The Dixmier-Douady Classes of Certain Groupoid $C^*$-Algebras with Continuous Trace

TL;DR

This paper provides explicit formulas for the Dixmier-Douady class of certain groupoid $C^*$-algebras with continuous trace, extending to more general extensions using a blow-up construction.

Contribution

It derives explicit formulas for the Dixmier-Douady invariant in the context of groupoid extensions and extends these results to broader classes of central extensions.

Findings

Explicit formula for Dixmier-Douady invariant for groupoid $C^*$-algebras

Extension of formulas to more general central extensions via blow-up construction

Application to étale equivalence relations

Abstract

Given a locally compact abelian group $G$, we give an explicit formula for the Dixmier--Douady invariant of the $C^*$-algebra of the groupoid extension associated to a \v{C}ech $2$-cocycle in the sheaf of germs of continuous $G$-valued functions. We then exploit the blow-up construction for groupoids to extend this to some more general central extensions of \'etale equivalence relations.