Topological Amenability

TL;DR

This paper establishes the equivalence between topological amenability of a group action, the injectivity of certain dual modules, and the existence of a G-invariant element in a specific double dual space, advancing the understanding of group actions.

Contribution

It provides a new characterization of topological amenability through module injectivity and invariant elements, linking topological dynamics with Banach module theory.

Findings

Equivalence between topological amenability and module injectivity.

Identification of a G-invariant element in the double dual space as a criterion.

Unification of concepts in topological dynamics and Banach modules.

Abstract

Let G be a locally compact topological group and X a compact space with continuous G-action. The main result of this essay states that the following statements are equivalent : 1) The action of G on X is topologically amenable ; 2) Every dual (G, X)-module of type C is a relatively injective Banach G-module ; 3) There is a G-invariant element in the double dual of C(X, L1(G)).