The equivalence relations of local homeomorphisms and Fell algebras (Q374070)

Recently, Huef, Pask and Sims introduced a Dixmier-Douady invariant for Fell algebras, see [\textit{A. an Huef} et al., J. Funct. Anal. 260, No. 5, 1543--1581 (2011; Zbl 1251.46028)]. In the paper under review, the authors study Fell algebras that arise as the groupoid \(C^*\)-algebra \(C^*(R(\psi))\) of a locally compact Hausdorff equivalence relation \(R(\psi)\) associated to a surjection \(\psi\) that is defined on a locally compact Hausdorff space. This is used to characterise up to Morita equivalence the separable Fell algebras with trivial Dixmier-Douady invariant, cf. Theorem 6.1. Twisted versions of \(C^*(R(\psi))\) are discussed, and are shown to provide examples of Fell algebras with nonvanishing Dixmier-Douady invariant. A characterisation of when the groupoid \(C^*\)-algebra \(C^*(G)\) associated to a principal groupoid \(G\) is a Fell algebra is obtained, see Theorem 5.1, and extends a result of \textit{R. J. Archbold} and \textit{D. W. B. Somerset} [Math. Scand. 73, No. 1, 81--111 (1993; Zbl 0792.46043)].