Spatiality of derivations on the algebra of $τ$-compact operators

Abstract

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