ADDITIVE DERIVATIONS ON ALGEBRAS OF MEASURABLE OPERATORS

Given a von Neumann algebra M we introduce so called central extension mix(M) of M. We show that mix(M) is a ∗-subalgebra in the algebra LS(M) of all locally measurable operators with respect to M, and this algebra coincides with LS(M) if and only if M does not admit type II direct summands. We prove that if M is a properly infinite von Neumann algebra then every additive derivation on the algebra mix(M) is inner. In particular each derivation on the algebra LS(M), where M is a type I∞ or a type III von Neumann algebra, is inner.