  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.
