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

Abstract: Let $\mathcal{A}$ and $\mathcal{B}$ be two $C^{*}$-algebras such that $\mathcal{B}$ is prime. In this paper, we investigate the additivity of map $\Phi$ from $\mathcal{A}$ onto $\mathcal{B}$ that are bijective unital and satisfies $$\Phi(AP+\eta PA^{*})=\Phi(A)\Phi(P)+\eta \Phi(P)\Phi(A)^{*},$$ 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 $|\eta|\neq1$, then $\Phi$ is additive. Moreover, if $\eta$ is rational then $\Phi$ is $\ast$-additive.