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 I1. We deduce that every 2-local Lie derivation δ on a finite von Neumann algebra M without central summands of type I1 can be written in the form δ(A)=AE−EA+h(A) for all A in M, where E is an element in M and h is a center-valued homogenous mapping which annihilates each commutator of M. In particular, every linear 2-local Lie derivation is a Lie derivation on a finite von Neumann algebra without central summands of type I1. We also show that every 2-local Lie derivation on a properly infinite von Neumann algebra is a Lie derivation.