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

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 ? : A ? B that satisfies ?({a, b}* +b*a) = {?(a),?(b)}* +?(b)*?(a) for all a, b ? A or satisfies ?({a,b}* +a*b) = {?(a),?(b)}* + ?(a)*?(b) for all a, b ? A is a *-ring isomorphism.