Nonlinear maps preserving the mixed Jordan triple $\eta$-$*$-product between factors

TL;DR

This paper characterizes nonlinear bijections between factor von Neumann algebras that preserve a specific mixed Jordan triple product, revealing they are essentially *-isomorphisms or their negatives under certain conditions.

Contribution

It provides a complete description of nonlinear maps preserving the mixed Jordan triple product between factors, extending known linear results to nonlinear maps.

Findings

For $ ext{eta}=1$, such maps are linear or conjugate linear *-isomorphisms, or their negatives.

For $ ext{eta} eq 1$ with $ ext{phi}(I)=1$, maps are either linear or conjugate linear *-isomorphisms.

The results unify the structure of maps preserving the mixed Jordan triple product in von Neumann factors.

Abstract

Let $\mathcal{A}$ and $\mathcal{B}$ be two factor von Neumann algebras and $\eta$ be a non-zero complex number. A nonlinear bijective map $\phi:\mathcal A\rightarrow\mathcal B$ has been demonstrated to satisfy $$\phi([A,B]_{*}^{\eta}\diamond_{\eta} C)=[\phi(A),\phi(B)]_{*}^{\eta}\diamond_{\eta}\phi(C)$$ for all $A,B,C\in\mathcal A.$ If $\eta=1,$ then $\phi$ is a linear $*$-isomorphism, a conjugate linear $*$-isomorphism, the negative of a linear $*$-isomorphism, or the negative of a conjugate linear $*$-isomorphism. If $\eta\neq 1$ and satisfies $\phi(I)=1,$ then $\phi$ is either a linear $*$-isomorphism or a conjugate linear $*$-isomorphism.