Unitary perturbations of masas in type $II_1$ factors

Details Unitary perturbations of masas in type $II_1$ factors Unitary perturbations of masas in type $II_1$ factors

2001-11-30

arxiv.org/abs/math/0111330v1

Mathematics

Operator Algebras

latex file, 29 pages

Scientific paper

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 $\alpha$-strongly singular, as defined in the above referenced paper, for a universal value of $\alpha$.

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