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 $n\geq 3$ and a free ergodic probability
measure preserving action of $SL_n(\mathbb Z)$ on a standard nonatomic
probability space $(X,\mu)$, write $M=((L^\infty(X,\mu)\rtimes SL_n(\mathbb
Z))\,\overline{\otimes}\, R$, where $R$ is the hyperfinite II$_1$ factor. We
show that whenever $M$ is represented as a von Neumann algebra on some Hilbert
space $\mathcal H$ and $N\subseteq\mathcal B(\mathcal H)$ is sufficiently close
to $M$, then there is a unitary $u$ on $\mathcal H$ 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
$L^\infty(X,\mu)\rtimes\Gamma$ 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.