An Estimate of the Depth from an Intermediate Subfactor

Abstract

We show that for a triple $K\subset N\subset M$ of type ${\mathrm{II}}_1$ factors the depth of the inclusion “$K\subset M$” is not greater than the maximum of depths of the inclusions “$K\subset N$” and “$N\subset M$” provided there is such a factor $P$, that the diagram $\begin{array}{ccc}P& \subset & M\\ \cup & & \cup \\ K& \subset & N\end{array}$ is commuting and co-commuting square (or a non-degenerate commuting square) of type ${\mathrm{II}}_1$ factors.