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Journal of Operator Theory

Volume 81, Issue 2, Spring 2019  pp. 407-431.

The Dixmier-Douady classes of certain groupoid $C^{*}$-algebras with continuous trace

Authors: Marius Ionescu (1), Alex Kumjian (2), Aidan Sims (3), Dana P. Williams (4)
Author institution:(1) Department of Mathematics, United States Naval Academy, Annapolis, MD 21402 U.S.A.
(2) Department of Mathematics, University of Nevada, Reno NV 89557 U.S.A.
(3) School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia
(4) Department of Mathematics, Dartmouth College, Hanover, NH 03755-3551 U.S.A.

Summary: 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 Cech $2$-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.

Keywords: Dixmier--Douady class, groupoid $C^*$-algebras, continuous-trace $C^*$-algebras

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