Groups, Geometry, and Dynamics
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Published online: 2014-07-08
On inverse semigroup $C^*$-algebras and crossed productsDavid Milan and Benjamin Steinberg (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, étale 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