On inverse semigroup \(C^*\)-algebras and crossed products (Q399427)

Summary: 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.