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A finiteness result for commuting squares with large second relative commutant

Author: Remus Nicoara
Journal: Trans. Amer. Math. Soc. 364 (2012), 3685-3698
MSC (2010): Primary 46L37
Published electronically: February 16, 2012
MathSciNet review: 2901230
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Abstract: We prove that there exist only finitely many commuting squares of finite dimensional $ *$-algebras of fixed dimension, satisfying a ``large second relative commutant'' condition. We show this by studying the local minima of $ w\rightarrow \dim (A\cap wBw^*)$, where $ A,B$ are fixed subalgebras of some $ *$-algebra $ C$ and $ w\in C$ is a unitary.

When applied to lattices arising from subfactors satisfying a certain extremality-like condition, our result yields Ocneanu's finiteness theorem for the standard invariants of such finite depth subfactors.

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Additional Information

Remus Nicoara
Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-0612 – and – Institute of Mathematics of the Romanian Academy, 21 Calea Grivitei Street, 010702 Bucharest, Romania

Received by editor(s): August 19, 2010
Received by editor(s) in revised form: December 26, 2010
Published electronically: February 16, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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