$C^*$-algebras of inverse semigroups: amenability and weak containment

TL;DR

This paper explores the concept of weak containment as a form of amenability in inverse semigroups, establishing conditions under which inverse semigroups and related structures exhibit this property.

Contribution

It introduces weak containment as an appropriate notion of amenability for inverse semigroups and characterizes when such semigroups have this property, including graph inverse semigroups and Nica's inverse semigroup.

Findings

Inverse semigroup weak containment is equivalent to group amenability under certain conditions.

All graph inverse semigroups have weak containment.

Nica's inverse semigroup has weak containment if and only if the associated group is amenable.

Abstract

We argue that weak containment is an appropriate notion of amenability for inverse semigroups. Given an inverse semigroup $S$ and a homomorphism $\phi$ of $S$ onto a group $G$, we show, under an assumption on $\ker(\phi)$, that $S$ has weak containment if and only if $G$ is amenable and $\ker(\phi)$ has weak containment. Using Fell bundle amenability, we find a related result for inverse semigroups with zero. We show that all graph inverse semigroups have weak containment and that Nica's inverse semigroup $\mcT_{G,P}$ of a quasi-lattice ordered group $(G,P)$ has weak containment if and only if $(G,P)$ is amenable.