Amenable actions, invariant means and bounded cohomology

Description/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.