We describe the -algebra of an -unitary or strongly --unitary inverse semigroup as the partial crossed product of a commutative -algebra by the maximal group image of the inverse semigroup. We give a similar result for the -algebra of the tight groupoid of an inverse semigroup. We also study conditions on a groupoid -algebra to be Morita equivalent to a full crossed product of a commutative -algebra with an inverse semigroup, generalizing results of Khoshkam and Skandalis for crossed products with groups.
Cite this article
David Milan, Benjamin Steinberg, On inverse semigroup -algebras and crossed products. Groups Geom. Dyn. 8 (2014), no. 2, pp. 485–512DOI 10.4171/GGD/236