On inverse semigroup $C^*$-algebras and crossed products

Abstract

We describe the C � -algebra of an E-unitary or strongly 0- E-unitary inverse semigroup as the partial crossed product of a commu- tative C � -algebra by the maximal group image of the inverse semigroup. We give a similar result for the C � -algebra of the tight groupoid of an inverse semigroup. We also study conditions on a groupoid C � -algebra to be Morita equivalent to a full crossed product of a commutative C � - algebra with an inverse semigroup, generalizing results of Khoshkam and Skandalis for crossed products with groups.