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A stabilization theorem for Fell bundles over groupoids

Published online by Cambridge University Press:  17 October 2017

Marius Ionescu
Affiliation:
Department of Mathematics, United States Naval Academy, Annapolis, MD 21402, USA ([email protected])
Alex Kumjian
Affiliation:
Department of Mathematics, University of Nevada, Reno, NV 89557, USA ([email protected])
Aidan Sims
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia ([email protected])
Dana P. Williams
Affiliation:
Department of Mathematics, Dartmouth College, Hanover, NH 03755-3551, USA ([email protected])

Abstract

We study the C*-algebras associated with upper semi-continuous Fell bundles over second-countable Hausdorff groupoids. Based on ideas going back to the Packer–Raeburn ‘stabilization trick’, we construct from each such bundle a groupoid dynamical system whose associated Fell bundle is equivalent to the original bundle. The upshot is that the full and reduced C*-algebras of any saturated upper semi-continuous Fell bundle are stably isomorphic to the full and reduced crossed products of an associated dynamical system. We apply our results to describe the lattice of ideals of the C*-algebra of a continuous Fell bundle by applying Renault's results about the ideals of the C*-algebras of groupoid crossed products. In particular, we discuss simplicity of the Fell-bundle C*-algebra of a bundle over G in terms of an action, described by Ionescu and Williams, of G on the primitive-ideal space of the C*-algebra of the part of the bundle sitting over the unit space. We finish with some applications to twisted k-graph algebras, where the components of our results become more concrete.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2018 

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