Given a subfactor N of a II, factor M with the same identity, one defines the index of N in M as [M:N] = dimN(L2 (M)) = the Murray von Neumann coupling constant for N on the Hilbert space L2 (M), (= completion of M with respect to the inner product <a,b> = tr(ab*), tr being the unique normalized trace on M). The following result shows the interest of this notion.

Theorem

1. a) If [M:N] < 4 then there is an n ∈ ℤ, n ≥ 3, with [M:N] =4 cos2π/n.

2. b) For any real r ≥ 4 there is a pair N ⊆ M with [M:N] = r.

The “basic construction” of the theory is as follows. If N ⊆ M are finite von Neumann algebras and tr is a faithful normal normalized trace on M, one considers the von Neumann algebra <M,eN> = {M,eN}” on L2 (M,tr) where eN, is the orthogonal projection onto L2 (N,tr) (tr restricted to N). If N and M are factors then [M:N] < ∞ iff <M,eN> is a II1, factor and then [M:N]tr(eN) = 1.

Ocneanu has made great progress on classifying subfactors of the hyperfinite II1 factor with given index, which he will explain in his talk. It would appear that the classification is complete for index < 4.