Direct splitting method for the Baum-Connes conjecture

摘要

We develop a new method for studying the Baum Connes conjecture, which we call the direct splitting method. We introduce what we call property (gamma) for G-equivariant Kasparov cycles. We show that the existence of a G-equivariant Kasparov cycle with property (gamma) implies the split-injectivity of the assembly map mu(G)(A) for any separable G-C*-algebra A. We also show that if such a cycle exists, the assembly map mu(G)(A) is an isomorphism if and only if the cycle acts as the identity on the right-hand side group K-* (A(sic)(r)G) of the Baum-Connes conjecture. In a separate paper, with J. Brodzki, E. Guentner and N. Higson, we use this method to give a finite-dimensional proof of the Baum-Connes conjecture for groups which act properly and co-compactly on a finite-dimensional CAT(0)-cubical space. (C) 2019 Elsevier Inc. All rights reserved.