Nonlinear Lie type Derivations of Von Neumann Algebras

TL;DR

This paper proves that all nonlinear Lie n-derivations on certain von Neumann algebras can be expressed as a sum of an additive derivation and a central-valued map, clarifying their structure.

Contribution

It establishes that nonlinear Lie n-derivations on von Neumann algebras without type I_1 summands are of a standard form, extending previous linear results.

Findings

Nonlinear Lie n-derivations are of standard form on specified von Neumann algebras.

Every such derivation decomposes into an additive derivation and a central-valued map.

The result applies to von Neumann algebras with no central summands of type I_1.

Abstract

Let $A$ be a von Neumann algebra with no central summands of type $I_1$. We will show that every nonlinear Lie $n$-derivation on $A$ is of the standard form, i.e. it can be expressed as a sum of an additive derivation and a central-valued mapping which annihilates each $(n-1)$-th commutator of $A$.