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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
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.'

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  • [BS] R. B. Bapat and V. S. Sunder, On hypergroups of matrices, Linear and Multilinear Algebra 29 (1991), no. 2, 125–140. MR 1119446, 10.1080/03081089108818063
  • [C] A. Connes, Correspondences, hand-written notes.
  • [DR] Sergio Doplicher and John E. Roberts, Duals of compact Lie groups realized in the Cuntz algebras and their actions on 𝐶*-algebras, J. Funct. Anal. 74 (1987), no. 1, 96–120. MR 901232, 10.1016/0022-1236(87)90040-1
  • [J] V. F. R. Jones, Index for subfactors, Invent. Math. 72 (1983), no. 1, 1–25. MR 696688, 10.1007/BF01389127
  • [MP] J. R. McMullen and J. F. Price, Reversible hypergroups, Rend. Sem. Mat. Fis. Milano 47 (1977), 67–85 (1979) (English, with Italian summary). MR 526875, 10.1007/BF02925743
  • [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] Mihai Pimsner and Sorin Popa, Entropy and index for subfactors, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 1, 57–106. MR 860811
  • [P] Sorin Popa, A short proof of “injectivity implies hyperfiniteness” for finite von Neumann algebras, J. Operator Theory 16 (1986), no. 2, 261–272. MR 860346
  • [R] Kenneth A. Ross, Hypergroups and centers of measure algebras, Symposia Mathematica, Vol. XXII (Convegno sull’Analisi Armonica e Spazi di Funzioni su Gruppi Localmente Compatti, INDAM, Rome, 1976) Academic Press, London, 1977, pp. 189–203. MR 0511036
  • [W] H. Wenzl, Representations of Hecke algebras and subfactors, Thesis, Univ. of Pennsylvania, 1985.

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Article copyright: © Copyright 1992 American Mathematical Society