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PARAMETRIZED STRICT DEFORMATION QUANTIZATION OF C*-BUNDLES AND HILBERT C*-MODULES

Part of: Modules, bimodules and ideals Selfadjoint operator algebras Infinite-dimensional manifolds Symplectic geometry, contact geometry

Published online by Cambridge University Press:  18 May 2011

KEITH C. HANNABUSS  and
VARGHESE MATHAI
KEITH C. HANNABUSS
Affiliation:
Mathematical Institute, 24–29 St. Giles’, Oxford OX1 3LB, England Balliol College, Oxford OX1 3BJ, England (email: [email protected])
VARGHESE MATHAI*
Affiliation:
Department of Pure Mathematics, University of Adelaide, Adelaide SA 5005, Australia (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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In this paper, we review the parametrized strict deformation quantization of C*-bundles obtained in a previous paper, and give more examples and applications of this theory. In particular, it is used here to classify H3-twisted noncommutative torus bundles over a locally compact space. This is extended to the case of general torus bundles and their parametrized strict deformation quantization. Rieffel’s basic construction of an algebra deformation can be mimicked to deform a monoidal category, which deforms not only algebras but also modules. As a special case, we consider the parametrized strict deformation quantization of Hilbert C*-modules over C*-bundles with fibrewise torus action.

Keywords

parametrized strict deformation quantizationnoncommutative principal torus bundlesT-dualityHilbert module categories

MSC classification

Secondary: 58B34: Noncommutative geometry (à la Connes) 46L87: Noncommutative differential geometry 16D90: Module categories ; module theory in a category-theoretic context; Morita equivalence and duality 53D55: Deformation quantization, star products
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 90 , Issue 1 , February 2011 , pp. 25 - 38
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

Footnotes

The second author was supported by the Australian Research Council.

References

[1]Bouwknegt, P., Hannabuss, K. C. and Mathai, V., ‘C*-algebras in tensor categories’, Clay Math. Proc. 12 (2011), 127–165.Google Scholar
[2]Brain, S. and Landi, G., ‘Moduli spaces of noncommutative instantons: gauging away noncommutative parameters’, Oxford Quart. J., doi: 10.1093/qmath/haq036.CrossRefGoogle Scholar
[3]Connes, A., ‘C*-algèbres et géometrie différentielle’, C. R. Acad. Sci. Paris Ser. A–B 290(13) (1980), 599–604.Google Scholar
[4]Connes, A., ‘An analogue of the Thom isomorphism for crossed products of a C *-algebra by an action of R’, Adv. Math. 39(1) (1981), 31–55.CrossRefGoogle Scholar
[5]Crocker, D., Kumjian, A., Raeburn, I. and Williams, D. P., ‘An equivariant Brauer group and actions of groups on C*-algebras’, J. Funct. Anal. 146(1) (1997), 151–184.CrossRefGoogle Scholar
[6]Echterhoff, S., Nest, R. and Oyono-Oyono, H., ‘Principal non-commutative torus bundles’, Proc. Lond. Math. Soc. 99 (2009), 1–31.CrossRefGoogle Scholar
[7]Echterhoff, S. and Williams, D. P., ‘Crossed products by C0(X)-actions’, J. Funct. Anal. 158(1) (1998), 113–151.CrossRefGoogle Scholar
[8]Fell, J. M. G. and Doran, R. S., Representations of *-Algebras, Locally Compact Groups, and Banach *-Algebraic Bundles (Academic Press, New York, 1988).Google Scholar
[9]Hannabuss, K. C. and Mathai, V., ‘Noncommutative principal torus bundles via parametrised strict deformation quantization’, AMS Proc. Symp. Pure Math. 81 (2010), 133–148.CrossRefGoogle Scholar
[10]Kahn, P. J., ‘Symplectic torus bundles and group extensions’, New York J. Math. 11 (2005), 35–55.Google Scholar
[11]Kasparov, G., ‘Hilbert C*-modules: theorems of Stinespring and Voiculescu’, J. Operator Theory 4 (1980), 133–150.Google Scholar
[12]Kasparov, G., ‘Equivariant K-theory and the Novikov conjecture’, Invent. Math. 91 (1988), 147–201.CrossRefGoogle Scholar
[13]Kasprzak, P., ‘Rieffel deformation via crossed products’, J. Funct. Anal. 257 (2009), 1288–1332.CrossRefGoogle Scholar
[14]Landstad, M. B., ‘Duality theory for covariant systems’, Trans. Amer. Math. Soc. 248 (1979), 223–267.CrossRefGoogle Scholar
[15]Landstad, M. B., ‘Quantization arising from abelian subgroups’, Internat. J. Math. 5 (1994), 897–936.CrossRefGoogle Scholar
[16]Mathai, V. and Rosenberg, J., ‘T-duality for torus bundles via noncommutative topology’, Comm. Math. Phys. 253(3) (2005), 705–721.CrossRefGoogle Scholar
[17]Mathai, V. and Rosenberg, J., ‘T-duality for torus bundles with H-fluxes via noncommutative topology, II: the high-dimensional case and the T-duality group’, Adv. Theor. Math. Phys. 10(1) (2006), 123–158.CrossRefGoogle Scholar
[18]Rieffel, M. A., ‘Induced representations of C*-algebras’, Adv. Math. 13 (1974), 176–257.CrossRefGoogle Scholar
[19]Rieffel, M. A., ‘Applications of strong Morita equivalence to transformation group C *-algebras’, AMS Proc. Symp. Pure Math. 38 (1982), 299–310, part I.CrossRefGoogle Scholar
[20]Rieffel, M. A., ‘Proper actions of groups on C *-algebras’, in: Mappings of Operator Algebras (Philadelphia, PA, 1988), Progress in Mathematics, 84 (Birkhäuser, Boston, 1990), pp. 141–182.Google Scholar
[21]Rieffel, M. A., ‘Deformation quantization for actions of Rd’, Mem. Amer. Math. Soc. 106(506) (1993), 93.Google Scholar
[22]Rieffel, M. A., ‘Quantization and C*-algebras’, Contemp. Math. 167 (1994), 67–97.Google Scholar