Nonlinear $\xi$-skew-Jordan triple higher derivations on prime $\ast$-algebras

Let $\mathfrak{B}$ denote a unital prime $\ast$-algebra over the complex field $\mathbb{C}$, and let $\xi \in \mathbb{C}\setminus \{0,\pm 1\}$. This paper characterized a family $\mathrm{\Psi }=\{{\phi }_m{\}}_{m\in \mathbb{N}}$ of maps (not necessarily additive) from $\mathfrak{B}$ into itself that satisfy the functional identity

$${\phi }_m(A{\diamond }_{\xi }B{\diamond }_{\xi }C)=\sum _{i+j+k=m}{\phi }_i(A){\diamond }_{\xi }{\phi }_j(B){\diamond }_{\xi }{\phi }_k(C)$$

for all $A,B,C\in \mathfrak{B}$, where $A{\diamond }_{\xi }B=AB+\xi B{A}^{\ast }$. It was proved that $\mathrm{\Psi }$ satisfies the above identity if and only if $\mathrm{\Psi }$ is an additive higher $\ast$-derivation and ${\phi }_m(\xi A)=\xi {\phi }_m(A)$ holds for every $A\in \mathfrak{B}$. As an application, the result not only generalizes the structure of nonlinear $\xi$-skew-Jordan triple derivations on prime $\ast$-algebras, but also yields a description of nonlinear $\xi$-skew-Jordan triple higher derivations on several important operator algebras, including standard operator algebras and factor von Neumann algebras.