Amenability of group actions on compact spaces and the associated Banach algebras

For a topological group $G$, amenability can be characterized by the amenability of the convolution Banach algebra $L^1(G)$. Here a Banach algebra $A$ is called amenable if every bounded derivation from $A$ into any dual–type $A$–$A$–Banach bimodule is inner. We extend this classical result to the case of discrete group actions on compact Hausdorff spaces. By introducing a Banach algebra naturally associated with the action and adopting a suitably weakened notion of amenability for Banach algebras, we obtain an analogous characterization of amenable actions. As a lemma, we also proved a fixed–point property for amenable actions that strengthens the theorem of Dong and Wang (2015).