Perturbations of certain crossed product algebras by free groups

Given two von Neumann algebras M and N   acting on the same Hilbert space, d(M,N)$d(M,N)$ is defined to be the Hausdorff distance between their unit balls. The Kadison–Kastler problem asks whether two sufficiently close von Neumann algebras are spatially isomorphic. In this article, we show that if _P0$P_0$ is an injective von Neumann algebra with a cyclic tracial vector, G is a free group, α is a free action of G   on _P0$P_0$ and N   is a von Neumann algebra such that d(N,_P0?αG)<1/7×10⁻⁷$d(N,P_0{?}_{\alpha }G)<1/7\times {10}^{-7}$, then N   and _P0?αG$P_0{?}_{\alpha }G$ are spatially isomorphic. Suitable choices of the actions give the first examples of infinite noninjective factors for which this problem has a positive solution.