Description of Derivations on Measurable Operator Algebras of Type I

Abstract

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