Nonlinear bi-skew Lie triple higher derivations on prime $*$-algebras

Abstract

Let $\mathfrak{E}$ be a unital prime $\ast$-algebra. For any $ \mathscr{U}, \mathscr{V} \in \mathfrak{E}$, the product defined by $ \mathscr{U} \diamond \mathscr{V}=\mathscr{U}^{\ast} \mathscr{V}-\mathscr{V}^{\ast} \mathscr{U}$ is known as bi-skew Lie product of $\mathscr{U}$ and $\mathscr{V}$. This paper establishes that if a family $\Delta=\{\zeta_n\}_{ n \in \mathbb{N}}$ of mappings $\zeta_n : \mathfrak{E} \rightarrow \mathfrak{E}$ (not necessarily linear) on $\mathfrak{E}$ with $\zeta_{0} = id_{\mathfrak{E}}$ (the identity map on $\mathfrak{E}$), satisfies the relation $\zeta_n(\mathscr{U} \diamond \mathscr{V} \diamond \mathscr{W}) = \sum\limits_{p+q+r=n} \zeta_p(\mathscr{U}) \diamond \zeta_q(\mathscr{V}) \diamond \zeta_r(\mathscr{W})$ for all $\mathscr{U}, \mathscr{V}, \mathscr{W} \in \mathfrak{E}$ and for each $n \in \mathbb{N},$ then $\Delta$ is an additive $\ast$-higher derivation provided $\zeta_n\left(\frac{\beta\mathscr{I}}{2}\right)$ is self-adjoint for $\beta \in \{1, i\}.$