Structure of derivations on various algebras of measurable operators for type I von Neumann algebras

Given a von Neumann algebra M denote by S(M) and LS(M) respectively the algebras of all measurable and locally measurable operators affiliated with M. For a faithful normal semi-finite trace τ on M let S(M,τ) be the algebra of all τ-measurable operators from S(M). We give a complete description of all derivations on the above algebras of operators in the case of type I von Neumann algebra M. In particular, we prove that if M is of type I∞ then every derivation on LS(M) (resp. S(M) and S(M,τ)) is inner.