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

This paper provides a complete account of the results announced in this article.

### Publication Details

*Coauthors:* Jan Cameron, Erik Christensen, Allan Sinclair, Roger Smith and Alan Wiggins.

Duke Math. J. 163(14) (2014), 2639-268

*Download:* Journal website, arXiv:1209.4416, Glasgow ePrint:92094

## Arco, Italy, 2012

Photo: Iain McFadyen