## Intermediate subfactors with no extra structure

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- by Pinhas Grossman and Vaughan F. R. Jones
- J. Amer. Math. Soc.
**20**(2007), 219-265 - DOI: https://doi.org/10.1090/S0894-0347-06-00531-5
- Published electronically: May 10, 2006
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## Abstract:

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.## References

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## Bibliographic Information

**Pinhas Grossman**- Affiliation: Department of Mathematics, University of California at Berkeley, Berkeley, California 94720
- Email: pinhas@math.berkeley.edu
**Vaughan F. R. Jones**- Affiliation: Department of Mathematics, University of California at Berkeley, Berkeley, California 94720
- MR Author ID: 95565
- Email: vfr@math.berkeley.edu
- Received by editor(s): February 14, 2005
- Published electronically: May 10, 2006
- Additional Notes: The authors were supported in part by NSF Grant DMS04-01734; the second author was also supported by the Marsden fund UOA520 and the Swiss National Science Foundation
- © Copyright 2006
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc.
**20**(2007), 219-265 - MSC (2000): Primary 46L37
- DOI: https://doi.org/10.1090/S0894-0347-06-00531-5
- MathSciNet review: 2257402