Proper Kasparov Cycles and the Baum-Connes Conjecture

TL;DR

This paper develops a new approach to the Baum-Connes conjecture using proper Kasparov cycles, which simplifies the process by avoiding the construction of proper algebras and Dirac elements, and proves split-injectivity and surjectivity results.

Contribution

It introduces proper Kasparov cycles and Property (gamma), providing a novel method to analyze the Baum-Connes conjecture without requiring G-compact universal spaces.

Findings

Proper Kasparov cycles induce maps on K-theory factoring through the Baum-Connes left-hand side.

Existence of cycles with Property (gamma) implies split-injectivity of the assembly map.

Results on the surjectivity of the assembly map are obtained.

Abstract

We introduce the notion of proper Kasparov cycles for Kasparov's G-equivariant KK-theory for a general locally compact, second countable topological group G. We show that for any proper Kasparov cycle, its induced map on K-theory factors through the left-hand side of the Baum-Connes conjecture. 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 (gamma) removing the G-compact assumption on the universal space EG in the previous paper "Direct Splitting Method for the Baum-Connes Conjecture". We show that the existence of a cycle with Property (gamma) implies the split-injectivity of the Baum-Connes assembly map for all coefficients. We also obtain results concerning the surjectivity of the assembly map.