Deformation quantization and the Baum-Connes conjecture

TL;DR

This paper reviews how deformation quantization techniques, based on groupoid theory, can be applied to understand and approach the Baum-Connes conjecture, especially for singular and stratified Poisson spaces.

Contribution

It introduces a unified lemma for various deformation quantizations and proposes a novel approach to the Baum-Connes conjecture using twisted Weyl-Moyal quantization.

Findings

Unified framework for deformation quantization via amenable groupoids

Application to Baum-Connes conjecture for smooth groupoids

Strategy for quantizing stratified Poisson spaces

Abstract

Alternative titles of this paper would have been `Index theory without index' or `The Baum-Connes conjecture without Baum.' In 1989, Rieffel introduced an analytic version of deformation quantization based on the use of continuous fields of C*-algebras. We review how a wide variety of examples of such quantizations can be understood on the basis of a single lemma involving amenable groupoids. These include Weyl-Moyal quantization on manifolds, C*-algebras of Lie groups and Lie groupoids, and the E-theoretic version of the Baum-Connes conjecture for smooth groupoids as described by Connes in his book Noncommutative Geometry. Concerning the latter, we use a different semidirect product construction from Connes. This enables one to formulate the Baum-Connes conjecture in terms of twisted Weyl-Moyal quantization. The underlying mechanical system is a noncommutative desingularization of a stratified Poisson space, and the Baum-Connes conjecture actually suggests a strategy for quantizing such singular spaces.