Unitary perturbations of masas in type II₁ factors

2001/11/30 by Allan M. Sinclair, Sinclair, Allan M., Roger R. Smith +1
Mathematics · #46L10 #46L35 #FOS: Mathematics #Operator Algebras (math.OA) #math.OA #msc:46L10 #msc:46L35

paper · pdf · doi:10.48550/arxiv.math/0111330

latex file, 29 pages

arxiv created 2001/11/30 · arxiv updated 2009/11/30

Abstract

We investigate maximal abelian subalgebras (masas) in separably acting type II₁ factors. We use the notion of distance between masas which we introduced in an earlier paper in this archive, OA/0107075. The main result of the paper is to show that for any unitary u and a masa A, the distance between A and uAu* is comparable to the distance in 2-norm from u to the unitary normalizer N(A) of A. In qualitative terms, this says that a unitary almost normalizes A if and only if it is close to a normalizing unitary. Inequalities in the paper make this statement quantitatively precise. As a consequence, all singular masas in separably acting type II₁ factors are α-strongly singular, as defined in the above referenced paper, for a universal value of α.

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