2021/07/12 by Ionescu, Marius, Kumjian, Alex, Renault, Jean N. +2 · 1 citation
#20L05 #46L05 #FOS: Mathematics #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.2107.05776
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.