Two subfactors and the algebraic decompsition of bimodules over II1 factors

Ocneanu noted that the data for what is now known as a Turaev-Viro type TQFT is supplied by a subfactor N of nite index and depth of a II1 factor M . One takes all the irreducible bimodules arising in the decomposition of the tensor powers, in the sense of Connes ([3]), of the Hilbert space L(M) viewed as an N−N bimodule, or, in Connes' terminology, a correspondence. (We will use the term correspondence systematically to di erentiate between these and purely algebraic bimodules over algebras.) These can be N − N,N−M,M−N andM−M correspondences. To get a 3-manifold invariant one starts with a triangulation and assigns in an arbitrary way M or N to the vertices, appropriate correspondences to the edges connecting vertices and intertwiners between the three correspondences around a face. Each 3simplex then de nes a scalar by a clever composition of the intertwiners on its boundary. One then multiplies these scalars over all the simplices and sums over all ways of assigning M , N , bimodules and intertwiners to obtain the invariant of the three manifold. It is natural to ask the following questions about this procedure: (i) Does one have to introduce Hilbert spaces and the Connes tensor product or is the purely algebraic decomposition of tensor powers of M over N enough? (ii) Might there not be more than 2 factors involved, with correspondences between all of them? Problem (i) is a subtle problem and easily overlooked since intertwiners between bimodules preserve bounded vectors. In the simplest case of L(M) ∗Supported in part by NSF Grant DMS0401734, and the Swiss National Science Foundation.