Smooth Cartan triples and Lie twists over Hausdorff étale Lie groupoids
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910369900920832 |
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| author | Duwenig, Anna Sims, Aidan |
| author_facet | Duwenig, Anna Sims, Aidan |
| contents | We describe how to recover a Lie structure on a twist over a Hausdorff étale groupoid from functional-analytic data in the spirit of Connes' reconstruction theorem for manifolds. We first characterise when a smooth structure on the unit space of a Hausdorff étale groupoid can be extended to a Lie-groupoid structure on the whole groupoid. We introduce Lie twists over Hausdorff Lie groupoids, building on Kumjian's notion of a twist over a topological groupoid. We establish necessary and sufficient conditions on a family of sections of a twist over a Lie groupoid under which the twist can be made into a Lie twist so that all the specified sections are smooth. We use these results in the setting of twists over étale groupoids to describe conditions on a Cartan pair of C*-algebras and a family of normalisers of the subalgebra, under which Renault's Weyl twist for the pair can be made into a Lie twist for which the given normalisers correspond to smooth sections. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_09177 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Smooth Cartan triples and Lie twists over Hausdorff étale Lie groupoids Duwenig, Anna Sims, Aidan Operator Algebras Differential Geometry 46L87 (primary), 58H05, 58B34 (secondary) We describe how to recover a Lie structure on a twist over a Hausdorff étale groupoid from functional-analytic data in the spirit of Connes' reconstruction theorem for manifolds. We first characterise when a smooth structure on the unit space of a Hausdorff étale groupoid can be extended to a Lie-groupoid structure on the whole groupoid. We introduce Lie twists over Hausdorff Lie groupoids, building on Kumjian's notion of a twist over a topological groupoid. We establish necessary and sufficient conditions on a family of sections of a twist over a Lie groupoid under which the twist can be made into a Lie twist so that all the specified sections are smooth. We use these results in the setting of twists over étale groupoids to describe conditions on a Cartan pair of C*-algebras and a family of normalisers of the subalgebra, under which Renault's Weyl twist for the pair can be made into a Lie twist for which the given normalisers correspond to smooth sections. |
| title | Smooth Cartan triples and Lie twists over Hausdorff étale Lie groupoids |
| topic | Operator Algebras Differential Geometry 46L87 (primary), 58H05, 58B34 (secondary) |
| url | https://arxiv.org/abs/2309.09177 |