Exotic crossed products and the Baum-Connes conjecture

TL;DR

This paper investigates exotic crossed-product functors, characterizing those extendable to equivariant C*-algebra categories, and demonstrates their applications in KK-theory descent and the Baum-Connes conjecture for K-amenable groups.

Contribution

It characterizes exotic crossed-product functors that extend to correspondence categories and applies this to KK-theory descent and the Baum-Connes conjecture.

Findings

All crossed products by correspondence functors of K-amenable groups are KK-equivalent.

The minimal exact Morita compatible crossed-product functor extends to correspondences for separable G-C*-algebras.

Descent in KK-theory is possible for these functors, aiding the Baum-Connes conjecture.

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 extends to correspondences when restricted to separable G-C*-algebras. It therefore allows a descent in KK-theory for separable systems.