Direct splitting method for the Baum–Connes conjecture

Abstract

We develop a new method for studying the Baum-Connes conjecture, which we call the direct splitting method. We introduce what we call property (γ) for G-equivariant Kasparov cycles. We show that the existence of a G-equivariant Kasparov cycle with property (γ) implies the split-injectivity of the assembly map µ G A for any separable G-C * -algebra A. We also show that if such a cycle exists, the assembly map µ G A is an isomorphism if and only if the cycle acts as the identity on the right-hand side group K * (A ⋊ 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.

On the Baum–Connes Conjecture for Groups Acting on CAT(0)-Cubical Spaces

“…In particular, The equalities (1.2) are the heart of this theorem. Assuming these, and the properties of the various cycles outlined above, conclusions (i) and (ii) are immediate; the first is essentially the definition of K-amenability [Cun83], and the second is an application of Nishikawa's direct splitting method [Nis19]. As for the individual equalities comprising (1.2), the first is the main technical result of this piece, and is the content of Theorem 5.1.…”

“…Assuming G acts properly, these representations are weakly contained in the regular representation of G. The operator in the Julg-Valette cycle is not exactly G-equivariant (although, it is approximately G-equivariant, as required by the axioms). Finally, assuming the action is both proper and cocompact, the de Rham cycle satisfies property (γ) recently introduced by Nishikawa [Nis19]. The de Rham cycle will be defined and analyzed in detail in Sections 4 and 5 below.…”

Proper Kasparov Cycles and the Baum-Connes Conjecture

“…An alternative approach to the Baum-Connes conjecture, which we call the direct splitting method, was introduced in [Nis19]. In this previous work, under the assumption that a group G admits a G-compact model of the universal proper G-space EG, we defined the notion of Property (γ) for a cycle for the Kasparov ring R(G).…”

“…The purpose of this paper is to introduce a simple concept, that of a proper KK-cycle, explain its relevance to Kasparov's work, and streamline some of Kasparov's arguments using it. The same notion also allows us to upgrade the "direct splitting method" for the Baum-Connes conjecture introduced in [Nis19].…”

“…This allows us to upgrade the direct splitting method, a recent new approach to the Baum-Connes conjecture which, in contrast to the standard gamma element method (the Dirac dual-Dirac method), avoids the need of constructing proper algebras and the Dirac and the dual-Dirac elements. We introduce the notion of Kasparov cycles with Property (γ) removing the Gcompact assumption on the universal space EG in the previous paper [Nis19]. We show that the existence of a cycle with Property (γ) implies the split-injectivity of the Baum-Connes assembly map for all coefficients.…”