AMENABLE ACTIONS, INVARIANT MEANS AND BOUNDED COHOMOLOGY

Abstract

We show that amenability of an action of a discrete group on a compact space X is equivalent to vanishing of bounded cohomology for a class of Banach G-modules associated to the action, that can be viewed as analogs of continuous bundles of dual modules over the G-space X. In the case when the compact space is a point, our result reduces to a classic theorem of Johnson, characterising amenability of groups. In the case when the compact space is the Stone–Čech compactification of the group, we obtain a cohomological characterisation of exactness, or equivalently, Yu's property A for the group, answering a question of Higson.