Pushouts of extensions of groupoids by bundles of abelian groups


  • Marius Ionescu United States Naval Academy
  • Alex Kumjian University of Nevada, Reno
  • Jean N. Renault Universite D'Orleans et CNRS
  • Aidan Sims University of Wollongong https://orcid.org/0000-0002-1965-6451
  • Dana P. Williams Dartmouth College




groupoid, C*-algebra, pushout


We analyse extensions $\Sigma$ of groupoids G by bundles A of abelian groups. We describe a pushout construction for such extensions, and use it to describe the extension group of a given groupoid G by a given bundle A. There is a natural action of Sigma on the dual of A, yielding a corresponding transformation groupoid. The pushout of this transformation groupoid by the natural map from the fibre product of A with its dual to the Cartesian product of the dual with the circle is a twist over the transformation groupoid resulting from the action of G on the dual of A. We prove that the full C*-algebra of this twist is isomorphic to the full C*-algebra of $\Sigma$, and that this isomorphism descends to an isomorphism of reduced algebras. We give a number of examples and applications.


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How to Cite

Ionescu, M., Kumjian, A., Renault, J. N., Sims, A., & Williams, D. P. (2021). Pushouts of extensions of groupoids by bundles of abelian groups. New Zealand Journal of Mathematics, 52, 561–581. https://doi.org/10.53733/136



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