Mappings preserving product $ab\pm ba^{*}$ on alternative $W^{*}$-factors

TL;DR

This paper characterizes bijective mappings between alternative $W^{*}$-factors that preserve specific product forms, proving they must be *-ring isomorphisms.

Contribution

It establishes that mappings preserving the products $ab \uplus ba^{*}$ are exactly the *-ring isomorphisms between alternative $W^{*}$-factors.

Findings

Mappings preserving $ab+ba^{*}$ or $ab-ba^{*}$ are *-ring isomorphisms.

Characterization of structure-preserving maps on alternative $W^{*}$-factors.

Proof of equivalence between product-preserving maps and *-ring isomorphisms.

Abstract

Let $\mathcal{A}$ and $\mathcal{B}$ be two alternative $W^{*}$-factors. In this paper, we proved that a bijective mapping $\Phi :\mathcal{A}\rightarrow \mathcal{B}$ satisfies $\Phi (ab+ba^{*})=\Phi (a)\Phi (b)+\Phi (b)\Phi (a)^{*}$ (resp., $\Phi (ab-ba^{*})=\Phi (a)\Phi (b)-\Phi (b)\Phi (a)^{*}$), for all elements $a,b\in \mathcal{A}$, if and only if $\Phi $ is a $\ast $-ring isomorphism.