Pushouts of extensions of groupoids by bundles of abelian groups
| Autor: | Dana P. Williams, Aidan Sims, Jean Renault, Marius Ionescu, Alex Kumjian |
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| Rok vydání: | 2021 |
| Předmět: |
Pure mathematics
Algebra and Number Theory Group (mathematics) Applied Mathematics Mathematics - Operator Algebras Pushout Extension (predicate logic) Cartesian product symbols.namesake 46L05 20L05 Bundle Mathematics::Category Theory symbols FOS: Mathematics Geometry and Topology Isomorphism Twist Abelian group Operator Algebras (math.OA) Mathematics::Symplectic Geometry Analysis Mathematics |
| DOI: | 10.48550/arxiv.2107.05776 |
| Popis: | 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. 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 |
| Databáze: | OpenAIRE |
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