Nonlinear skew Lie triple centralizers (derivations) on *-algebras

Abstract

Let $\mathcal {A}$ be a unital $*$-algebra over the complex field $\mathbb{C}$. In this paper, we prove that   every nonlinear skew Lie (triple) centralizer on $\mathcal {A}$ is a linear $*$-centralizer.  Under some mild conditions on $\mathcal {A}$, we also prove that a map $\Phi$ on $\mathcal{A}$ is a nonlinear skew Lie triple derivation if and only if $\Phi$ is an additive $*$-derivation. As applications, nonlinear 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.