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;), write M =(L (X;) 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;) whenever the comparison map from the bounded to usual group cohomology vanishes in degree 2 for the module L 2 (X;). 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.