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1 November 2014 Kadison–Kastler stable factors
Jan Cameron, Erik Christensen, Allan M. Sinclair, Roger R. Smith, Stuart White, Alan D. Wiggins
Duke Math. J. 163(14): 2639-2686 (1 November 2014). DOI: 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 n3 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 NB(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.

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Jan Cameron. Erik Christensen. Allan M. Sinclair. Roger R. Smith. Stuart White. Alan D. Wiggins. "Kadison–Kastler stable factors." Duke Math. J. 163 (14) 2639 - 2686, 1 November 2014. https://doi.org/10.1215/00127094-2819736

Information

Published: 1 November 2014
First available in Project Euclid: 31 October 2014

zbMATH: 1314.46069
MathSciNet: MR3273580
Digital Object Identifier: 10.1215/00127094-2819736

Subjects:
Primary: 46L10

Rights: Copyright © 2014 Duke University Press

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Vol.163 • No. 14 • 1 November 2014
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