Index and flow of weights of factors of type III (Q1101929)

Let \({\mathcal M}\) be a factor of type III and \({\mathcal N}^ a \)subfactor of \({\mathcal M}\) of the same type. Let E be a conditional expectation (Index E\(<\infty)\) from \({\mathcal M}\) into \({\mathcal N}\). The authors investigate to show that \({\mathcal M}\) and \({\mathcal N}\) possess some similar properties in the type III set-up. In fact, they sketch a proof of the following theorem: Theorem. There exists a flow \((X,\{T_ t\}_{t\in R})\) satisfying (i) X is isomorphic to \(X_{{\mathcal M}}\times \{1,2,...,m\}\) (resp. \(X_{{\mathcal N}}\times \{1,2,...,n\})\) as a measure space for some positive integer m, \(m\leq Index E\) (resp. positive integer n, \(n\leq Index E)\), (ii) the projection map \(\pi_{{\mathcal M}}\) (resp. \(\pi_{{\mathcal M}})\) from X onto \(X_{{\mathcal M}}\) (resp. \(X_{{\mathcal N}})\) intertwines \(T_ t\) and \(T_ t^{{\mathcal M}}\) (resp. \(T_ t\) and \(T_ t^{{\mathcal N}})\): \[ T_ t^{{\mathcal M}}\circ \pi_{{\mathcal M}}=\pi_{{\mathcal M}}\circ T_ t,\quad T_ t^{{\mathcal N}}\circ \pi_{{\mathcal N}}=\pi_{{\mathcal N}}\circ T_ t,\quad t\in R. \]