L-2-cohomology for von Neumann algebras

We study L-2-Betti numbers for von Neumann algebras, as defined by D. Shlyakhtenko and A. Connes in [CoS]. We give a definition of L-2-cohomology and show how the study of the first L-2-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 L-2-Betti number of a tracial von Neumann algebra coincides with the corresponding number for an arbitrary weakly dense sub-C -algebra. Along the way, we prove some results about the dimension function of modules over rings of affiliated operators which are of independent interest.