Non-additive families satisfying a mixed bi-skew higher derivation identity on $ \star $-algebras

Let $ \mathscr{A} $ be a unital $ \star $-algebra over the complex field that possesses a nontrivial projection $ \mathscr{P} $ satisfying the faithfulness property $ \mathscr{X} \mathscr{A} \mathscr{P} = 0 \Rightarrow \mathscr{X} = 0 $ (and the analogous condition for $ I- \mathscr{P} $). Suppose a family $ \{\Delta_n\}_{n = 0}^{\infty} $ of (not necessarily additive) maps $ \Delta_n: \mathscr{A}\to \mathscr{A} $ where $ \Delta_0 = \mathrm{id}_{ \mathscr{A}} $ satisfies the mixed bi-skew Jordan–Lie higher derivation relation

$ \begin{align*} \Delta_n\bigl([ \mathscr{X}\bullet \mathscr{Y},\, \mathscr{Z}]_\star\bigr) & = [\Delta_n( \mathscr{X})\bullet \mathscr{Y},\, \mathscr{Z}]_\star + [ \mathscr{X}\bullet \Delta_n( \mathscr{Y}),\, \mathscr{Z}]_\star+ [ \mathscr{X}\bullet \mathscr{Y},\,\Delta_n( \mathscr{Z})]_\star \\ &\quad + \sum\limits_{\substack{r+s+t = n\\ 0\leq r,s,t\leq n-1}} \bigl[\Delta_r( \mathscr{X})\bullet\Delta_s( \mathscr{Y}),\,\Delta_t( \mathscr{Z})\bigr]_\star \end{align*} $

for all $ \mathscr{X}, \mathscr{Y}, \mathscr{Z}\in \mathscr{A} $, and each $ n\ge 1 $. We prove that all maps $ \Delta_n $ are additive and preserve the involution. Consequently, the family $ \{\Delta_n\}_{n = 0}^{\infty} $ forms an additive $ \star $-higher derivation with respect to the ternary product $ [\, \cdot\bullet\cdot\, ,\cdot]_\star $. Direct consequences for prime $ \star $-algebras, factor von Neumann algebras, and standard operator algebras are presented, and an example is provided to illustrate the theory.