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Transactions of the American Mathematical Society
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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PII: S 0002-9947(1992)1049618-6
Article copyright: © Copyright 1992 American Mathematical Society