1-cohomology and rigidity in II(1) factors

We will investigate closable derivations on von Neumann algebras and their associated completely positive semigroups. We will be primarily interested in properties of a von Neumann algebra which imply that certain classes of closable derivations must vanish. In this way we obtain a characterization of property (T) for von Neumann algebras in terms of 1-cohomology similar to the Delorme-Guichardet Theorem for groups.;Investigating derivations into the coarse correspondence we then introduce the notion of L2-rigidity for von Neumann algebras, a generalization of property (T) which can be viewed as an analogue for the vanishing of 1-cohomology into the left regular representation of a group. We show that L2-rigidity passes to normalizers and is satisfied by nonamenable II1 factors which are non-prune, have property Gamma, or are weakly rigid. As a consequence we obtain that if M is a free product of diffuse von Neumann algebras, or if M = LGamma where Gamma is a finitely generated group with b21 (Gamma) > 0, then any nonamenable regular subfactor of M is prime and does not have properties Gamma or (T). In particular this gives a new approach for showing primeness of all nonamenable subfactors of a free group factor thus recovering a well known recent result of N. Ozawa.