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

Let $\mathcal{A}$ and $\mathcal{B}$ be two alternative $W^{\ast }$-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\mathrm{.},$\Phi (ab-ba^{})=\Phi (a)\Phi (b)-\Phi (b)\Phi (a)^{}$), for all elements $a,b\in \mathcal{A}$, if and only if $\mathrm{\Phi }$ is a $\ast$-ring isomorphism.