Amenable actions of inverse semigroups

TL;DR

This paper characterizes when actions of inverse semigroups on spaces are amenable, linking the amenability of the spectrum action to all actions of the semigroup, with implications for groupoid theory.

Contribution

It establishes an equivalence between the amenability of the spectrum action and all actions of a countable inverse semigroup, providing a new criterion for amenability.

Findings

Spectrum action is amenable iff all actions are amenable

Provides a characterization of amenability for inverse semigroup actions

Connects semigroup actions to groupoid amenability theory

Abstract

We say that an action of a countable discrete inverse semigroup on a locally compact Hausdorff space is amenable if its groupoid of germs is amenable in the sense of Anantharaman-Delaroche and Renault. We then show that for a given inverse semigroup $S$, the action of $S$ on its spectrum is amenable if and only if every action of $S$ is amenable.