Cameron, J., Christensen, E., Sinclair, A. M., Smith, R. R., White, S. and Wiggins, A. D. (2014) Kadison-Kastler stable factors. Duke Mathematical Journal, 163(14), pp. 2639-2686. (doi: 10.1215/00127094-2819736)
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Publisher's URL: http://dx.doi.org/10.1215/00127094-2819736
Abstract
A conjecture of Kadison and Kastler from 1972 asks whether sufficiently close operator algebras in a natural uniform sense must be small unitary perturbations of one another. For n≥3 and a free, ergodic, probability measure-preserving action of SL<sub>n</sub>(Z) on a standard nonatomic probability space (X,μ), write M=(L<sup>∞</sup>(X,μ)⋊SL<sub>n</sub>(Z))⊗¯¯¯R, where R is the hyperfinite II1-factor. We show that whenever M is represented as a von Neumann algebra on some Hilbert space H and N⊆B(H) is sufficiently close to M, then there is a unitary u on H close to the identity operator with uMu∗=N. This provides the first nonamenable class of von Neumann algebras satisfying Kadison and Kastler’s conjecture. We also obtain stability results for crossed products L<sup>∞</sup>(X,μ)⋊Γ whenever the comparison map from the bounded to usual group cohomology vanishes in degree 2 for the module L<sup>2</sup>(X,μ). In this case, any von Neumann algebra sufficiently close to such a crossed product is necessarily isomorphic to it. In particular, this result applies when Γ is a free group.
Item Type: | Articles |
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Status: | Published |
Refereed: | Yes |
Glasgow Author(s) Enlighten ID: | White, Professor Stuart |
Authors: | Cameron, J., Christensen, E., Sinclair, A. M., Smith, R. R., White, S., and Wiggins, A. D. |
College/School: | College of Science and Engineering > School of Mathematics and Statistics > Mathematics |
Journal Name: | Duke Mathematical Journal |
Publisher: | Duke University Press |
ISSN: | 0012-7094 |
ISSN (Online): | 1547-7398 |
Copyright Holders: | Copyright © 2014 Duke University Press |
Publisher Policy: | Reproduced in accordance with the copyright policy of the publisher |
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