Characterization of non-linear mixed bi-skew Jordan triple higher derivations on prime $*$-algebras

Abstract

Let $\mathcal{A}$ be a prime $*$-algebra. For any $A, B\in \mathcal{A}$, define a new product $A\bullet B=AB^*+B A^*$. Let $\delta$ be a non-linear map satisfying $\delta(A \circ B \bullet C)=\delta(A) \circ B \bullet C + A \circ \delta(B) \bullet C + A \circ B \bullet \delta(C)$ for all $A, B, C \in \mathcal{A}$. In this manuscript, we show that $\delta$ is an additive $*$-derivation. Furthermore, we also discuss above result for higher derivable maps on $\mathcal{A}$.