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A stabilization theorem for Fell bundles over groupoids

  • Marius Ionescu (a1), Alex Kumjian (a2), Aidan Sims (a3) and Dana P. Williams (a4)

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.

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A stabilization theorem for Fell bundles over groupoids

  • Marius Ionescu (a1), Alex Kumjian (a2), Aidan Sims (a3) and Dana P. Williams (a4)

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