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Characteristic invatiant of tensor product actions and actions on crossed products

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  09 April 2009

Yukako Miwa  and
Yoshikazu Katayama
Yoshikazu Katayama
Affiliation:
Division of Mathematical Sciences, Osaka-kyoiku University, Kashiwara, Osaka 582, Japan e-mail: [email protected]
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Abstract

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The first purpose of this paper is to give a tensor product formula of the characteristic invariant and modular invariant for a tensor product action of a discrete group G on AFD factors. The second purpose is to describe a characteristic invariant and modular invariant of the extended action to a crossed product in terms of the original invariants.

MSC classification

Secondary: 46L40: Automorphisms
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 73 , Issue 3 , December 2002 , pp. 357 - 376
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Connes, A., ‘Une classification des facteurs de type III’, Ann. Sci. École Norm. Sup. 6 (1973), 133–252.CrossRefGoogle Scholar
[2]Connes, A., ‘Almost periodic states and factors of type III1’, J. Funct. Anal. 16 (1974), 415–445.CrossRefGoogle Scholar
[3]Connes, A., ‘Outer conjugacy classes of automorphisms of factors’, Ann. Sci. École Norm. Sup. 8 (1975), 383–419.CrossRefGoogle Scholar
[4]Connes, A., ‘Classification of injective factors, cases II1 IIλ, λ ≠ 1’, Ann. of Math. (2) 104 (1976), 73–115.CrossRefGoogle Scholar
[5]Connes, A., ‘On the classification of von Neumann algebras and their automorphisms’, Symposia Marhematica, vol. 20 (INDAM, Rome, 1975) (Academic Press, London, 1976), pp. 435–478.Google Scholar
[6]Connes, A., ‘Periodic automorphisms of the hyperfinite factor of type II1’, Acta Sci. Math. 39 (1977), 39–66.Google Scholar
[7]Connes, A. and Takesaki, M., ‘The flow of weights on factors of type III’, Tôhoku Math. J. 29 (1977), 473–555.CrossRefGoogle Scholar
[8]Haagerup, U. and Størmer, E., ‘Pointwise inner automorphisms of von Neumann algebras with an appendix by C. Sutherland’, J. Funct. Anal. 92 (1990), 177–201.CrossRefGoogle Scholar
[9]Jones, V. F. R., Actions of finite groups on the hyperfinite type II1 factor, Mem. Amer. Math. Soc. 237 (Amer. Math. Soc., 1980).Google Scholar
[10]Jones, V. F. R. and Takesaki, M., ‘Actions of compact abelian groups on semifinite injective factors’, Acta Math. 153 (1984), 213–258.CrossRefGoogle Scholar
[11]Katayama, Y., Sutherland, C. E. and Takesaki, M., ‘The characteristic square of a factor and the cocycle conjugacy of discrete group actions on factors’, Invent. Math. 132 (1998), 331–380.CrossRefGoogle Scholar
[12]Kawahigashi, Y., ‘One parameter automorphism groups othe hyperfinite type II1 factor’, J. Operator Theory 25 (1991), 37–59.Google Scholar
[13]Kawahigashi, Y., Sutherland, C. E. and Takesaki, M., ‘The structure of the automorphism group of an injective factor and the cocycle conjugacy of discrete abeian group actions’, Acta Math. 169 (1992), 105–130.CrossRefGoogle Scholar
[14]Kawahigashi, Y. and Takesaki, M., ‘Compact abelian group actions on injective factors’, J. Funct. Anal. 105 (1992), 112–128.CrossRefGoogle Scholar
[15]Krieger, W., ‘On ergodic flows and the isomorphism of factors’, Math. Ann. 223 (1976), 19–70.CrossRefGoogle Scholar
[16]Miwa, Y., Remarks on tensor formulae of characteristic invariant and modular invariant for action on AFD factors (M.Sc. Thesis, Osaka Kyoiku University, 1999).Google Scholar
[17]Ocneanu, A., Actions of discrete amenable groups on factors, Lecture Notes in Math. 1138 (Springer, Berlin, 1985).Google Scholar
[18]Sekine, Y., ‘Flows of weights of crossed products of type III factors by discrete groups’, Publ. Res. Inst. Math. Sci. 26 (1990), 655–666.CrossRefGoogle Scholar
[19]Sutherland, C. E. and Takesaki, M., ‘Actions of discrete amenable groups and groupoids on von Neumann algebras’, Publ. Res. Inst. Math. Sci. 21 (1985), 1087–1120.CrossRefGoogle Scholar
[20]Sutherland, C. E. ‘Actions of discrete amenable groups on injective factors of type IIIλ, λ ≠ 1’, Pacific J. Math. 137 (1989), 405–444.CrossRefGoogle Scholar
[21]Takesaki, M., ‘Duality for crossed products and the structure of von Neumann algebras of type III’, Acta Math. 131 (1973), 249–310.CrossRefGoogle Scholar