Amenable semigroups and nonexpansive dynamical systems

TL;DR

This paper characterizes the amenability of semitopological semigroups through fixed point properties of nonexpansive actions, providing a complete answer to a longstanding question and proposing new approaches to related problems.

Contribution

It offers a complete characterization of semitopological semigroups with a left invariant mean on WAP(S) and introduces a novel approach to Lau's problem on amenability.

Findings

Characterization of amenability via fixed point properties.

Affirmative answer to Lau and Zhang's question.

Partial results on Day-Mitchell's characterization in specific settings.

Abstract

We characterize amenability of subspaces of $C(S)$, where $S$ is a semitopological semigroup, in terms of fixed point properties of nonexpansive actions. In particular, we give a complete characterization of a semitopological semigroup with a left invariant mean on WAP(S) that answers a question of A.T.-M. Lau and Y. Zhang in the affirmative. We also propose a new approach to Lau's problem concerning a counterpart of Day-Mitchell's characterization of amenable semigroups and show some partial results, in the case of weak$^{\ast }$ compact convex sets with the Radon-Nikod\'{y}m property, and in the duals of $M$-embedded Banach spaces.