Additivity of maps preserving Jordan ${\eta }_{\ast }$-products on $C^{\ast }$-algebras

Let $\mathcal{A}$ and $\mathcal{B}$ be two $C^{\ast }$-algebras such that $\mathcal{B}$ is prime. In this paper, we investigate the additivity of map $\mathrm{\Phi }$ from $\mathcal{A}$ onto $\mathcal{B}$ that are bijective unital and satisfies $$\mathrm{\Phi }(AP+\eta P{A}^{\ast })=\mathrm{\Phi }(A)\mathrm{\Phi }(P)+\eta \mathrm{\Phi }(P)\mathrm{\Phi }(A{)}^{\ast },$$ for all $A\in \mathcal{A}$ and $P\in \{P_1,I_{\mathcal{A}}-P_1\}$ where $P_1$ is a nontrivial projection in $\mathcal{A}$. Let $\eta$ be a non-zero complex number such that $\mathrm{\mid }\eta \mathrm{\mid }\mathrm{\ne }1$, then $\mathrm{\Phi }$ is additive. Moreover, if $\eta$ is rational then $\mathrm{\Phi }$ is $\ast$-additive.