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

Abstract

Let $\mathcal{A}$ be a unital associative algebra and $\mathcal{M}$ be an $\mathcal{A}$-bimodule. A linear mapping $\varphi$ from $\mathcal{A}$ into an $\mathcal{A}$-bimodule $\mathcal{M}$ is called a Lie derivation if $\varphi[A,B]=[\varphi(A),B]+[A,\varphi(B)]$ for each $A,B$ in $\mathcal{A}$, and $\varphi$ is called a \emph{local Lie derivation} if for every $A$ in $\mathcal{A}$, there exists a Lie derivation $\varphi_{A}$ (depending on $A$) from $\mathcal{A}$ into $\mathcal{M}$ such that $\varphi(A)=\varphi_{A}(A)$. In this paper, we prove that every local Lie derivation on von Neumann algebras is a Lie derivation; and we show that if $\mathcal M$ is a type I von Neumann algebra with atomic lattice of projections, then every local Lie derivation on $LS(\mathcal M)$ is a Lie derivation.