A NOTE ON INTERMEDIATE SUBFACTORS

In this note we prove that if N subset-of M subset-of P is an inclusion of II1, factors with finite Jones index such that N subset-of P has finite depth, then N subset-of M and M subset-of P have finite depth. We show this result by studying the iterated basic constructions for M subset-of P and N subset-of P. In particular our proof gives detailed information about the graphs for N subset-of M resp. M subset-of P. Furthermore, we give an abstract characterization of intermediate subfactors in terms of Jones projections in N' and P1, where N subset-of P subset-of P1 is the basic construction for N subset-of P and give examples showing that if N subset-of M and M subset-of P have finite depth, then N subset-of P does not necessarily have finite depth.