Groups, Geometry, and Dynamics

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Volume 8, Issue 2, 2014, pp. 485–512
DOI: 10.4171/GGD/236

Published online: 2014-07-08

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

David Milan[1] and Benjamin Steinberg[2]

(1) The University of Texas at Tyler, USA
(2) City College of New York, USA

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.

Keywords: Crossed products, inverse semigroups, ├ętale groupoids, partial actions

Milan David, Steinberg Benjamin: On inverse semigroup $C^*$-algebras and crossed products. Groups Geom. Dyn. 8 (2014), 485-512. doi: 10.4171/GGD/236