2015/12/18 by Ionescu, Marius, Kumjian, Alex, Sims, Aidan +1
#46L05 #46L55 #FOS: Mathematics #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.1512.06046
We study the C^*-algebras associated to upper-semicontinuous 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-semicontinuous 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 the first and last named authors, 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.