Pushouts of extensions of groupoids by bundles of abelian groups
Marius Ionescu, Alex Kumjian, Jean N. Renault, Aidan Sims, Dana P., Williams

TL;DR
This paper investigates extensions of groupoids by abelian group bundles, introduces a pushout construction for these extensions, and relates the resulting structures to $C^*$-algebras, providing new insights and examples.
Contribution
It introduces a pushout construction for extensions of groupoids by abelian groups and relates these to $C^*$-algebra isomorphisms, advancing the understanding of such extensions.
Findings
The full $C^*$-algebra of the constructed twist is isomorphic to that of the extension.
The isomorphism between full $C^*$-algebras descends to reduced algebras.
Examples and applications illustrating the theory are provided.
Abstract
We analyse extensions of groupoids by bundles of abelian groups. We describe a pushout construction for such extensions, and use it to describe the extension group of a given groupoid by a given bundle . There is a natural action of on the dual of , yielding a corresponding transformation groupoid. The pushout of this transformation groupoid by the natural map from the fibre product of 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 on the dual of . We prove that the full -algebra of this twist is isomorphic to the full -algebra of , and that this isomorphism descends to an isomorphism of reduced algebras. We give a number of examples and applications.
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Taxonomy
TopicsAdvanced Operator Algebra Research · Advanced Topics in Algebra · Homotopy and Cohomology in Algebraic Topology
