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Quantization and Morita equivalence for constant Dirac structures on tori. (English) Zbl 1068.46044

Summary: We define a \(C^*\)-algebraic quantization of constant Dirac structures on tori and prove that \(O(n,n|\mathbb{Z})\)-equivalent structures have Morita equivalent quantizations. This completes and extends from the Poisson case a theorem of Rieffel and Schwarz.

MSC:

46L65 Quantizations, deformations for selfadjoint operator algebras
81S10 Geometry and quantization, symplectic methods
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References:

[1] J. Block & E. Getzler, Quantization of foliations, Vol. 1, 2, World Scientific, 1992, p. 471-4870812.58028 · Zbl 0812.58028
[2] A. Connes, Noncommutative Geometry, Academic Press, 19940818.460761303779 · Zbl 0818.46076
[3] A. Connes, M.R. Douglas & A. Schwarz, Noncommutative Geometry and Matrix Theory: Compactification on Tori, J. High Energy Phys (1998)1018.810521613978 · Zbl 1018.81052
[4] T.J. Courant, Dirac manifolds, Trans. A.M.S319 (1990) p. 631-6610850.70212998124 · Zbl 0850.70212
[5] G. A. Elliott, On the K-theory of the \(C*C^*\) algebras generated by a projective representation of a torsion-free discrete abelian group, Pitman, 1984, p. 157-1840542.46030 · Zbl 0542.46030
[6] G.A. Elliott & H. Li, Morita equivalence of smooth noncommutative tori, e-print, math.OA/0311502 · Zbl 1137.46030
[7] [7] , Kronecker foliation, \(D1\)-branes and Morita equivalence of noncommutative two-tori, J. High Energy Phys8 (2002) no.50 · Zbl 1226.81097
[8] M. Kontsevich, Homological algebra of mirror symmetry., Vol. 1, 2, Birkhäuser, 1995, p. 120-1390846.53021 · Zbl 0846.53021
[9] H. Li, Strong Morita equivalence of higher-dimensional noncommutative tori, e-print. To appear J. Reine Angew. Math., math.OA/03031231063.460572099203 · Zbl 1063.46057
[10] F. Lizzi & R. Szabo, Noncommutative Geometry and String Duality, J. High Energy Phys., 1999 · Zbl 0994.81111
[11] P.S. Muhly, J.N. Renault & D.P. Williams, Equivalence and isomorphism for groupoid \(C*C^*\)-algebras, J. Operator Theory17 (1987) p. 3-220645.46040873460 · Zbl 0645.46040
[12] S. Mukai, Duality between D(X)\(D(X)\) and D(X^)\(D(\widehat{X})\) with its application to Picard sheaves, Nagoya Math. J.81 (1981) p. 153-1750417.14036607081 · Zbl 0417.14036
[13] M.A. Rieffel, Morita equivalence for \(C*C^*\)-algebras and \(W*W^*\)-algebras, J. Pure Appl. Algebra5 (1974) p. 51-960295.46099367670 · Zbl 0295.46099
[14] M.A. Rieffel, \(C*C^*\)-algebras associated with irrational rotations, Pacific. J. Math.93 (1981) p. 415-4290499.46039623572 · Zbl 0499.46039
[15] M.A. Rieffel, Projective modules over higher-dimensional non-commutative noncommutative tori, Canadian J. Math40 (1988) p. 257-3380663.46073941652 · Zbl 0663.46073
[16] M.A. Rieffel, Deformation quantization of Heisenberg manifolds, Commun. Math. Phys122 (1989) p. 531-5620679.460551002830 · Zbl 0679.46055
[17] M.A. Rieffel & A. Schwarz, Morita equivalence of multidimensional noncommutative tori, Int. J. Math10 (1999) p. 289-2990968.460601687145 · Zbl 0968.46060
[18] A. Schwarz, Morita equivalence and duality, Lett. Math. Phys50 (1999) p. 309-3210967.580041663471 · Zbl 0967.58004
[19] X. Tang, Deformation Quantization of Pseudo Symplectic (Poisson) Groupoids, e-print, math.QA/040537805051278 · Zbl 1119.53061
[20] [20] , Symplectic groupoids, geometric quantization, and irrational rotation algebras, MSRI Series, Springer, 1991, p. 281-290 · Zbl 0731.58031
[21] P. Xu, Noncommutative Poisson algebras, Amer. J. Math116 (1994) p. 101-1250797.580121262428 · Zbl 0797.58012
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