Amenable actions, invariant means and bounded cohomology

TL;DR

This paper establishes equivalences between topological amenability, invariant means, and bounded cohomology vanishing for group actions, extending classical group amenability results and characterizing exactness via cohomology.

Contribution

It provides a cohomological characterization of topological amenability and exactness for group actions, generalizing Johnson's theorem and answering Higson's question.

Findings

Topological amenability is equivalent to the existence of an invariant mean.

Vanishing bounded cohomology characterizes amenability and exactness.

The results unify group and action properties through cohomological conditions.

Abstract

We show that topological amenability of an action of a countable discrete group on a compact space is equivalent to the existence of an invariant mean for the action. We prove also that this is equivalent to vanishing of bounded cohomology for a class of Banach G-modules associated to the action, as well as to vanishing of a specific cohomology class. In the case when the compact space is a point our result reduces to a classic theorem of B.E. Johnson characterising amenability of groups. In the case when the compact space is the Stone-\v{C}ech compactification of the group we obtain a cohomological characterisation of exactness for the group, answering a question of Higson.