Minimizing indices of conditional expectations onto a subfactor (Q1822733)

Let M be a factor and N a subfactor of M. Let \({\mathcal E}(M,N)\) be the set of all faithful normal conditional expectations from M onto N. The index E of \(E\in {\mathcal E}(M,N)\) was introduced by Kosaki making use of Connes' spatial theory and Haagerup's theory of operator valued weight. If Index E\(<\infty\) for some \(E\in {\mathcal E}(M,N)\), it is shown that there exists a unique \(E_ 0\in {\mathcal E}(M,N)\) such that Index \(E_ 0=\min \{Index E:\) \(E\in {\mathcal E}(M,N)\}\) and such \(E_ 0\) is characterized. When M and N are of type \(II_ 1\), the relation between Index \(E_ 0\) and Jones' Index [M:N] is established when the latter is finite.