2020
DOI: 10.1016/j.aim.2020.107314
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A Going-Down principle for ample groupoids and the Baum-Connes conjecture

Abstract: We study a Going-Down (or restriction) principle for ample groupoids and its applications. The Going-Down principle for locally compact groups was developed by Chabert, Echterhoff and Oyono-Oyono and allows to study certain functors, that arise in the context of the topological K-theory of a locally compact group, in terms of their restrictions to compact subgroups. We extend this principle to the class of ample Hausdorff groupoids using Le Gall's groupoid equivariant version of Kasparov's bivariant KK-theory.… Show more

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Cited by 9 publications
(35 citation statements)
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“…This question has been addressed before in the case of groups by Echterhoff, Lück, Phillips and Walters in [10] and Gillaspy treated the cases of transformation groups, higher-rank graphs and group bundles in a series of articles [16,17,18]. Using the machinery developed by the author in [5] this article presents a unified approach and considerable extension of the above mentioned results.…”
Section: Introductionmentioning
confidence: 93%
See 2 more Smart Citations
“…This question has been addressed before in the case of groups by Echterhoff, Lück, Phillips and Walters in [10] and Gillaspy treated the cases of transformation groups, higher-rank graphs and group bundles in a series of articles [16,17,18]. Using the machinery developed by the author in [5] this article presents a unified approach and considerable extension of the above mentioned results.…”
Section: Introductionmentioning
confidence: 93%
“…Before proceeding to the precise form of the Going-Down principle we are going to use, the reader may wish to recall the definitions of Le Gall's KK G -theory [28], the topological K-theory K top * (G; A) of a groupoid with coefficients in a G-algebra A, and the Baum-Connes assembly map µ A : K top * (G; A) → K * (A r G) [44]. See also [6,5] for a detailed overview. Theorem 4.5.…”
Section: Homotopies Of Twistsmentioning
confidence: 99%
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“…Induced actions. We recall now from [1] the notion of induced action to a groupoid from a subgroupoid action. Let G be a locally compact groupoid with space of units X and open source and range maps, let H be a relatively clopen subgroupoid of G with space of units Y and let Z be a (left) H-space with anchor map p :…”
Section: Groupoid Actionsmentioning
confidence: 99%
“…Before proceeding to the precise form of the Going-Down principle we are going to use, the reader may wish to recall the definitions of Le Gall's KK G -theory [28], the topological K-theory K top * (G; A) of a groupoid with coefficients in a G-algebra A, and the Baum-Connes assembly map µ A : K top * (G; A) → K * (A⋊ r G) [44]. See also [6,5] for a detailed overview. Theorem 4.5.…”
Section: Homotopies Of Twistsmentioning
confidence: 99%