(This post is cross post on https://mathoverflow.net/questions/405965/densely-defined-derivations-in-von-neumann-algebrain-norm-topology)
We know that all derivation on von Neumann algebra are inner. How about unbounded derivation? If we restrict the domain of the derivation is dense, do we have any unbounded derivation on abelian von Neumann algebra? I am mostly interested in the case of $l^{\infty}(\mathbb{N})$ and $L^{\infty}([0,1],\mu)$.
(Here, "unbounded derivation" means that the domain of derivation is at least dense in the algebra, which have nothing to do with the norm)