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Strong singularity for subfactors
Author(s) -
Grossman Pinhas,
Wiggins Alan
Publication year - 2010
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdp137
Subject(s) - mathematics , singularity , pure mathematics , subalgebra , unitary state , isolated singularity , mathematical analysis , algebra over a field , political science , law
We examine the notion of α‐strong singularity for subfactors of a II 1 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 Jones index singular subfactor of the hyperfinite II 1 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 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.