On Preserving Properties of Linear Maps on C*-algebras

Let A and B be two unital C ∗ -algebras and φ : A → 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 φ is unital, B is commutative and V (φ(a) ∗φ(b)) ⊆ V (a ∗ b) for all a, b ∈ A, then φ is a ∗-homomorphism. It is also shown that if φ(|ab|) = |φ(a)φ(b)| for all a, b ∈ A, then φ is a unital ∗-homomorphism.