2002/10/08 by N.P. Landsman, N. P. Landsman
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebra over a field #Algebraic structures and combinatorial models #Algorithm #Canonical quantization #Conjecture #Crossed product #Geometric quantization #Group (periodic table) #Lie group #Mathematics #Noncommutative geometry #Physics #Pure mathematics #Quantization (signal processing) #Quantum #Quantum gravity #Quantum mechanics #Semidirect product #math-ph #math.KT #math.MP #msc:46L80 #msc:53D55
paper · pdf · doi:10.1007/s00220-003-0838-0
21 pages
arxiv created 2002/10/08 · openalex publication_date 2003/06/01 · arxiv updated 2009/11/30 · openalex created_date 2021/02/01 · openalex updated_date 2026/08/05
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.