Exotic crossed products and the Baum–Connes conjecture

Alcides Buss; Siegfried Echterhoff; Rufus Willett

doi:10.1515/crelle-2015-0061

Published by De Gruyter October 21, 2015

Exotic crossed products and the Baum–Connes conjecture

Alcides Buss , Siegfried Echterhoff and Rufus Willett

From the journal Journal für die reine und angewandte Mathematik (Crelles Journal)

https://doi.org/10.1515/crelle-2015-0061

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Abstract

We study general properties of exotic crossed-product functors and characterise those which extend to functors on equivariant C*-algebra categories based on correspondences. We show that every such functor allows the construction of a descent in KK-theory and we use this to show that all crossed products by correspondence functors of K-amenable groups are KK-equivalent. We also show that for second countable groups the minimal exact Morita compatible crossed-product functor used in the new formulation of the Baum–Connes conjecture by Baum, Guentner and Willett ([‘Expanders, exact crossed products, and the Baum–Connes conjecture’, preprint 2013]) extends to correspondences when restricted to separable G-C*-algebras. It therefore allows a descent in KK-theory for separable systems.

Funding source: Deutsche Forschungsgemeinschaft

Award Identifier / Grant number: SFB 878

Funding source: National Science Foundation

Award Identifier / Grant number: DMS 1401126

Funding statement: The authors were supported by Deutsche Forschungsgemeinschaft (SFB 878, Groups, Geometry & Actions), by CNPq/CAPES – Brazil, and by the US NSF (DMS 1401126).

Acknowledgements

Part of the work on this paper took place during visits of the second author to the Federal University of Santa Catarina, and of the third author to the Westfälische Wilhelms-Universität, Münster. We would like to thank these institutions for their hospitality.

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Received: 2014-09-16

Revised: 2015-07-04

Published Online: 2015-10-21

Published in Print: 2018-07-01

Exotic crossed products and the Baum–Connes conjecture

Abstract

Acknowledgements

References

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