Measure continuous derivations on von Neumann algebras and applications to L2-cohomology

We prove that norm continuous derivations from a von Neumann algebra into the algebra of operators affiliated with its tensor square are automatically continuous for both the strong operator topology and the measure topology. Furthermore, we prove that the first continuous L2-Betti number scales quadratically when passing to corner algebras and derive an upper bound given by Shen's generator invariant. This, in turn, yields vanishing of the first continuous L2-Betti number for \twoone factors with property (T), for finitely generated factors with non-trivial fundamental group and for factors with property Gamma.