On inverse semigroup $C^{\ast }$-algebras and crossed products

Abstract

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

arXiv commentWe correct an error in the proof of Theorem 3.5 that was pointed out to us by Alcides Buss. No results are changed