Document Type : Research Paper


1 Department of Multimedia, Tabriz Islamic Art University, Tabriz, Iran.

2 Department of Mathematics, Sahand University of Technology, Sahand Street, Tabriz, Iran.


Let $A$ and $B$ be two unital $C^{*}$-algebras and $\varphi:A \rightarrow B$ be a linear map. In this paper, we investigate the structure of linear maps between two $C^{*}$-algebras that preserve a certain property or relation. In particular, we show that if $\varphi$ is unital, $B$ is commutative and $V(\varphi(a)^{*}\varphi(b))\subseteq V(a^{*}b)$ for all $a,b\in A$, then $\varphi$ is a $*$-homomorphism. It is also shown that if $\varphi(|ab|)=|\varphi(a)\varphi(b)|$ for all $a,b\in A$, then $\varphi$ is a unital $*$-homomorphism.


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