The automatic additivity of $\xi-$Lie derivations on von Neumann algebras

TL;DR

This paper proves that nonlinear $\xi$-Lie derivations on certain von Neumann algebras are necessarily additive derivations, extending understanding of derivation structures in operator algebras.

Contribution

It establishes the additivity of nonlinear $\xi$-Lie derivations on von Neumann algebras without type I$_1$ summands, a new result in operator algebra theory.

Findings

Nonlinear $\xi$-Lie derivations are additive derivations.

The result applies to von Neumann algebras with no type I$_1$ summands.

Provides a characterization of $\xi$-Lie derivations in this setting.

Abstract

Let ${\mathcal M}$ be a von Neumann algebra with no central summands of type I$_1$. It is shown that every nonlinear $\xi-$Lie derivation ($\xi\neq 1$) on $\mathcal M$ is an additive derivation.