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

A {\it W$^*$-representation} of a II$_1$ subfactor $N\subset M$ with finite Jones index, $[M:N]<\infty$, is a non-degenerate commuting square embedding of $N\subset M$ into an inclusion of atomic von Neumann algebras $\oplus_{i\in I} \Cal B(\Cal K_i)=\Cal N \subset^{\Cal E} \Cal M=\oplus_{j\in J} \Cal B(\Cal H_j)$. We undertake here a systematic study of this notion, first introduced in [P92], giving examples and considering invariants such as the (bipartite) {\it inclusion graph} $\Lambda_{\Cal N \subset \Cal M}$, the {\it coupling vector} $(\text{\rm dim}(_M\Cal H_j))_j$ and the {\it RC-algebra} (relative commutant) $M'\cap \Cal N$, for which we establish some basic properties. We then prove that if $N\subset M$ admits a W$^*$-representation $\Cal N\subset^{\Cal E}\Cal M$, with the expectation $\Cal E$ preserving a semifinite trace on $\Cal M$, such that there exists a norm one projection of $\Cal M$ onto $M$ commuting with $\Cal E$, a property of $N\subset M$ that we call {\it weak injectivity/amenability}, then $[M:N]$ equals the square norm of the inclusion graph $\Lambda_{\Cal N \subset \Cal M}$.