Editing Von Neumann algebra (section)

==Non-amenable factors==
Von Neumann algebras of type I are always amenable, but for the other types there are an uncountable number of different non-amenable factors, which seem very hard to classify, or even distinguish from each other. Nevertheless, [[Dan-Virgil Voiculescu|Voiculescu]] has shown that the class of non-amenable factors coming from the group-measure space construction is '''disjoint''' from the class coming from group von Neumann algebras of free groups. Later [[Narutaka Ozawa]] proved that group von Neumann algebras of [[hyperbolic group]]s yield [[Prime number|prime]] type II<sub>1</sub> factors, i.e. ones that cannot be factored as tensor products of type II<sub>1</sub> factors, a result first proved by Leeming Ge for free group factors using Voiculescu's [[free probability theory|free entropy]]. Popa's work on fundamental groups of non-amenable factors represents another significant advance. The theory of factors "beyond the hyperfinite" is rapidly expanding at present, with many new and surprising results; it has close links with [[Grigory Margulis|rigidity phenomena]] in [[geometric group theory]] and [[ergodic theory]].