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 factor. After developing some basic facts concerning bimodules over II 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 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 . 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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DOI:
https://doi.org/10.1090/S0002-9947-1992-1049618-6

Article copyright:
© Copyright 1992
American Mathematical Society