Exotic crossed products and the Baum–Connes conjecture

By Alcides Buss, S. Echterhoff, Rufus Willett. Journal für die Reine und Angewandte Mathematik, 2014

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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 K K-theory and we use this to show that all crossed products by correspondence functors of K-amenable groups are K K-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 K K-theory for separable systems.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/60bed19a61945ea801a0e899ebbee17058d722a5