Deformation Quantization and the Baum–Connes Conjecture

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.