# Journal of Operator Theory

Volume 63, Issue 2, Spring 2010 pp. 317-332.

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

**Author institution:**Department of Mathematics, University of Nebraska, Lincoln, NE 68588-0130, U.S.A.

**Summary:**We argue that weak containment is an appropriate notion of am\-enability 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.

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