A finiteness result for commuting squares with large second relative commutant

Nicoara, Remus

doi:10.1090/S0002-9947-2012-05532-2

A finiteness result for commuting squares with large second relative commutant
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Trans. Amer. Math. Soc. 364 (2012), 3685-3698 Request permission

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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