A remark on the minimal index of subgfactors
Let \(M\supset N\) be factors with a faithful normal conditional expectation \(E: M\to N\) satisfying \(\text{Ind } E<\infty\). Let \(N\subset M\subset M_ 1\subset M_ 2\subset\dots\) be the Jones tower of basic constructions, and \(E_ k: M_ k\to M_{k-1}\) be the dual expectation constructed canonically from \(E\). The authors prove tht, if \(E\) is minimal, then so is the composition \(E\circ E_ 1\circ \dots\circ E_ k: M_ k\to N\). To achieve this result, they use the tensoring by a factor of type \(\text{III}_ 1\) to reduce the problem to the \(\text{II}_ 1\) case so that \textit{M. Pimsner} and \textit{S. Popa's} result in Trans. Am. Math. Soc. 310, No. 1, 127-133 (1988; Zbl 0706.46047) can be applied.
factors with a faithful normal conditional expectation
scientific article; zbMATH DE number 65351
scientific article