Abstract
ForM a smooth manifold equipped with a Poisson bracket, we formulate aC*-algebra framework for deformation quantization, including the possibility of invariance under a Lie group of diffeomorphisms preserving the Poisson bracket. We then show that the much-studied non-commutative tori give examples of such deformation quantizations, invariant under the usual action of ordinary tori. Going beyond this, the main results of the paper provide a construction of invariant deformation quantizations for those Poisson brackets on Heisenberg manifolds which are invariant under the action of the Heisenberg Lie group, and for various generalizations suggested by this class of examples. Interesting examples are obtained of simpleC*-algebras on which the Heisenberg group acts ergodically.
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Communicated by A. Connes
This work was supported in part by National Science Foundation grant DMS 8601900
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Rieffel, M.A. Deformation quantization of Heisenberg manifolds. Commun.Math. Phys. 122, 531–562 (1989). https://doi.org/10.1007/BF01256492
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DOI: https://doi.org/10.1007/BF01256492