Bing Yang, Xiaochun Fang. Source: Annals of Functional Analysis, Volume 10, Number 2, 242--251.Abstract:
In this article we characterize the form of each 2-local Lie derivation on a von Neumann algebra without central summands of type ${I_{1}}$ . We deduce that every 2-local Lie derivation $\delta$ on a finite von Neumann algebra $\mathcal{M}$ without central summands of type ${I_{1}}$ can be written in the form $\delta(A)=AE-EA+h(A)$ for all $A$ in $\mathcal{M}$ , where $E$ is an element in $\mathcal{M}$ and $h$ is a center-valued homogenous mapping which annihilates each commutator of $\mathcal{M}$ . In particular, every linear 2-local Lie derivation is a Lie derivation on a finite von Neumann algebra without central summands of type ${I_{1}}$ . We also show that every 2-local Lie derivation on a properly infinite von Neumann algebra is a Lie derivation.