A Note on Nonlinear Mixed (bi-Skew, skew Lie) Triple Derivations on $\ast$-Algebras

Abstract

Let \(\mathfrak{A}\) be a unital \(\ast\)-algebra containing a non-trivial projection. We prove that if a map \(\Lambda : \mathfrak{A} \to \mathfrak{A}\) satisfies
\[
\Lambda( [[ \mathscr{L},\mathscr{M}]_\bullet, \mathscr{N}]_\ast) = [[ \Lambda(\mathscr{L}),\mathscr{M}]_\bullet, \mathscr{N}]_\ast + [[ \mathscr{L},\Lambda(\mathscr{M})]_\bullet, \mathscr{N}]_\ast + [[ \mathscr{L},\mathscr{M}]_\bullet, \Lambda(\mathscr{N})]_\ast
\]
for all \(\mathscr{L}, \mathscr{M}, \mathscr{N} \in \mathfrak{A},\) then \(\Lambda\) is additive. Moreover, if \(\Lambda(\mathfrak{I})\) is self-adjoint, then \(\Lambda\) is a \(\ast\)-derivation. Additionally, as an application, we can also apply our results to factor von Neumann algebras, standard operator algebras, and prime \(\ast\)-algebras.