L2-Cohomology for Von Neumann Algebras

We study L2-Betti numbers for von Neumann algebras, as defined by D. Shlyakhtenko and A. Connes in [CoS]. We give a definition of L2-cohomology and show how the study of the first L2-Betti number can be related to the study of derivations with values in a bi-module of affiliated operators. We show several results about the possibility of extending derivations from sub-algebras and about uniqueness of such extensions. In particular, we show that the first L2-Betti number of a tracial von Neumann algebra coincides with the corresponding number for an arbitrary weakly dense sub-C*-algebra.