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Pushouts of extensions of groupoids by bundles of abelian groups
- Publication Year :
- 2021
-
Abstract
- 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.<br />Comment: 23 pages; the pushout construction and some of the other results were initially posted in Section 4 of [arxiv:2001.01312 v1], but the paper was subsequently split into two. arXiv admin note: text overlap with arXiv:2001.01312
- Subjects :
- Mathematics - Operator Algebras
46L05, 20L05
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2107.05776
- Document Type :
- Working Paper