A note on nonlinear skew Lie triple derivation between Prime ∗-algebras

Abstract

Recently, Li et al proved that $\mathrm{\Phi }$ which satisfies the following condition on factor von Neumann algebras
$$\mathrm{\Phi }([[A,B{]}_{\ast },C{]}_{\ast })=[[\mathrm{\Phi }(A),B{]}_{\ast },C{]}_{\ast }+[[A,\mathrm{\Phi }(B){]}_{\ast },C{]}_{\ast }+[[A,B{]}_{\ast },\mathrm{\Phi }(C){]}_{\ast }$$
where $[A,B{]}_{\ast }=AB-B{A}^{\ast }$ for all $A,B\in \mathcal{A}$, is additive $\ast$-derivation. In this short note we show the additivity of $\mathrm{\Phi }$ which satisfies the above condition on prime $\ast$-algebras.