Nonlinear η-*-Jordan n-Derivation on *-Algebras

Let $\mathcal{A}$ be a unital *-algebra with the unit I over the complex field $\mathbb{C}$ and let $\eta \ne 0,\pm 1$ be a complex number. For any $A,B\in \mathcal{A}$, $A{\diamond }_{\eta }B=AB+\eta B{A}^{*}$ is referred to as the $\eta$-Jordan *-product. Suppose that $n\ge 3$ is a fixed positive integer. In this study, it is shown that if a map $\phi :\mathcal{A}\to \mathcal{A}$ satisfies $\phi (A_1{\diamond }_{\eta }{A}_2{\diamond }_{\eta }\dots {\diamond }_{\eta }{A}_n)={\sum }_{k=1}^n{A}_1{\diamond }_{\eta }\dots {\diamond }_{\eta }{A}_{k-1}{\diamond }_{\eta }\phi \left(A_k\right){\diamond }_{\eta }{A}_{k+1}{\diamond }_{\eta }\dots {\diamond }_{\eta }{A}_n$ for all $A_1,A_2\dots A_{n-3}\in \{I,iI\}$ and $A_{n-2},A_{n-1},A_n\in \mathcal{A}$, then $\phi$ is an additive *-derivation and $\phi \left(\eta A\right)=\eta \phi \left(A\right)$ for all $A\in \mathcal{A}$, where i is the imaginary unit. In application, characterizations of prime *-algebras, von Neumann algebras with no central summands of type $I_1$ and factor von Neumann algebras are obtained.