$W^{\ast }$-representations of subfactors and restrictions on the Jones index

Abstract

A $W^{\ast }$-representation of a ${\mathrm{II}}_1$ 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 ${\oplus }_{i\in I}\mathcal{B}({\mathcal{K}}_i)=\mathcal{N}{\subset }^{\mathcal{E}}\mathcal{M}={\oplus }_{j\in J}\mathcal{B}({\mathcal{H}}_j)$. 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 ${\Lambda }_{\mathcal{N}\subset \mathcal{M}}$, the coupling vector $(\dim ({}_M{\mathcal{H}}_j){)}_j$ and the RC-algebra (relative commutant) $M^{\prime }\cap \mathcal{N}$, for which we establish some basic properties. We then prove that if $N\subset M$ admits a $W^{\ast }$-representation $\mathcal{N}{\subset }^{\mathcal{E}}\mathcal{M}$, with the expectation $\mathcal{E}$ preserving a semifinite trace on $\mathcal{M}$, such that there exists a norm one projection of $\mathcal{M}$ onto $M$ commuting with $\mathcal{E}$, a property of $N\subset M$ that we call weak injectivity/amenability, then $[M:N]$ equals the square norm of the inclusion graph ${\Lambda }_{\mathcal{N}\subset \mathcal{M}}$.