Exotic crossed products and the Baum–Connes conjecture

We study general properties of exotic crossed-product functors and characterise those which extend to functors on equivariant -algebra categories based on correspondences. We show that every such functor allows the construction of a descent in -theory and we use this to show that all crossed products by correspondence functors of -amenable groups are -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 -algebras. It therefore allows a descent in -theory for separable systems.