Characterization of Lie Derivations on von Neumann Algebras

Abstract

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) $\xi=1$, $L=\varphi+f$, where $\varphi$ 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 $\varphi$ such that $L(A)=\varphi(A)+L(I)A$ for all $A$; (3) $\xi=-1$, $L$ is a Jordan derivation; (4) $\xi$ is rational and $\xi\not=0, \pm1$, $L$ is an additive derivation; (5) $\xi$ is not rational, there exists an additive derivation $\varphi$ satisfying $\varphi(\xi I)=\xi L(I)$ such that $L(A)=\varphi(A) + L(I)A$ 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 $f:{\mathcal M}\rightarrow{\mathcal Z}({\mathcal M})$ 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) $\xi\not=0,1$, $L(A)=AT-TA$ for all $A \in{\mathcal M}$.