# Non-linear $\ast$-Jordan triple derivation on prime $\ast$-algebras

## TL;DR

This paper investigates non-linear maps on prime $ extit{*}$-algebras that preserve a specific triple $ extit{*}$-Jordan derivation structure, proving additivity and conditions under which they are $ extit{*}$-derivations.

## Contribution

It establishes that such maps are additive and, under certain conditions, are $ extit{*}$-derivations, extending understanding of structure-preserving maps in prime $ extit{*}$-algebras.

## Key findings

- Preservation of triple $ extit{*}$-Jordan derivation implies additivity.
- Additional conditions ensure the map is a $ extit{*}$-derivation.
- Results apply to prime $ extit{*}$-algebras with specific self-adjoint properties.

## Abstract

Let $\mathcal{A}$ be a prime $\ast$-algebra and $\Phi$ preserves triple $\ast$-Jordan derivation on $\mathcal{A}$, that is, for every $A,B \in \mathcal{A}$, $$\Phi(A\diamond B \diamond C)=\Phi(A)\diamond B\diamond C+A\diamond \Phi(B)\diamond C+A\diamond B\diamond \Phi(C)$$ where $A\diamond B = AB + BA^{\ast}$ then $\Phi$ is additive. Moreover, if $\Phi(\alpha I)$ is self-adjoint for $\alpha\in\{1,i\}$ then $\Phi$ is a $\ast$-derivation.