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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

$ {\rm II}\sb 1$ factors, their bimodules and hypergroups


Author: V. S. Sunder
Journal: Trans. Amer. Math. Soc. 330 (1992), 227-256
MSC: Primary 46L35; Secondary 20N99, 43A62, 46L10, 46L55
DOI: https://doi.org/10.1090/S0002-9947-1992-1049618-6
MathSciNet review: 1049618
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Abstract: In this paper, we introduce a notion that we call a hypergroup; this notion captures the natural algebraic structure possessed by the set of equivalence classes of irreducible bifinite bimodules over a II$ _{1}$ factor. After developing some basic facts concerning bimodules over II$ _{1}$ factors, we discuss abstract hypergroups. To make contact with the problem of what numbers can arise as index-values of subfactors of a given II$ _{1}$ factor with trivial relative commutant, we define the notion of a dimension function on a hypergroup, and prove that every finite hypergroup admits a unique dimension function, we then give some nontrivial examples of hypergroups, some of which are related to the Jones subfactors of index $ 4{\cos ^2}\pi /(2n + 1)$. In the last section, we study the hypergroup invariant corresponding to a bifinite module, which is used, among other things, to obtain a transparent proof of a strengthened version of what Ocneanu terms 'the crossed-product remembering the group.'


References [Enhancements On Off] (What's this?)

  • [BS] R. B. Bapat and V. S. Sunder, On hypergroups of matrices, Linear and Multilinear Algebra 29 (1991), 125-140. MR 1119446 (92k:20137)
  • [C] A. Connes, Correspondences, hand-written notes.
  • [DR] S. Doplicher and J. E. Roberts, Duals of compact Lie groups realized in the Cuntz algebra and their actions on $ {C^{\ast}}$-algebras, J. Funct. Anal. 74 (1987), 96-120. MR 901232 (89a:22011)
  • [J] V. F. R. Jones, Index for subfactors, Invent. Math. 71 (1983), 1-25. MR 696688 (84d:46097)
  • [MP] J. R. McMullen and J. F. Price, Reversible hypergroups, Rend. Sem. Mat. Fis. Milano 47 (1977), 67-85. MR 526875 (80c:20010)
  • [O$ _{1}$] A. Ocneanu, Subalgebras are canonically fixed-point algebras, Amer. Math. Soc. Abstracts 6 (1986), 822-99-165.
  • [O$ _{2}$] -, A Galois theory for von Neumann algebras, preprint 1985.
  • [O$ _{3}$] -, Quantized groups, string algebras and Galois theory for operator algebras, Operator Algebras and Applications, Vol. 2 (Warwick 1987), LMS Lecture Notes Ser., vol. 136, Cambridge Univ. Press, 1988, pp. 119-172.
  • [PP] M. Pimsner and S. Popa, Entropy and index for subfactors, Ann. Sci. École Norm. Sup. 19 (1986), 57-106. MR 860811 (87m:46120)
  • [P] S. Popa, Correspondences, INCREST preprint 1986. MR 860346 (87m:46115)
  • [R] K. A. Ross, Hypergroups and centers of measure algebras, 1st Naz. di Alta Mat., Symposia Math. 22 (1977), 189-203 MR 0511036 (58:23344)
  • [W] H. Wenzl, Representations of Hecke algebras and subfactors, Thesis, Univ. of Pennsylvania, 1985.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1992-1049618-6
Article copyright: © Copyright 1992 American Mathematical Society

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