Complete description of derivations on $\tau$-compact operators for type I von Neumann algebras

Abstract

Given a type I von Neumann algebra $M$ with a faithful normal semi-finite trace $\tau,$ let $S_0(M, \tau)$ be the algebra of all $\tau$-compact operators affiliated with $M.$ We give a complete description of all derivations on the algebra $S_0(M, \tau).$ In particular, we prove that if $M$ is of type I$_\infty$ then every derivation on $S_0(M, \tau)$ is spatial.