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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Intermediate subfactors with no extra structure
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by Pinhas Grossman and Vaughan F. R. Jones PDF
J. Amer. Math. Soc. 20 (2007), 219-265 Request permission

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
  • 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