Journal of Operator Theory

Volume 65, Issue 1, Winter 2011 pp. 197-210.

Enveloping algebras of partial actions as groupoid $C^*$-algebras

Authors: R. Exel (1), T. Giordano (2), and D. Goncalves (3)
Author institution: (1) Departamento de Matematica, Universidade Federal de Santa Catarina, Florianopolis, 88040-900, Brazil
(2) Department of Mathematics and Statistics, University of Ottawa, Ottawa, K1N 6N5, Canada
(3) Departamento de Matematica, Universidade Federal de Santa Catarina, Florianopolis, 88040-900, Brazil

Summary: We describe the enveloping $C^*$-algebra associated to a partial action of a countable discrete group on a locally compact space as a groupoid $C^*$-algebra (more precisely as a $C^*$-algebra from an equivalence relation) and we use our approach to show that, for a large class of partial actions of $\Z$ on the Cantor set, the enveloping $C^*$-algebra is an AF-algebra. We also completely characterize partial actions of a countable discrete group on a compact space such that the enveloping action acts in a Hausdorff space.

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