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

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.