An Estimate of the Depth from an Intermediate Subfactor

Abstract

We show that for a triple KaN^M of type II j factors the depth of the inclusion AcAf' is not greater than the maximum of depths of the inclusions A'cTV and N<=.Af provided there P ci Af is such a factor P, that the diagram u u is commuting and co-commuting square (or a Ka N non-degenerate commuting square) of type II j factors. In [B] D. Bisch proved, that if depth of inclusion Kc:M of two type II1 factors is finite then for any intermediate subfactor N, the depths of Kc-W and of TVcM are finite too. In this note we give a partial converse to the assertion. Similar result was obtained recently in the case of depth two irreducible inclusions in [S] by T. Sano, who used a different method. After the work had been completed, the author learned about another, much shorter proof of Theorem 6 below, based on the bimodule technique ([K]).