PI-1-FACTORS, THEIR BIMODULES AND HYPERGROUPS

In this paper, we introduce a notion that we call a hypergroup; this n otion captures the natural algebraic structure possessed by the set of equivalence classes of irreducible bifinite bimodules over a II1 fact or. After developing some basic facts concerning bimodules over II1, f actors, we discuss abstract hypergroups. To make contact with the prob lem of what numbers can arise as index-values of subfactors of a given II1 factor with trivial relative commutant, we define the notion of a dimension function on a hypergroup, and prove that every finite hyper group admits a unique dimension function. we then give some nontrivial examples of hypergroups, some of which are related to the Jones subfa ctors of index 4 cos2-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 gro up."