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

Source 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.