Additivity of maps preserving products $AP\pm P{A}^{\ast }$ on $C^{\ast }$-algebras

Abstract

Let $\mathcal{A}$ and $\mathcal{B}$ be two prime $C^{\ast }$-algebras. In this paper, we investigate the additivity of map $\Phi$ from $\mathcal{A}$ onto $\mathcal{B}$ that are bijective unital and satisfies $$\Phi (AP+\lambda P{A}^{\ast })=\Phi (A)\Phi (P)+\lambda \Phi (P)\Phi (A{)}^{\ast },$$ for all $A\in \mathcal{A}$ and $P\in \{P_1,I_{\mathcal{A}}-P_1\}$ where $P_1$ is a nontrivial projection in $\mathcal{A}$ and $\lambda \in \{-1,+1\}$. Then, $\Phi$ is $\ast$-additive.

10 pages, Accepted for publishing in Mathematica Slovaca