Nonlinear $\ast$-Jordan triple derivation on prime $\ast$-algebras

Abstract

Let $\mathcal{𝒜}$ be a prime $\ast$-algebra and suppose that $\mathrm{\Phi }$ preserves triple $\ast$-Jordan derivation on $\mathcal{𝒜}$, that is, for every $A,B\in \mathcal{𝒜}$,

$$\mathrm{\Phi }\left(A◇B◇C\right)=\mathrm{\Phi }\left(A\right)◇B◇C+A◇\mathrm{\Phi }\left(B\right)◇C+A◇B◇\mathrm{\Phi }\left(C\right),$$

where $A◇B=AB+B{A}^{\ast }$. Then $\mathrm{\Phi }$ is additive. Moreover, if $\mathrm{\Phi }\left(\alpha I\right)$ is selfadjoint for $\alpha \in \left\{1,i\right\}$, then $\mathrm{\Phi }$ is a $\ast$-derivation.