Nonlinear $\xi$-bi-skew Lie derivations on prime $\ast$-algebras

Abstract

Let $\mathcal{A}$ be a factor von Neumann algebra acting on a complex Hilbert space $H$ withdim$\mathcal{A}>1$ and $\xi$ be a nonzero scalar. We prove that a map $\varphi: \mathcal{A}\rightarrow \mathcal{A}$ satisfies
$\varphi([A, B]^{\xi}_{\diamond})=[\varphi(A), B]^{\xi}_{\diamond}+[A, \varphi(B)]^{\xi}_{\diamond}$
for all $A, B\in \mathcal{A}$ if and only if $\varphi$ is an additive $\ast$-derivation
and $\varphi(\xi A)=\xi\varphi(A)$ for all $A\in \mathcal{A}$, where $[A, B]^{\xi}_{\diamond}=AB^{\ast}-\xi BA^{\ast}$.