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On Preserving Properties of Linear Maps on $C^{*}$-algebras

Journal: Sahand Communications in Mathematical Analysis (Vol.17, No. 1)

Publication Date:

Authors : ; ;

Page : 125-137

Keywords : Absolute value preserving; $*$-homomorphism; Unitary preserving; numerical range;

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Abstract

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 $a,bin A$, then $varphi$ is a $*$-homomorphism. It is also shown that if $varphi(|ab|)=|varphi(a)varphi(b)|$ for all $a,bin A$, then $varphi$ is a unital $*$-homomorphism.

Last modified: 2020-06-16 17:02:43