Notes on Derivations on Algebras of Measurable Operators

Derivations on algebras of (unbounded) operators affiliated with a von Neumann algebra $\mathscr M$ are considered. Let $\mathscr A$ be one of the algebras of measurable operators, of locally measurable operators, and of $\tau$-measurable operators. The von Neumann algebras $\mathscr M$ of type I for which any derivation on $\mathscr A$ is inner are completely described in terms of properties of central projections. It is also shown that any derivation on the algebra $LS(\mathscr M)$ of all locally measurable operators affiliated with a properly infinite von Neumann algebra $\mathscr M$ vanishes on the center $LS(\mathscr M)$.