Amenable and inner amenable actions and approximation properties for crossed products by locally compact groups

TL;DR

This paper explores amenable and inner amenable actions of locally compact groups on von Neumann and C*-algebras, establishing new characterizations of injectivity and nuclearity, and relating these properties to approximation properties of crossed products.

Contribution

It introduces a new notion of inner amenability for group actions and generalizes existing results on injectivity and nuclearity for crossed products by locally compact groups.

Findings

Characterization of injectivity for crossed products of von Neumann algebras.

Relation between amenability of actions and nuclearity of crossed products.

Applications to approximation properties and answering recent open questions.

Abstract

Amenable actions of locally compact groups on von Neumann algebras are investigated by exploiting the natural module structure of the crossed product over the Fourier algebra of the acting group. The resulting characterisation of injectivity for crossed products generalises a result of Anantharaman-Delaroche on discrete groups. Amenable actions of locally compact groups on $C^*$-algebras are investigated in the same way, and amenability of the action is related to nuclearity of the corresponding crossed product. A survey is given to show that this notion of amenable action for $C^*$-algebras satisfies a number of expected properties. A notion of inner amenability for actions of locally compact groups is introduced, and a number of applications are given in the form of averaging arguments, relating approximation properties of crossed product von Neumann algebras to properties of the components of the underlying $w^*$-dynamical system. We use these results to answer a recent question of Buss-Echterhoff-Willett.