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

Abstract

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 $\Phi$ from $\mathcal{A}$ onto $\mathcal{B}$ that are bijective unital and satisfies $$\Phi (AP+\eta P{A}^{\ast })=\Phi (A)\Phi (P)+\eta \Phi (P)\Phi (A{)}^{\ast },$$ for all $A\in \mathcal{A}$ and $P\in \{P_1,I_{\mathcal{A}}-P_1\}$ where $P_1$ is a nontrivial projection in $\mathcal{A}$. Let $\eta$ be a non-zero complex number such that $|\eta |\ne 1$, then $\Phi$ is additive. Moreover, if $\eta$ is rational then $\Phi$ is $\ast$-additive.

10 pages, Accepted in Bulletin of Iranian Mathematics Society. arXiv admin note: substantial text overlap with arXiv:1405.4611