Additivity of maps preserving products $AP\pm PA^{*}$ on $C^{*}$-algebras

Abstract

Let $\mathcal{A}$ and $\mathcal{B}$ be two prime $C^{*}$-algebras. In this paper, we investigate the additivity of map $\Phi$ from $\mathcal{A}$ onto $\mathcal{B}$ that are bijective unital and satisfies $$\Phi(AP+\lambda PA^{*})=\Phi(A)\Phi(P)+\lambda \Phi(P)\Phi(A)^{*},$$ for all $A\in\mathcal{A}$ and $P\in\{P_{1},I_{\mathcal{A}}-P_{1}\}$ where $P_{1}$ is a nontrivial projection in $\mathcal{A}$ and $\lambda\in\{-1,+1\}$. Then, $\Phi$ is $*$-additive.