Wiggins, AD (2014) Kadison-Kastler stable factors. Duke Mathematical Journal, 163 (14). pp. 2639-2686. ISSN 0012-7094

摘要: 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. This paper provides a complete account of the results announced in [6].