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 $*$-homomorph...

The paper investigates linear functionals ϕ : ℝn → ℝ preserving a set ℳ⊆ ℝ, i.e., such that ϕ(v) ∈ ℳ for any vector u with components in ℳ. For various types of subsets real numbers, characterizations the preserve them are obtained. In particular, sets ℤ,ℚ, ℤ+,ℚ+,ℝ+, several infinite integers, bounded and unbounded intervals, finite numbers considered. A characterization also allows one to desc...