A result on mixed skew and $\eta$-Jordan triple derivations on prime $*$-algebras

Abstract

Let $\mathcal{A}$ be a prime $*$-algebra. In this paper, we establish under some mild conditions that every non-linear mixed $\eta-$Jordan triple $*$-derivation i.e., a non-linear map $\psi: \mathcal{A}\rightarrow \mathcal{A}$ satisfying $$\psi(\Aa \circ_\eta \Ba \bullet \Ca)=\psi(\Aa) \circ_\eta \Ba \bullet \Ca + \Aa \circ_\eta \psi(\Ba) \bullet \Ca + \Aa \circ_\eta \Ba \bullet \psi(\Ca)$$for any $\Aa, \Ba, \Ca \in \mathcal{A}$, is a linear $*$-derivation.