Nonlinear mixed Jordan triple *-derivations on *-algebras

Abstract

Let $\mathcal{A}$ be a unital $\ast$-algebra. For $A,B\in \mathcal{A}$, define by $[A,B{]}_{\ast }=AB-B{A}^{\ast }$ and $A\bullet B=AB+B{A}^{\ast }$ the new products of $A$ and $B$. In this paper, under some mild conditions on $\mathcal{A}$, it is shown that a map $\Phi :\mathcal{A}\to \mathcal{A}$ satisfies $\Phi ([A\bullet B,C{]}_{\ast })=[\Phi (A)\bullet B,C{]}_{\ast }+[A\bullet \Phi (B),C{]}_{\ast }+[A\bullet B,\Phi (C){]}_{\ast }$ for all $A,B,C\in \mathcal{A}$ if and only if $\Phi$ is an additive $\ast -$derivation. In particular, we apply the above result to prime $\ast$-algebras, von Neumann algebras with no central summands of type $I_1$, factor von Neumann algebras and standard operator algebras.