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A characterization of normal extensions for subfactors


Author: Tamotsu Teruya
Journal: Proc. Amer. Math. Soc. 120 (1994), 781-783
MSC: Primary 46L37
DOI: https://doi.org/10.1090/S0002-9939-1994-1207542-7
MathSciNet review: 1207542
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Abstract: Let $ N \subset M \subset L$ be a tower of factors. If $ L$ is a crossed product $ N{ \rtimes _\alpha }G$ of $ N$ by an outer action $ \alpha $ of a finite group $ G$ on $ N$ then it is well known that there exists a subgroup $ H$ of $ G$ such that $ M = N{ \rtimes _{{\alpha _{{\vert _H}}}}}H$. We prove in this paper that $ H$ is a normal subgroup of $ G$ if and only if there exist a finite group $ F$ and an outer action $ \beta $ of $ F$ on $ M$ such that $ L = M{ \rtimes _\beta }F$.


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  • [GHJ] F. Goodman, P. de la Harpe, and V. F. R. Jones, Coxeter graphs and towers of algebras, Math. Sci. Res. Inst. Publ., vol. 14, Springer-Verlag, New York, 1989. MR 999799 (91c:46082)
  • [J1] V. F. R. Jones, Actions of finite groups on the hyperfinite type $ {\operatorname{II} _1}$ factor, Mem. Amer. Math. Soc., no 237, Amer. Math. Soc., Providence, RI, 1980. MR 587749 (81m:46094)
  • [J2] -, Index for subfactors, Invent. Math. 72 (1983). MR 696688 (84d:46097)
  • [K] H. Kosaki, Characterization of crossed product (properly infinite case), Pacific J. Math. 137 (1989), 159-167. MR 983334 (90k:46139)
  • [KY] H. Kosaki and S. Yamagami, Irreducible bimodules associated with crossed product algebras, Internat. J. Math. 3 (1992), 661-676. MR 1189679 (94f:46087)
  • [NT] N. Nakamura and Z. Takeda, A Galois theory for finite factors, Proc. Japan Acad. 36 (1960), 258-260. MR 0123925 (23:A1246)
  • [O] A. Ocneanu, Quantum symmetry, differential geometry of finite graphs and classification of subfactors, University of Tokyo seminary notes, 1991.
  • [PP] M. Pimsner and S. Popa, Entropy and index for subfactors, Ann. Sci. École Norm. Sup. (4) 19 (1986), 57-106. MR 860811 (87m:46120)
  • [S] Colin E. Sutherland, Cohomology and extensions of von Neumann algebras. I, II, Publ. Res. Inst. Math. Sci. 16 (1980), 105-133, 135-174. MR 574031 (81k:46067)
  • [T] Z. Takeda, On the normal basis theorem of the Galois theory for finite factors, Proc. Japan Acad. 37 (1961). MR 0130874 (24:A728)
  • [W] Y. Watatani, Index for $ {C^{\ast}}$-subalgebras, Mem. Amer. Math. Soc., no. 424, Amer. Math. Soc., Providence, RI, 1990. MR 996807 (90i:46104)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1994-1207542-7
Keywords: Factor, outer action, crossed product, Galois theory, fixed point algebra, conditional expectation
Article copyright: © Copyright 1994 American Mathematical Society

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