Intermediate subfactors with no extra structure

摘要

If N C P, Q C M are type II factors with N n M=cid and [N:N] < ∞ we show that restrictions on the standard invariants of the elementary inclusions N C P, N C Q, P C M and Q C M imply drastic restrictions on the indices and angles between the subfactors. In particular we show that if these standard invariants are trivial and the conditional expectations onto and do not commute, then 6 is 6 or 6+4. In the former case N is the fixed point algebra for an outer action of S3 on M and the angle is , and in the latter case the angle is P and an example may be found in the GHJ subfactor family. The techniques of proof rely heavily on planar algebras.