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.
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Received: 30 April 2002 / Accepted: 2 October 2002 Published online: 17 April 2003
RID="⋆"
ID="⋆" Supported by a Fellowship from the Royal Netherlands Academy of Arts and Sciences (KNAW).
Communicated by H. Araki, D. Buchholz and K. Fredenhagen
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Landsman, N. Deformation Quantization and the Baum–Connes Conjecture. Commun. Math. Phys. 237, 87–103 (2003). https://doi.org/10.1007/s00220-003-0838-0
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DOI: https://doi.org/10.1007/s00220-003-0838-0