Kadison–Kastler conjecture for von Neumann algebras

Let $\mathcal H$ be a Hilbert space, let $\mathcal B(\mathcal H)$ denote the bounded operators on $\mathcal H$, and let $d(M,N)$ be the Hausdorff distance between the unit balls of von Neumann algebras $M,N\subseteq\mathcal B(\mathcal H)$, measured in the operator norm. Kadison–Kastler conjecture. For all $\varepsilon>0$, there exists $\delta>0$ with the property that if $M,N\subseteq\mathcal B(\mathcal H)$ are von Neumann algebras with $d(M,N)<\delta$, then there exists a unitary operator $u$ on $\mathcal H$ with $uMu^*=N$ and $\|u-\operatorname{id}_{\mathcal H}\|<\varepsilon$. The conjecture asserts that sufficiently close von Neumann algebras are related by a small unitary perturbation. It was established for amenable von Neumann algebras, while the paper describes examples of non-amenable algebras satisfying it; the general status is not established by the supplied text.