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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

Intermediate subfactors with no extra structure


Authors: Pinhas Grossman and Vaughan F. R. Jones
Journal: J. Amer. Math. Soc. 20 (2007), 219-265
MSC (2000): Primary 46L37
Published electronically: May 10, 2006
MathSciNet review: 2257402
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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.


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Additional 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
Email: vfr@math.berkeley.edu

DOI: http://dx.doi.org/10.1090/S0894-0347-06-00531-5
PII: S 0894-0347(06)00531-5
Keywords: Subfactors, planar algebras, intermediate subfactors
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
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.