Mappings preserving sum of products a◊b + b*a (resp., a*◊b + ab*) on ∗-algebras

Abstract

Let A and B be two prime complex ∗-algebras. We proved that every bijective mapping Φ : A → B satisfying Φ(a ◊+ b∗ a) = Φ(a)◊Φ(b) + Φ(b)∗Φ(a) (resp., Φ(a∗ ◊b + ab∗) = Φ(a)∗ ◊Φ(b) + Φ(a)Φ(b)∗), where a ◊b = ab + ba∗, for all elements a, b ∈ A, is a ∗-ring isomorphism.