University of MaraghehSahand Communications in Mathematical Analysis2322-580717120200101On Preserving Properties of Linear Maps on $C^{*}$-algebras1251373733610.22130/scma.2019.107553.607ENFatemehGolfarshchiDepartment of Multimedia, Tabriz
Islamic Art University, Tabriz, Iran.Ali AsgharKhalilzadehDepartment of Mathematics, Sahand University of Technology, Sahand Street, Tabriz, Iran.Journal Article20190508Let $A$ 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.https://scma.maragheh.ac.ir/article_37336_5e75596460a797fec56dbf1fd1ff1242.pdf