Intermediate Subfactors with No Extra Structure

Abstract

If $N\subset P,Q\subset M$ are type II_1 factors with $N^{\prime }\cap M=Cid$ and $[M:N]$ finite we show that restrictions on the standard invariants of the elementary inclusions $N\subset P$, $N\subset Q$, $P\subset M$ and $Q\subset M$ 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 $P$ and $Q$ do not commute, then $[M:N]$ is 6 or $6+4\sqrt{2}$. In the former case $N$ is the fixed point algebra for an outer action of $S_3$ on $M$ and the angle is $\pi /3$, and in the latter case the angle is $co{s}^{-1}(\sqrt{2}-1)$ and an example may be found in the GHJ subfactor family. The techniques of proof rely heavily on planar algebras.

51 pages, 65 figures