Perturbations of subalgebras of type II1 factors
| Autor: | Allan M. Sinclair, Sorin Popa, Roger R. Smith |
|---|---|
| Jazyk: | angličtina |
| Předmět: |
Discrete mathematics
Pure mathematics Partial isometry von Neumann algebras Mathematics::Operator Algebras Center (group theory) Type (model theory) Perturbations Centralizer and normalizer Unitary state Measure (mathematics) Factors symbols.namesake Von Neumann algebra symbols Jones projection Normalizing unitary Affiliated operator Subfactors Analysis Mathematics |
| Zdroj: | Journal of Functional Analysis. (2):346-379 |
| ISSN: | 0022-1236 |
| DOI: | 10.1016/j.jfa.2004.01.010 |
| Popis: | In this paper we consider two von Neumann subalgebras B0 and B of a type II1 factor N. For a map φ on N, we define||φ||∞,2=sup{||φ(x)||2:||x||⩽1},and we measure the distance between B0 and B by the quantity ||EB0−EB||∞,2. Under the hypothesis that the relative commutant in N of each algebra is equal to its center, we prove that close subalgebras have large compressions which are spatially isomorphic by a partial isometry close to 1 in the ||·||2-norm. This hypothesis is satisfied, in particular, by masas and subfactors of trivial relative commutant. A general version with a slightly weaker conclusion is also proved. As a consequence, we show that if A is a masa and u∈N is a unitary such that A and uAu∗ are close, then u must be close to a unitary which normalizes A. These qualitative statements are given quantitative formulations in the paper. |
| Databáze: | OpenAIRE |
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