Crossed product approach to equivariant localization algebras

TL;DR

This paper connects the gamma element method with Roe and Yu's localization algebras for the Baum--Connes conjecture, showing their equivalence and applying this to extend proofs for CAT(0)-cubical groups.

Contribution

It establishes an equivalence between the gamma element approach and controlled algebraic methods, and extends Baum--Connes conjecture proofs to non-cocompact CAT(0)-cubical groups.

Findings

The forget-control map equals the Baum--Connes assembly map.

The gamma element method can be described via controlled algebraic perspective.

The proof of Baum--Connes conjecture with coefficients is extended to non-cocompact CAT(0)-cubical groups.

Abstract

The goal of this article is to provide a bridge between the gamma element method for the Baum--Connes conjecture (the Dirac dual-Dirac method) and the controlled algebraic approach of Roe and Yu (localization algebras). For any second countable, locally compact group G, we study the reduced crossed product algebras of the representable localization algebras for proper G-spaces. We show that the naturally defined forget-control map is equivalent to the Baum--Connes assembly map for any locally compact group G and for any coefficient G-C*-algebra B. We describe the gamma element method for the Baum--Connes conjecture from this controlled algebraic perspective. As an application, we extend the recent new proof of the Baum--Connes conjecture with coefficients for CAT(0)-cubical groups to the non-cocompact setting.