Minimizing Indices of Conditional Expectations onto a Subfactor

Abstract

For a pair of factors $M\supseteq N$, let $\mathcal{E}(M,N)$ be the set of all conditional expectations from $M$ onto $N$. We characterize $E_0\in \mathcal{E}(M,N)$ whose index is the minimum of $\{\mathrm{Index}E:E\in \mathcal{E}(M,N)\}$. When $M\supseteq N$ are ${\mathrm{II}}_1$ factors, we establish the relation between $\mathrm{Index}{E}_0$ and $[M:N]$.