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Waldmann, Stefan

The Picard groupoid in deformation quantization. (English) Zbl 1065.53070

Lett. Math. Phys. 69, Spec. Iss., 223-235 (2004).
The paper under review is divided into six sections. Section 1 is a short introduction where the author clarifies the picture of his paper. Section 2 presents some motivations for the study of *-representations. The notion of positivity makes the object of section 3. Section 4 deals with the study of pre-Hilbert modules and their tensor products. In section 5 he presents the strong Morita equivalence and its corresponding Picard groupoid. Finally, the last section proves how some Morita invariants can be seen as arising from actions of the Picard groupoid. In conclusion, a very nice paper.
Reviewer: Mircea Puta (Timişoara)

Cited in 3 Documents

MSC:

53D55 Deformation quantization, star products

Keywords:

deformation quantization; Morita equivalence; Picard grupoid
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\textit{S. Waldmann}, Lett. Math. Phys. 69, 223--235 (2004; Zbl 1065.53070)
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References:

[1] Ara, P.: Morita equivalence for rings with involution, Algebra Represent. Theory 2 (1999), 227-247. · Zbl 0944.16005
[2] Ara, P.: Morita equivalence and Pedersen Ideals, Proc. Amer. Math. Soc. 129(4) (2000), 1041-1049. · Zbl 1058.46038
[3] Bass, H.: Algebraic K-theory, W. A. Benjamin, New York, 1968.
[4] Bayen, F., Flato, M., Frønsdal, C., Lichnerowicz, A. and Sternheimer, D.: Deformation theory and quantization, Ann. Phys. 111 (1978), 61-151. · Zbl 0377.53024
[5] Beer, W.: On Morita equivalence of nuclear C*-algebras, J. Pure Appl. Algebra. 26(3) (1982), 249-267. · Zbl 0528.46042
[6] B’enabou, J.: Introduction to bicategories, In: Reports of the Midwest Category Seminar, Springer-Verlag, 1967, pp. 1-77. · Zbl 1375.18001
[7] Bertelson, M., Cahen, M. and Gutt, S.: Equivalence of star products, Classical Quantum Gravity 14 (1997), A93-A107. · Zbl 0881.58021
[8] Bordemann, M., Neumaier, N., Pflaum, M. J. and Waldmann, S.: On representations of star product algebras over cotangent spaces on Hermitian line bundles, J. Funct. Anal. 199 (2003), 1-47. · Zbl 1038.53087
[9] Bordemann, M. and Waldmann, S.: Formal GNS construction and states in deformation quantization, Comm. Math. Phys. 195 (1998), 549-583. · Zbl 0989.53057
[10] Brown, L. G., Green, P. and Rieffel, M.: Stable isomorphism and strong Morita equivalence of C*-algebras, Pacific J. Math. 71 (1977), 349-363. · Zbl 0362.46043
[11] Bursztyn, H.: Semiclassical geometry of quantum line bundles and Morita equivalence of star products. Internet. Math. Res. Notices 2002(16) 2002, 821-846. · Zbl 1031.53120
[12] Bursztyn, H. and Radko, O.: Gauge equivalence of Dirac structures and symplectic groupoids, Ann. Inst. Fourier 53 (2003), 309-337. · Zbl 1026.58019
[13] Bursztyn, H. and Waldmann, S.: *-Ideals and formal Morita equivalence of *-algebras, Internet. J. Math. 12(5) (2001), 555-577. · Zbl 1111.46303
[14] Bursztyn, H. and Waldmann, S.: Algebraic Rieffel induction, formal Morita equivalence and applications to deformation quantization, J. Geom. Phys. 37 (2001), 307-364. · Zbl 1039.46052
[15] Bursztyn, H. and Waldmann, S.: Bimodule deformations, Picard groups and contravariant connections, K-Theory 25 (2002), 1-37. · Zbl 0998.19004
[16] Bursztyn, H. and Waldmann, S.: The characteristic classes of Morita equivalent star products on symplectic manifolds, Comm. Math. Phys. 228 (2002), 103-121. · Zbl 1036.53068
[17] Bursztyn, H. and Waldmann, S.: Completely positive inner products and strong Morita equivalence, Preprint (FR-THEP 2003/12) math.QA/0309402. To appear in Pacific J. Math. · Zbl 1111.53071
[18] Bursztyn, H. and Weinstein, A.: Picard groups in Poisson geometry. Preprint math.SG/0304048, 2003. To appear in Moscow Math. J. · Zbl 1068.53055
[19] Cattaneo, A. and Felder, G.: A path integral approach to the Kontsevich quantization formula, Comm. Math. Phys. 212 (2000), 591-611. · Zbl 1038.53088
[20] DeWilde, M. and Lecomte, P. B. A.: Existence of star-products and of formal deformations of the Poisson lie algebra of arbitrary symplectic manifolds, Lett. Math. Phys. 7 (1983), 487-496. · Zbl 0526.58023
[21] Dito, G. and Sternheimer, D.: Deformation quantization: genesis, developments and metamorphoses, In: G. Halbout (ed.), Deformation Quantization, Vol. 1, IRMA Lectures in Math. Theoret. Phys., Walter de Gruyter, Berlin, 2002, pp. 9-54. · Zbl 1014.53054
[22] Fedosov, B. V.: A simple geometrical construction of deformation quantization, J. Differential Geom. 40 (1994), 213-238. · Zbl 0812.53034
[23] Gutt, S.: Variations on deformation quantization, In: G. Dito and D. Sternheimer (eds), Conf’erence Mosh’e Flato 1999. Quantization, Deformations, and Symmetries, Math. Phys. Stud. 21, Kluwer Acad. Publ., Dordrecht, 2000, pp. 217-254. · Zbl 0997.53068
[24] Haag, R.: Local Quantum Physics, 2nd edn, Springer-Verlag, Berlin, 1993. · Zbl 0792.46059
[25] Jur?o, B., Schupp, P. and Wess, J.: Noncommutative line bundles and Morita equivalence, Lett. Math. Phys. 61 (2002), 171-186. · Zbl 1036.53070
[26] Kontsevich, M.: Deformation quantization of Poisson manifolds, I, Preprint q-alg/9709040, 1997.
[27] Lance, E. C.: Hilbert C*-Modules. A Toolkit for Operator Algebraists, London Math. Soc. Lecture Note Ser. 210, Cambridge University Press, Cambridge, 1995. · Zbl 0822.46080
[28] Landsman, N. P.: Mathematical Topics between Classical and Quantum Mechanics, Springer Monogr. in Math., Springer-Verlag, Berlin, 1998. · Zbl 0923.00008
[29] Morita, K.: Duality for modules and its applications to the theory of rings with minimum condition, Sci. Rep. Tokyo Kyoiku Daigaku Sect. A 6 (1958), 83-142. · Zbl 0080.25702
[30] Nest, R. and Tsygan, B.: Algebraic index theorem, Comm. Math. Phys. 172 (1995), 223-262. · Zbl 0887.58050
[31] Omori, H., Maeda, Y. and Yoshioka, A.: Weyl manifolds and deformation quantization, Adv. Math. 85 (1991), 224-255. · Zbl 0734.58011
[32] Raeburn, I. and Williams, D. P.: Morita Equivalence and Continuous-trace C*-Algebras, Math. Surveys Monogr. 60, Amer. Math. Soc., Providence, RI, 1998. · Zbl 0922.46050
[33] Rieffel, M. A.: Induced representations of C*-algebras, Adv. Math. 13 (1974), 176-257. · Zbl 0284.46040
[34] Rieffel, M. A.: Morita equivalence for C*-algebras and W*-algebras, J. Pure Appl. Math. 5 (1974), 51-96. · Zbl 0295.46099
[35] Schmüdgen, K.: Unbounded Operator Algebras and Representation Theory, Vol. 37: Operator Theory: Advances and Applications, Birkh”auser-Verlag, Basel, 1990. · Zbl 0697.47048
[36] Weinstein, A. and Xu, P.: Hochschild cohomology and characteristic classes for starproducts, In: A. Khovanskij, A. Varchenko, and V. Vassiliev (eds), Geometry of Differential Equations, Amer. Math. Soc., Providence, 1998, pp. 177-194. · Zbl 0956.53055
[37] Xu, P.: Morita equivalence of Poisson manifolds, Comm. Math. Phys. 142 (1991), 493-509. · Zbl 0746.58034
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