New York Journal of Mathematics
Volume 19 (2013) 367-394


Lisa Orloff Clark, Astrid an Huef, and Iain Raeburn

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Published: June 27, 2013
Keywords: Fell algebra; continuous-trace algebra; Dixmier-Douady invariant; the C*-algebra of a local homeomorphism; groupoid C*-algebra
Subject: 46L55

We study the groupoid C*-algebra associated to the equivalence relation induced by a quotient map on a locally compact Hausdorff space. This C*-algebra is always a Fell algebra, and if the quotient space is Hausdorff, it is a continuous-trace algebra. We show that the C*-algebra of a locally compact, Hausdorff and principal groupoid is a Fell algebra if and only if the groupoid is one of these relations, extending a theorem of Archbold and Somerset about étale groupoids. The C*-algebras of these relations are, up to Morita equivalence, precisely the Fell algebras with trivial Dixmier-Douady invariant as recently defined by an Huef, Kumjian and Sims. We use twisted groupoid algebras to provide examples of Fell algebras with nontrivial Dixmier-Douady invariant.

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Department of Mathematics and Statistics, University of Otago, PO Box 56, Dunedin 9054, New Zealand.