Formulae and continuity for the index of subfactors

Abstract:

Let $N \subset M$ be an inclusion of $II_{1}$-factors, $\tau$ the trace state of $M$, and $\mathcal {P}(M)$, $\mathcal {P}(N)$ the set of projections in $M$ and $N$, respectively. We prove that the Jones index for the inclusion is \begin{equation*}\begin {split} [M : N ]&= sup_{e \in \mathcal {P}(M) \backslash \{0\}} inf_{p \in \mathcal {P}(N) \backslash \{0\}} \frac {\tau (p)}{\tau (ep)}\\ &=sup_{e \in \mathcal {P}(M) \backslash \{0\}} inf \{ \frac {\tau (p)}{\tau (ep)} : p \in \mathcal {P}(N), e \preceq p \} . \end{split}\end{equation*} This formula is exploited to obtain continuity results for the index. In particular, we obtain a formula for the index which expresses $[M:N]$ in terms of the positions of $N_{i}$ and $M_{j} , i,j \in \mathbb {N}$, in $M$, when $N_{1} \subset N_{2} \subset N_{3} \subset \cdots$ and $M_{1} \subset M_{2} \subset M_{3} \subset \cdots$ are finite-dimensional $C^{\ast }$-subalgebras with dense union in $N$ and $M$, respectively.