The structure of 2-local Lie derivations on von Neumann algebras

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.