Amenability of group actions on compact spaces and the associated Banach algebras
Abstract
The amenability of group actions on topological spaces generalizes amenability of groups and has applications in group theory, such as characterizing $C^{*}$--exact groups. We characterized it in terms of Banach algebras. For a topological group $G$ , group amenability can be characterized through the amenability of the convolution Banach algebra $L^1(G)$. Here a Banach algebra $A$ is called amenable if all derivations from $A$ to any dual--type $A$--$A$--Banach bimodules are inner. We extended this result to discrete group actions on compact Hausdorff spaces, using a certain Banach algebra arising from the action and a weakened amenability condition of Banach algebras. We also proved a fixed point characterization of amenable actions, which improves the result by Dong and Wang (2015).