L)LP, and the equivariant versions of injectivity and the WEP.

Motivation & Objective

• To define and study a new notion of amenability for actions of locally compact groups on C*-algebras, extending the discrete group case.
• To investigate the relationship between this amenability and approximation properties, especially in the context of weak containment and crossed product structure.
• To resolve the equivalence between topological amenability and measurewise amenability for second countable G-spaces, using a recent result by Bearden and Crann.
• To characterize when the maximal and reduced crossed products of a C*-algebra coincide, particularly for actions on commutative C*-algebras.
• To explore connections between amenability and regularity properties such as nuclearity, exactness, and the (L)LP in the equivariant setting.

Proposed method

• Define amenability for a G-C*-algebra (A, α) via the existence of a G-equivariant conditional expectation from L∞(G) ⊗̂ A** to A**.
• Use the enveloping G-von Neumann algebra construction to lift the action to a von Neumann algebra setting where amenability can be analyzed via conditional expectations.
• Establish permanence properties of amenability under various operations, including ideals, quotients, and direct limits.
• Prove that for commutative C*-algebras, amenability is equivalent to measurewise amenability, leveraging the result of Bearden and Crann.
• Apply the weak quasi-central approximation property (wQAP) to analyze Fell bundle structures and their relation to amenability.
• Use the Haagerup standard form and injective representations to study commutant amenability and its relation to weak containment.

Experimental results

Research questions

• RQ1 Is the new notion of amenability for locally compact group actions on C*-algebras equivalent to measurewise amenability for commutative C*-algebras?
• RQ2 Does the agreement of maximal and reduced crossed products for C₀(X) ⋊ G hold if and only if the action of G on X is amenable, for exact locally compact groups?
• RQ3 Can the weak containment property be satisfied by a non-amenable action of a discrete group?
• RQ4 Is there an intrinsic approximation property equivalent to commutant amenability for a G-C*-algebra?
• RQ5 Does commutant amenability of an action on a nuclear C*-algebra imply nuclearity of the reduced crossed product?

Key findings

• The new notion of amenability for actions of locally compact groups on C*-algebras is equivalent to measurewise amenability when the C*-algebra is commutative.
• For an exact locally compact group G acting on a locally compact space X, the full and reduced crossed products C₀(X) ⋊max G and C₀(X) ⋊red G coincide if and only if the action is amenable.
• The analogue of Matsumura's theorem fails for noncommutative C*-algebras: amenability of the action does not imply agreement of maximal and reduced crossed products.
• Amenability of an action on a type I C*-algebra with Hausdorff spectrum implies strong amenability, extending a result from the commutative case.
• Commutant amenability does not imply amenability, and there exist non-amenable actions with the weak containment property.
• Nuclearity of the reduced crossed product A ⋊red G implies nuclearity of A, but does not imply amenability of the action unless G is in a specific class (e.g., inner amenable groups).