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Crossed products of Hilbert C*-bimodules by bundles

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  09 April 2009

Tsuyoshi Kajiwara  and
Yasuo Watatani
Tsuyoshi Kajiwara
Affiliation:
Department of Environmental and Mathematical Sciences Okayama UniversityTsushima, 700, Japan
Yasuo Watatani
Affiliation:
Graduate School of Mathematics Kyushu UniversityRopponmatsu Fukuoka, 810, Japan
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Abstract

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We present the definition of crossed products of Hilbert C*-bimodules by Hilbert bundles with commuting finite group actions and finite dimensional fibers. This is a general construction containing the bundle construction and crossed products of Hilbert C*-bimodule by finite groups. We also study the structure of endomorphism algebras of the tensor products of bimodules. We also define the multiple crossed products using three bimodules containing more than 2 bundles and show the associativity law. Moreover, we present some examples of crossed product bimodules easily computed by our method.

MSC classification

Secondary: 46L35: Classifications of $C^*$-algebras 46L55: Noncommutative dynamical systems
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 64 , Issue 1 , February 1998 , pp. 119 - 135
Copyright
Copyright © Australian Mathematical Society 1998

References

[B]Blackadar, B., K-theory for operator algebras, Math. Sci. Res. Inst. Publ. 5 (Springer, Berlin, 1986).CrossRefGoogle Scholar
[CK]Choda, M. and Kosaki, H., ‘Strongly outer actions for an inclusion of factors’, J. Funct. Anal. 122 (1994), 315–332.CrossRefGoogle Scholar
[Co]Combs, F., ‘Crossed products and Morita equivalence’, Proc. London Math. Soc. 49 (1984), 289–306.CrossRefGoogle Scholar
[CMW]Curto, R. E., Muhly, P. S. and Williams, D. P., ‘Cross products of strongly Morita equivalent C*-algebras’, Proc. Amer. Math. Soc. 90 (1984), 315–332.Google Scholar
[GDJ]Goodman, F. M., de la Harpe, P. and Jones, V. F. R., Coxeter graphs and towers of algebras, Math. Sci. Res. Inst. Publ. 14 (Springer, Berlin, 1989).CrossRefGoogle Scholar
[J]Jones, V., ‘Index for subfactors’, Invent. Math. 72 (1983), 1–15.CrossRefGoogle Scholar
[K]Kajiwara, T., ‘Remarks on strongly Morita equivalent C*-crossed products’, Math. Japan. 32 (1987), 257–260.Google Scholar
[KW1]Kajiwara, T. and Watatani, Y., ‘Jones index theory by Hilbert C*-bimodules and K-theory’, preprint.Google Scholar
[KW2]Kajiwara, T. and Watatani, Y., ‘Crossed products of Hilbert C*-bimodules by countable discrete groups’, Proc. Amer. Math. Soc., to appear.Google Scholar
[Kay]Kajiwara, T. and Yamagami, S., ‘Irreducible bimodules associated with crossed product algebras II’, Pacific J. Math. 171 (1995), 209–229.CrossRefGoogle Scholar
[KoY]Kosaki, H. and Yamagami, S., ‘Irreducible bimodules associated with crossed product algebras’, Internat. J. Math. 3 (1992), 661–676.CrossRefGoogle Scholar
[O]Ocneanu, A., ‘Quantized symmetry, differential geometry of finite graphs and classification of subfactors’, (Notes by Kawahigashi, Y.), University of Tokyo Seminar Notes 45 (1991).Google Scholar
[W]Watatani, Y., Index for C*-subalgebras, Memoris Amer. Math. Soc. 424 (Amer. Math. Soc., Providence, 1990).Google Scholar
[Y1]Yamagami, S., ‘A note on Ocneanu's approach to Jones index theory’, Internat. J. Math. 4 (1993), 859–871.CrossRefGoogle Scholar
[Y2]Yamagami, S., ‘On Ocneanu's characterization of crossed products’, preprint.Google Scholar
[Y3]Yamagami, S., ‘Frobinus reciprocity in tensor categories’, preprint.Google Scholar