Unitary perturbations of masas in type $II_1$ factors

Allan M. Sinclair; Roger R. Smith

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Unitary perturbations of masas in type $II_1$ factors

Allan M. Sinclair·Roger R. Smith

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Abstract

We investigate maximal abelian subalgebras (masas) in separably acting type $II_1$ 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 $II_1$ factors are $α$-strongly singular, as defined in the above referenced paper, for a universal value of $α$.

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BibTeX RIS

Allan M. Sinclair, Roger R. Smith. 2001-11-30. Unitary perturbations of masas in type $II_1$ factors. https://arxiv.org/abs/math/0111330

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