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Twisted crossed products by coactions

Part of: Selfadjoint operator algebras Locally compact groups and their algebras

Published online by Cambridge University Press:  09 April 2009

John Phillips  and
Iain Raeburn
John Phillips
Affiliation:
Department of Mathematics and Statistics, University of Victoria, P.O. Box 3045, Victoria, B.C. V8W 3P4, Canada
Iain Raeburn
Affiliation:
Department of Mathematics, University of Newcastle, Newcastle, New South Wales, 2308, Australia
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Abstract

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We consider coactions of a locally compact group G on a C*-algebra A, and the associated crossed product C*-algebra A× G. Given a normal subgroup N of G, we seek to decompose A× G as an iterated crossed product (A× G/ N) × N, and introduce notions of twisted coaction and twisted crossed product which make this possible. We then prove a duality theorem for these twisted crossed products, and discuss how our results might be used, especially when N is abelian.

MSC classification

Secondary: 46L55: Noncommutative dynamical systems 22D25: $C^*$-algebras and $W^*$-algebras in relation to group representations 22D35: Duality theorems
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 56 , Issue 3 , June 1994 , pp. 320 - 344
Copyright
Copyright © Australian Mathematical Society 1994

References

[1]Bédos, E., ‘Discrete groups and simple C*-algebras’, Math. Proc. Cambridge Phil. Soc. 109 (1991), 521–537.CrossRefGoogle Scholar
[2]Blackadar, B., K-theory for operator algebras, MSRI Publications 5 (Springer, New York, 1986).CrossRefGoogle Scholar
[3]Brown, L. G., Green, P. and Rieffel, M. A., ‘Stable isomorphism and strong Morita equivalence of C*-algebras’, Pacific J. Math. 71 (1977), 349–363.CrossRefGoogle Scholar
[4]Busby, R. C. and Smith, H. A., ‘Representations of twisted group algebras’, Trans. Amer. Math. Soc. 149 (1970), 503–537.CrossRefGoogle Scholar
[5]Connes, A., ‘An analogue of the Thom isomorphism for crossed products of C*-algebras’, Adv. in Math. 39 (1981), 31–55.CrossRefGoogle Scholar
[6]de Brabanter, M., ‘Decomposition theorems for certain C*-crossed products’, Math. Proc. Cambridge Phil. Soc. 94 (1983), 265–275.CrossRefGoogle Scholar
[7]Green, P., ‘The local structure of twisted covariance algebras’, Acta Math. 140 (1978), 191–250.CrossRefGoogle Scholar
[8]Herz, C., ‘Le rapport entre l'algèbre A p d'un groupe et d'un sous-groupe’, C. R. Acad. Sci. Paris Sér.1 Math. A271 (1970), 244–246.Google Scholar
[9]Iwasawa, K., ‘On some types of topological groups’, Ann. of Math. 50 (1949), 507–558.CrossRefGoogle Scholar
[10]Katayama, Y., ‘Takesaki's duality for a nondegenerate coaction’, Math. Scand. 55 (1985), 141–151.CrossRefGoogle Scholar
[11]Lanstad, M. B., ‘Ergodic actions of nonabelian compact groups’, in: Ideas and methods in mathematical analysis, stochastics and applications (ed. Albeverio, S. et al. ) (Cambridge University Press, London, 1992) pp. 365–388.Google Scholar
[12]Landstad, M. B., Philips, J., Raeburn, I. and Sutherland, C. E., ‘Representations of crossed products by coactions and principal bundles’, Trans. Amer. Math. Soc. 299 (1987), 747–784.CrossRefGoogle Scholar
[13]Mansfield, K., ‘Induced representations of crossed products by coactions’, J. Funct. Anal. 97 (1991), 112–161.CrossRefGoogle Scholar
[14]Moore, C. C., ‘Group extensions and cohomology for locally compact groups, III’, Trans. Amer. Math. Soc. 221 (1976), 1–33.CrossRefGoogle Scholar
[15]Nakagami, Y. and Takesaki, M., Duality for crossed products of von Neumann algebras, Lecture Notes in Math. 731 (Springer, Berlin, 1979).CrossRefGoogle Scholar
[16]Olesen, D. and Raeburn, I., ‘Pointwise unitary automorphism groups’, J. Funct. Anal. 93 (1990), 278–309.CrossRefGoogle Scholar
[17]Packer, J. A. and Raeburn, I., ‘Twisted crossed products of C*-algebras’, Math. Proc. Cambridge Phil. Soc. 106 (1989), 293–311.CrossRefGoogle Scholar
[18]Packer, J. A. and Raeburn, I., ‘Twisted crossed products of C*-algebras, II’, Math. Ann. 287 (1990), 595–612.CrossRefGoogle Scholar
[19]Pedersen, G. K., C*-algebras and their automorphism groups (Academic Press, London, 1979).Google Scholar
[20]Quigg, J. C., ‘Duality for reduced twisted crossed products of C*-algebras’, Indiana Univ. Math. J. 35 (1986), 549–571.CrossRefGoogle Scholar
[21]Quigg, J. C., ‘Full C*-crossed product duality’, J. Austral. Math. Soc. (Series A) 50 (1991), 34–52.CrossRefGoogle Scholar
[22]Raeburn, I., ‘On crossed products by coactions and their representation theory’, Proc. London Math. Soc. 64 (1992), 625–652.CrossRefGoogle Scholar
[23]Rieffel, M. A., Unitary representations of group extensions: an algebraic approach to the theory of Mackey and Blattner, Adv. in Math. Suppl. Studies 4 (Academic Press, New York, 1979) pp. 43–82.Google Scholar
[24]Takesaki, M., ‘Covariant representations of C*-algebras and their locally compact automorphism groups’, Acta Math. 119 (1967), 273–303.CrossRefGoogle Scholar
[25]Wassermann, A., ‘Ergodic actions of compact groups on operator algebras, II: classification of full multiplicity ergodic actions’, Canad. J. Math. 40 (1988), 1482–1527.CrossRefGoogle Scholar