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Pushouts of extensions of groupoids by bundles of abelian groups

Journal Article


Abstract


  • We analyse extensions σ 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 σ 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 σ, and that this isomorphism descends to an isomorphism of reduced algebras. We give a number of examples and applications.

Publication Date


  • 2021

Citation


  • 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. doi:10.53733/136

Scopus Eid


  • 2-s2.0-85117925885

Web Of Science Accession Number


Start Page


  • 561

End Page


  • 581

Volume


  • 52

Abstract


  • We analyse extensions σ 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 σ 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 σ, and that this isomorphism descends to an isomorphism of reduced algebras. We give a number of examples and applications.

Publication Date


  • 2021

Citation


  • 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. doi:10.53733/136

Scopus Eid


  • 2-s2.0-85117925885

Web Of Science Accession Number


Start Page


  • 561

End Page


  • 581

Volume


  • 52