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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References
Arnal, D.: *-products and representations of nilpotent groups. Pacific J. Math.114, 285–308 (1984)
Basart, H., Lichnerowicz, A.: Conformal symplectic geometry, deformations, rigidity and geometrical (KMS) conditions. Lett. Math. Phys.10, 167–177 (1985)
Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., Sternheimer, D.: Deformation theory and quantization, I, II. Ann. Phys.110, 61–110, 111–151 (1978)
Bayen, F., Fronsdal, C.: Quantization on the sphere. J. Math. Phys.22, 1345–1349 (1981)
Bratteli, O., Elliott, G. A., Goodman, F. M., Jorgensen, P. E. T.: Smooth Lie group actions on non-commutative tori, preprint
Cahen, M., Gutt, S.: Non localité d'une deformation symplectique sur la sphereS 2. Bull. Soc. Math. Belg.36, 207–214 (1984)
Connes, A.:C*-algébres et géometrie differentielle. C. R. Acad. Sci. Paris290, 599–604 (1980)
——,: Non-commutative differential geometry. Pub. I.H.E.S.62, 257–360 (1986)
Dixmier, J.: LesC*-algèbres et leurs représentations, 2nd ed., Paris: Gauthier-Villars 1969
Green, P.: C*-algebras of transformation groups with smooth orbit space. Pacific J. Math.72, 71–97 (1977)
Jorgensen, P. E. T., Moore, R. T.: Operator commutation relations. Dordrecht: D. Reidel 1984
Lichnerowicz, A.: In: Deformations and quantization, geometry and physics. Modugno M. (ed.). pp. 103–116. Bologna: Pilagora Editrice Bologna 1983.
Moreno, C.: Invariant star products and representations of compact semisimple Lie groups. Lett. Math. Phys.12, 217–229 (1986)
Moreno, C., Ortega-Navarro, P.: *-Products onD 1(C),S 2 and related spectral analysis. Lett. Math. Phys.7, 181–193 (1983)
Muhly, P. S., Williams, D. P.: Transformation groupC*-algebras with continuous trace, II. J. Operator. Theory11, 109–124 (1984)
Olesen, D., Pedersen, G. K.: Applications of the Connes spectrum toC*-dynamical systems, II. J. Funct. Anal.36, 18–32 (1980)
Olesen, D., Pedersen, G. K., Takesaki, M.: Ergodic actions of compact Abelian groups. J. Operator Theory3, 237–269 (1980)
Packer, J. A.:C*-algebras corresponding to projective representations of discrete Heisenberg groups. J. Operator Theory18, 42–66 (1987)
--,: Strong Morita equivalence for HeisenbergC*-algebras and the positive cones of theirK 0-groups. Canadian J. Math. (to appear)
--,: Twisted groupC*-algebras corresponding to nilpotent discrete groups. Math. Scand. (to appear)
Packer, J. A., Raeburn, I.: The structure of twisted groupC*-algebras, preliminary version
Pedersen, G. K.:C*-algebras and their automorphism groups. Lond. Math. Soc. Monographs vol.14, London: Academic Press 1979
Podles, P.: Quantum spheres. Lett. Math. Phys.14, 193–202 (1987)
Raeburn, I., Williams, D. P.: Pull-backs ofC*-algebras and crossed products by certain diagonal actions. Trans. A.M.S.287, 755–777 (1985)
Rieffel, M. A.: Unitary representations of group extensions; an algebraic approach to the theory of Mackey and Blattner. Stud. Anal. Adv. Math. [Suppl.] Ser.4, 43–82 (1979)
——,: Applications of strong Morita equivalence to transformation groupC*-algebras. Operator Algebras Appl. Kadison, R. V.: (ed.). pp. 299–310. Proc. Symp. Pure Math. vol.38, Providence, RI: American Mathematical Society 1982
——,: Projective modules over higher dimensional non-commutative tori. Canadian J. Math.40, 257–338 (1988)
--,: Continuous fields ofC*-algebras from group cocycles and actions, Math. Ann. (to appear)
--,: Proper actions of groups onC*-algebras, preprint
Vey, J.: Déformation du crochet de Poisson sur une variété symplectique. Commun. Math. Helv.50, 421–454 (1975)
Wassermann, A. J.: Ergodic actions of compact groups on operator algebras, III: Classification forSU(2). Invent. Math.93, 309–354 (1988)
Weinstein, A.: Poisson geometry of the principal series and nonlinearizable structures. J. Diff. Geom.25, 55–73 (1987)
Williams, D. P.: The topology of the primitive ideal space of transformation groupC*-algebras and C. C. R. transformation groupC*-algebras. Trans. A.M.S.266, 335–359 (1981)
Woronowicz, S. L.: TwistedSU(2) group. An example of a non-commutative differential calculus. Pub. R.I.M.S. Kyoto University23, 117–181 (1987)
——,: Compact matrix pseudogroups. Commun. Math. Phys.111, 613–665 (1987)
——,: Tannaka-Krein duality for compact matrix pseudogroups. TwistedSU(N) groups, Invent. Math.93, 35–76 (1988)
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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