$C^{\ast }$-algebras of inverse semigroups: amenability and weak containment

Abstract

We argue that weak containment is an appropriate notion of amenability for inverse semigroups. Given an inverse semigroup $S$ and a homomorphism $\varphi$ of $S$ onto a group $G$, we show, under an assumption on $\ker (\varphi )$, that $S$ has weak containment if and only if $G$ is amenable and $\ker (\varphi )$ 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.

14 pages. Corrected an error in Proposition 3.1 (thanks to the referee for finding this). This has led to a slight restructuring of the paper and an extra hypothesis in the main theorem