Bönicke, C. and Proietti, V. (2024) Categorical approach to the Baum-Connes conjecture for étale groupoids. Journal of the Institute of Mathematics of Jussieu, (doi: 10.1017/S1474748023000531) (Early Online Publication)
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Abstract
We consider the equivariant Kasparov category associated to an étale groupoid, and by leveraging its triangulated structure we study its localization at the "weakly contractible" objects, extending previous work by R. Meyer and R. Nest. We prove the subcategory of weakly contractible objects is complementary to the localizing subcategory of projective objects, which are defined in terms of "compactly induced" algebras with respect to certain proper subgroupoids related to isotropy. The resulting "strong" Baum-Connes conjecture implies the classical one, and its formulation clarifies several permanence properties and other functorial statements. We present multiple applications, including consequences for the Universal Coefficient Theorem, a generalized "Going-Down" principle, injectivity results for groupoids that are amenable at infinity, the Baum-Connes conjecture for group bundles, and a result about the invariance of K-groups of twisted groupoid C∗-algebras under homotopy of twists.
| Item Type: | Articles |
|---|---|
| Additional Information: | The first author was supported by the Alexander von Humboldt Foundation. The second author was supported by: Science and Technology Commission of Shanghai Municipality (grant No. 18dz2271000), Foreign Young Talents’ grant (National Natural Science Foundation of China), CREST Grant Number JPMJCR19T2 (Japan), Marie Sk lodowska-Curie Individual Fellowship (project number 101063362). |
| Status: | Early Online Publication |
| Refereed: | Yes |
| Glasgow Author(s) Enlighten ID: | Bonicke, Dr Christian |
| Authors: | Bönicke, C., and Proietti, V. |
| College/School: | College of Science and Engineering > School of Mathematics and Statistics > Mathematics |
| Journal Name: | Journal of the Institute of Mathematics of Jussieu |
| Publisher: | Cambridge University Press |
| ISSN: | 1474-7480 |
| ISSN (Online): | 1475-3030 |
| Published Online: | 02 January 2024 |
| Copyright Holders: | Copyright: © The Author(s), 2023 |
| First Published: | First published in Journal of the Institute of Mathematics of Jussieu 2024 |
| Publisher Policy: | Reproduced under a Creative Commons licence |
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