The equivalence relations of local homeomorphisms and Fell algebras

2012/07/11 by Clark, Lisa Orloff, Huef, Astrid an, Raeburn, Iain · 1 citation
#46L55 #FOS: Mathematics #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1207.2540

Abstract

We study the groupoid C*-algebras 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 etale 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 non-trivial Dixmier-Douady invariant.

Cited by

• A new characterization of (pre)liminary C*-algebras

Related

• Equivalence and Exact Groupoids
• Equivalence of Fell Systems and their Reduced C*-Algebras
• The representation theory of C*-algebras associated to groupoids
• C^*-algebras of Fell bundles over étale groupoids
• AF Embeddability of the C*-Algebra of a Deaconu-Renault Groupoid