• DOI:10.1017/S0308210517000129
• Corpus ID: 55975926

A stabilization theorem for Fell bundles over groupoids

@article{Ionescu2015AST,
  title={A stabilization theorem for Fell bundles over groupoids},
  author={Marius Ionescu and Alex Kumjian and Aidan Sims and Dana P. Williams},
  journal={Proceedings of the Royal Society of Edinburgh: Section A Mathematics},
  year={2015},
  volume={148},
  pages={79 - 100}
}

• Marius Ionescu, A. Kumjian, +1 author Dana P. Williams
• Published 18 December 2015
• Mathematics
• Proceedings of the Royal Society of Edinburgh: Section A Mathematics

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.