Mappings preserving sum of products {a, b}∗ + b ∗a or preserving {a, b}∗ + a ∗ b on generalized matrix ∗-rings

Abstract

Let A be a generalized matrix ∗-ring and let B be an unital ∗-ring. For a, b ∈ A, we define {a, b}∗ = ab+ba∗ . In this paper, we prove that, under some additional conditions, a bijective map φ : A → B that satisfies φ({a, b}∗ + b ∗a) = {φ(a), φ(b)}∗ + φ(b) ∗φ(a) for all a, b ∈ A or satisfies φ({a, b}∗ + a ∗ b) = {φ(a), φ(b)}∗ + φ(a) ∗φ(b) for all a, b ∈ A is a ∗-ring isomorphism.