Kadison-Kastler stable factors

TY - JOUR

T1 - Kadison-Kastler stable factors

AU - Cameron, Jan

AU - Christensen, Erik

AU - Sinclair, Allan M.

AU - Smith, Roger R.

AU - White, Stuart A.

AU - Wiggins, Alan D.

PY - 2014

Y1 - 2014

N2 - 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.

AB - 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.

U2 - 10.1215/00127094-2819736

DO - 10.1215/00127094-2819736

M3 - Journal article

SN - 0012-7094

VL - 163

SP - 2639

EP - 2686

JO - Duke Mathematical Journal

JF - Duke Mathematical Journal

IS - 14

ER -