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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