THE EQUIVALENCE RELATIONS OF LOCAL HOMEOMORPHISMS AND FELL ALGEBRAS

Abstract. We study the groupoid C∗-algebras C∗(R(ψ)) associated to the equivalence relation R(ψ) induced by a quotient map ψ: Y → X. If Y is Hausdorff then C∗(R(ψ)) is a Fell algebra, and if both Y and X are Hausdorff then C∗(R(ψ)) has continuous trace. We show that the C∗-algebra C∗(G) of a locally compact, Hausdorff and principal groupoid G is a Fell algebra if and only if G is topologically isomorphic to some R(ψ), extending a theorem of Archbold and Somerset. The algebras C∗(R(ψ)) 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 a twisted analogue of C∗(R(ψ)) to provide examples of Fell algebras with non-trivial Dixmier-Douady invariant. 1