Nonlinear maps preserving Jordan $\eta$-$\ast$-$n$-products

Abstract

Let $\eta \ne -1$ be a non-zero complex number, and let $\varphi$ be a not necessarily linear bijection between two von Neumann algebras, one of which has no central abelian projections preserving the Jordan $\eta$-$\ast$-$n$-product. It is showed that $\varphi$ is a linear $\ast$-isomorphism if $\eta$ is not real and $\varphi$ is the sum of a linear $\ast$-isomorphism and a conjugate linear $\ast$-isomorphism if $\eta$ is real.