Description of Derivations on Measurable Operator Algebras of Type I

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 $M\mathrm{.}$ We give a complete description of all derivations on the algebra $L(M,\tau )\mathrm{.}$ In particular, we prove that if $M$ is of type I${}_{\mathrm{\infty }}$ then every derivation on $L(M,\tau )$ is inner.