W*-representations of subfactors and restrictions on the Jones index

A W *-representation of a II1 subfactor N subset of M with finite Jones index, [M : N] < infinity, is a non-degenerate commuting square embedding of N subset of M into an inclusion of atomic von Neumann algebras circle plus B-i is an element of I (K-i) = N subset of(epsilon) M = circle plus(j is an element of J) B (H-j). We undertake here a systematic study of this notion, first introduced by the author in 1992, giving examples and considering invariants such as the (bipartite) inclusion graph Lambda(N subset of M), the coupling vector (dim(M H-j ))(j) and the RC-algebra (relative commutant) M' boolean AND N , for which we establish some basic properties. We then prove that if N subset of M admits a W*-representation N subset of(epsilon) M, with the expectation epsilon preserving a semifinite trace on M, such that there exists a norm one projection of M onto M commuting with epsilon, a property of N subset of M that we call weak injectivity/amenability, then [M : N] equals the square norm of the inclusion graph Lambda(N subset of M).