A stabilization theorem for Fell bundles over groupoids. Issue 1 (17th October 2017)

• Record Type: Journal Article
• Authors: Ionescu, Marius; Kumjian, Alex; Sims, Aidan; Williams, Dana P.
• Is Part Of: Proceedings. Volume 148:Issue 1(2018)
• Page Start: 79
• Page End: 100
• Publication Date: 2017-10-17
• Subjects: C*-algebra, -- groupoid, -- Fell bundle, -- stabilization, -- Morita equivalence, -- crossed product; Primary 46L05, -- 46L55, -- Secondary 46L08; Mathematics -- Periodicals; 510
• DOI: 10.1017/S0308210517000129
• Languages: English
• ISSNs: 0308-2105

Abstract

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.