$C^*$-algebras of inverse semigroups: amenability and weak containment
Abstract
Description
We argue that weak containment is an appropriate notion of amenability for inverse semigroups. Given an inverse semigroup $S$ and a homomorphism $ϕ$ of $S$ onto a group $G$, we show, under an assumption on $\ker(ϕ)$, that $S$ has weak containment if and only if $G$ is amenable and $\ker(ϕ)$ 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
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