Kadison-Kastler stable factors

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 SLn(Z) on a standard nonatomic probability space (X,μ), write M=(L(X,μ)⋊SLn(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(X,μ)⋊Γ whenever the comparison map from the bounded to usual group cohomology vanishes in degree 2 for the module L2(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
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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Project CodeAward NoProject NamePrincipal InvestigatorFunder's NameFunder RefLead Dept
596411The Cuntz Semigroup and the Fine Structure of Nuclear C*-AlgebrasStuart WhiteEngineering & Physical Sciences Research Council (EPSRC)EP/I019227/1M&S - MATHEMATICS