Kadison–Kastler stable factors
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© 2014. 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 n .Z on a standard nonatomic probability space (X;μ), write M =(L ∞ (X;μ)⋊ SL n (Z))⊗ R, where R is the hyperfinite II 1 -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 L 2 (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.
author list (cited authors)
Cameron, J., Christensen, E., Sinclair, A. M., Smith, R. R., White, S., & Wiggins, A. D.
complete list of authors
Cameron, Jan||Christensen, Erik||Sinclair, Allan M||Smith, Roger R||White, Stuart||Wiggins, Alan D