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

Let $\mathcal{A}$ be a unital associative algebra and $\mathcal{M}$ be an $\mathcal{A}$-bimodule. A linear mapping $\phi$ from $\mathcal{A}$ into an $\mathcal{A}$-bimodule $\mathcal{M}$ is called a Lie derivation if $\phi [A,B]=[\phi (A),B]+[A,\phi (B)]$ for each $A,B$ in $\mathcal{A}$, and $\phi$ is called a \emph{local Lie derivation} if for every $A$ in $\mathcal{A}$, there exists a Lie derivation ${\phi }_A$ (depending on $A$) from $\mathcal{A}$ into $\mathcal{M}$ such that $\phi (A)={\phi }_A(A)$. 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.