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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.

@article{landsman_deformation_2003, title = {Deformation quantization and the {Baum}-{Connes} conjecture}, volume = {237}, issn = {0010-3616, 1432-0916}, url = {http://arxiv.org/abs/math-ph/0210015}, doi = {10.1007/s00220-003-0838-0}, 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.}, language = {en}, number = {1}, urldate = {2020-12-14}, journal = {Communications in Mathematical Physics}, author = {Landsman, N. P.}, month = jun, year = {2003}, note = {arXiv: math-ph/0210015}, keywords = {46L80, 53D55, Mathematical Physics, Mathematics - K-Theory and Homology}, pages = {87--103}, }

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