Mappings preserving product $ab\pm b{a}^{\ast }$ on alternative $W^{\ast }$-factors

Abstract

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