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A stabilization theorem for Fell bundles over groupoids

Journal Article


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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.

UOW Authors


  •   Ionescu, Marius (external author)
  •   Kumjian, Alexander (external author)
  •   Sims, Aidan
  •   Williams, Dana P. (external author)

Publication Date


  • 2018

Citation


  • Ionescu, M., Kumjian, A., Sims, A. & Williams, D. (2018). A stabilization theorem for Fell bundles over groupoids. Proceedings of the Royal Society of Edinburgh Section A Mathematics, 148 (1), 79-100.

Scopus Eid


  • 2-s2.0-85032192883

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=2223&context=eispapers1

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers1/1221

Number Of Pages


  • 21

Start Page


  • 79

End Page


  • 100

Volume


  • 148

Issue


  • 1

Place Of Publication


  • United Kingdom

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.

UOW Authors


  •   Ionescu, Marius (external author)
  •   Kumjian, Alexander (external author)
  •   Sims, Aidan
  •   Williams, Dana P. (external author)

Publication Date


  • 2018

Citation


  • Ionescu, M., Kumjian, A., Sims, A. & Williams, D. (2018). A stabilization theorem for Fell bundles over groupoids. Proceedings of the Royal Society of Edinburgh Section A Mathematics, 148 (1), 79-100.

Scopus Eid


  • 2-s2.0-85032192883

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=2223&context=eispapers1

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers1/1221

Number Of Pages


  • 21

Start Page


  • 79

End Page


  • 100

Volume


  • 148

Issue


  • 1

Place Of Publication


  • United Kingdom