Non-linear $\ast$-Jordan derivations on von Neumann algebras

TL;DR

This paper investigates the structure of non-linear $ ext{ extasterisk}- ext{Jordan}$ derivations on factor von Neumann algebras, showing they are additive $ ext{ extasterisk}- ext{derivations}$, thus extending understanding of algebraic derivation properties.

Contribution

It proves that non-linear $ ext{ extasterisk}- ext{Jordan}$ derivations on factor von Neumann algebras are necessarily additive $ ext{ extasterisk}- ext{derivations}$, revealing their algebraic structure.

Findings

Non-linear $ ext{ extasterisk}- ext{Jordan}$ derivations are additive $ ext{ extasterisk}- ext{derivations}$ on factor von Neumann algebras.

The result extends the understanding of derivation structures in operator algebras.

The proof relies on properties of the algebra and the derivation definition.

Abstract

Let $\mathcal{A}$ be a factor von Neumann algebra and $\phi$ be the $\ast$-Jordan derivation on $A$, that is, for every $A,B \in \mathcal{A}$, $\phi(A\diamond_{1} B) = \phi(A)\diamond_{1} B + A\diamond_{1}\phi( B)$ where $A\diamond_{1} B = AB + BA^{\ast}$, then $\phi$ is additive $\ast$-derivation.