Characterization of Lie Derivations on von Neumann Algebras

Let $\mathcal{M}$ be a von Neumann algebra without central summands of type $I_1$ and $\xi \in \mathbb{C}$ a scalar. It is shown that an additive map $L$ on $\mathcal{M}$ satisfies $L(AB-\xi BA)=L(A)B-\xi BL(A)+L(B)A-\xi AL(B)$ whenever $A,B\in \mathcal{M}$ with $AB=0$ if and only if one of the following statements holds: (1) $\xi =1$, $L=\phi +f$, where $\phi$ is an additive derivation on $\mathcal{M}$ and $f$ is an additive map from $\mathcal{M}$ into its center vanishing on $[A,B]$ with $AB=0$; (2) $\xi =0$, $L(I)\in \mathcal{Z}(\mathcal{M})$ and there exists an additive derivation $\phi$ such that $L(A)=\phi (A)+L(I)A$ for all $A$; (3) $\xi =-1$, $L$ is a Jordan derivation; (4) $\xi$ is rational and $\xi\not=0, \pm1$,$L$isanadditivederivation;(5)$\xi$ is not rational, there exists an additive derivation $\phi$ satisfying $\phi (\xi I)=\xi L(I)$ such that $L(A)=\phi (A)+L(I)A$ for all $A\in \mathcal{M}$. A linear map $L$ on $\mathcal{M}$ satisfies $L(AB-\xi BA)=L(A)B-\xi BL(A)+L(B)A-\xi AL(B)$ whenever $A,B\in \mathcal{M}$ with $AB=0$ if and only if there exists a $T\in \mathcal{M}$ and a linear map $f:\mathcal{M}\to \mathcal{Z}(\mathcal{M})$ vanishing on $[A,B]$ with $AB=0$ such that (i) $\xi =1$, $L(A)=AT-TA+f(A)$ for all $A\in \mathcal{M}$; (ii) $\xi =0$, $L(I)\in \mathcal{Z}(\mathcal{M})$ and $L(A)=AT-(T-L(I))A$ for all $A\in \mathcal{M}$; (iii) $\xi ≠0,1$, $L(A)=AT-TA$ for all $A\in \mathcal{M}$.