Nonlinear mixed triple derivable mapping on prime ∗-algebras

Abstract

Let $\mathcal A$ be a prime $\ast$-algebra with dim$\mathcal A>1$. In this paper, we prove that a map $\phi:\mathcal A\rightarrow\mathcal A$ satisfies $\phi([A\diamond B,C])=[\phi(A)\diamond B,C]+[A\diamond\phi(B),C]+[A\diamond B,\phi(C)]$ for all $A,B,C\in\mathcal A$ if and only if there exists an element $\lambda\in\mathcal{Z}_{S}(\mathcal{A})$  such that $\phi(A)=d(A)+\mathrm i \lambda A$, where $d:\mathcal A\rightarrow\mathcal A$ is an additive $\ast$-derivation and $A\diamond B=AB^{\ast}+BA^{\ast}$. Also, we give the structure of this map on factor von Neumann algebras.