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 and a free, ergodic, probability measure-preserving action of on a standard nonatomic probability space , write , where is the hyperfinite II1-factor. We show that whenever is represented as a von Neumann algebra on some Hilbert space and is sufficiently close to , then there is a unitary on close to the identity operator with . This provides the first nonamenable class of von Neumann algebras satisfying Kadison and Kastler’s conjecture.
We also obtain stability results for crossed products whenever the comparison map from the bounded to usual group cohomology vanishes in degree for the module . 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 is a free group.
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