Local Derivations on Algebras of Measurable Operators

The paper is devoted to local derivations on the algebra Formula: see text of τ-measurable operators affiliated with a von Neumann algebra Formula: see text and a faithful normal semi-finite trace τ. We prove that every local derivation on Formula: see text which is continuous in the measure topology, is in fact a derivation. In the particular case of type I von Neumann algebras, they all are inner derivations. It is proved that for type I finite von Neumann algebras without an abelian direct summand, and also for von Neumann algebras with the atomic lattice of projections, the continuity condition on local derivations in the above results is redundant. Finally we give necessary and sufficient conditions on a commutative von Neumann algebra Formula: see text for the algebra Formula: see text to admit local derivations which are not derivations.