Spatiality of derivations on the algebra of $\tau$-compact operators

This paper is devoted to derivations on the algebra $S_0(M, \tau)$ of all $\tau$-compact operators affiliated with a von Neumann algebra $M$ and a faithful normal semi-finite trace $\tau.$ The main result asserts that every $t_\tau$-continuous derivation $D:S_0(M, \tau)\rightarrow S_0(M, \tau)$ is spatial and implemented by a $\tau$-measurable operator affiliated with $M$, where $t_\tau$ denotes the measure topology on $S_0(M, \tau)$. We also show the automatic $t_\tau$-continuity of all derivations on $S_0(M, \tau)$ for properly infinite von Neumann algebras $M$. Thus in the properly infinite case the condition of $t_\tau$-continuity of the derivation is redundant for its spatiality.