A Stabilization Theorem for Fell Bundles over groupoids
| Autor: | Ionescu, Marius, Kumjian, Alex, Sims, Aidan, Williams, Dana P. |
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| Rok vydání: | 2015 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: Accepted for publication in Proc. Roy. Soc. Edinburgh Sect. A. We improved Corollary 3.12 following the anonymous referee suggestion |
| Databáze: | arXiv |
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