Local Lie derivations on von Neumann algebras and algebras of locally measurable operators

Abstract

Let A be a unital associative algebra and M be an A-bimodule. A linear mappingand ϕ is called a local Lie derivation if for every A in A, there exists a Lie derivation ϕ A (depending on A) from A into M such that ϕ(A) = ϕ A (A). In this paper, we prove that every local Lie derivation on von Neumann algebras is a Lie derivation; and we show that if M is a type I von Neumann algebra with atomic lattice of projections, then every local Lie derivation on LS(M) is a Lie derivation.