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A categorical approach to the Baum-Connes conjecture for étale groupoids

Lookup NU author(s): Dr Christian BönickeORCiD

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Abstract

We consider the equivariant Kasparov category associated to an étalegroupoid, and by leveraging its triangulated structure we study its localization atthe “weakly contractible” objects, extending previous work by Meyer and Nest.We prove that the subcategory of weakly contractible objects is complementary to thelocalizing subcategory of projective objects, which are defined in terms of “compactlyinduced” algebras with respect to certain proper subgroupoids related to isotropy.The resulting “strong” Baum–Connes conjecture implies the classical one, and itsformulation clarifies several permanence properties and other functorial statements.We present multiple applications, including consequences for the Universal CoefficientTheorem, a generalized “Going-Down” principle, injectivity results for groupoids thatare amenable at infinity, the Baum-Connes conjecture for group bundles, and a resultabout the invariance of K-groups of twisted groupoid C*-algebras under homotopy oftwists.

Publication metadata

Author(s): Bönicke C, Proietti V

Publication type: Article

Publication status: Published

Journal: Journal of the Institute of Mathematics of Jussieu

Year: 2024

Pages: Epub ahead of print

Online publication date: 02/01/2024

Acceptance date: 27/11/2023

Date deposited: 27/11/2023

ISSN (print): 1474-7480

ISSN (electronic): 1475-3030

Publisher: Cambridge University Press

URL: https://doi.org/10.1017/S1474748023000531

DOI: 10.1017/S1474748023000531

ePrints DOI: 10.57711/984w-7545

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