Minimizing Indices of Conditional Expectations onto a Subfactor

Abstract

For a pair of factors M⊇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⊇N are \mathrm{II}_1 factors, we establish the relation between \mathrm{Index}\ E_0 and [M:N] .