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 commutative 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.