The nonlinear mixed bi-skew Lie triple derivations on *-algebras

Abstract

Let $\mathcal {A}$ be a unital $*$-algebra. In this paper, under some mild conditions on $\mathcal {A}$, it is shown that a map $\Phi:\mathcal{A}\rightarrow\mathcal{A}$ is the nonlinear mixed bi-skew Lie triple derivation if and only if $\Phi$ is an additive $*$-derivation. As applications, the nonlinear mixed bi-skew Lie triple derivations on prime $\ast$-algebras, von Neumann algebras with no central summands of type $I_1$, factor von Neumann algebras and standard operator algebras are characterized.