Intermediate subfactors with no extra structure

Grossman, Pinhas¹ ; Jones, Vaughan¹

¹ Department of Mathematics, University of California at Berkeley, Berkeley, California 94720

Journal of the American Mathematical Society, Tome 20 (2007) no. 1, pp. 219-265

Résumé

If $N\subseteq P,Q\subseteq M$ are type II$_1$ factors with $N’\cap M =\mathbb C id$ and $[M:N]\infty$ we show that restrictions on the standard invariants of the elementary inclusions $N\subseteq P$, $N\subseteq Q$, $P\subseteq M$ and $Q\subseteq M$ 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 $\cos ^{-1}(\sqrt 2-1)$ and an example may be found in the GHJ subfactor family. The techniques of proof rely heavily on planar algebras.

Détail

DOI : 10.1090/S0894-0347-06-00531-5

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