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Strong Singularity for Subfactors

Pinhas Grossman·Alan Wiggins
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Abstract

We examine the notion of $α$-strong singularity for subfactors of a \IIi factor, which is a metric quantity that relates the distance between a unitary in the factor and a subalgebra with the distance between that subalgebra and its unitary conjugate. Through planar algebra techniques, we demonstrate the existence of a finite index singular subfactor of the hyperfinite \IIi factor that cannot be strongly singular with $α=1$, in contrast to the case for masas. Using work of Popa, Sinclair, and Smith, we show that there exists an absolute constant $0<c<1$ such that all singular subfactors are $c$-strongly singular. Under the hypothesis of 2-transitivity, we prove that finite index subfactors are $α$-strongly singular with a constant that tends to 1 as the Jones Index tends to infinity and infinite index subfactors are 1-strongly singular. Finally, we give a proof that proper finite index singular subfactors do not have the weak asymptotic homomorphism property relative to the containing factor.

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Pinhas Grossman, Alan Wiggins. 2008-10-23. Strong Singularity for Subfactors. https://doi.org/10.1112/blms%2Fbdp137

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