Kadison–Kastler stable factors

Abstract

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 $n\ge 3$ and a free, ergodic, probability measure-preserving action of ${SL}_n\left(\mathbb{Z}\right)$ on a standard nonatomic probability space $(X,\mu )$, write $M=\left(L^{\infty }\right(X,\mu \left)⋊{SL}_n\right(\mathbb{Z}\left)\right)\overline{\otimes }R$, where $R$ is the hyperfinite II₁-factor. We show that whenever $M$ is represented as a von Neumann algebra on some Hilbert space $\mathcal{H}$ and $N\subseteq \mathcal{B}\left(\mathcal{H}\right)$ is sufficiently close to $M$, then there is a unitary $u$ on $\mathcal{H}$ close to the identity operator with $uM{u}^{\ast }=N$. This provides the first nonamenable class of von Neumann algebras satisfying Kadison and Kastler’s conjecture.

We also obtain stability results for crossed products $L^{\infty }(X,\mu )⋊\Gamma$ whenever the comparison map from the bounded to usual group cohomology vanishes in degree $2$ for the module $L^2(X,\mu )$. 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 $\Gamma$ is a free group.