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

TL;DR

This paper characterizes bijections between von Neumann algebras that preserve a specific Jordan product, showing they are essentially *-isomorphisms or sums of such, depending on the parameter ta.

Contribution

It establishes conditions under which maps preserving Jordan ta-st-n-products are linear st-isomorphisms or sums thereof, depending on ta's nature.

Findings

ta not real: ta-bijective maps are linear st-isomorphisms.

ta real: ta-bijective maps are sums of linear and conjugate linear st-isomorphisms.

The result applies to von Neumann algebras with no central abelian projections.

Abstract

Let $\eta\neq -1$ be a non-zero complex number, and let $\phi$ 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 $\phi$ is a linear $\ast$-isomorphism if $\eta$ is not real and $\phi$ is the sum of a linear $\ast$-isomorphism and a conjugate linear $\ast$-isomorphism if $\eta$ is real.