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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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$\textrm {II}_ 1$ factors, their bimodules and hypergroups
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by V. S. Sunder
Trans. Amer. Math. Soc. 330 (1992), 227-256
DOI: https://doi.org/10.1090/S0002-9947-1992-1049618-6
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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 $\text {II}_{1}$ factor. After developing some basic facts concerning bimodules over $\text {II}_{1}$ factors, we discuss abstract hypergroups. To make contact with the problem of what numbers can arise as index-values of subfactors of a given $\text {II}_{1}$ 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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Bibliographic Information
  • © Copyright 1992 American Mathematical Society
  • 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