Jordan Maps on Standard Operator Algebras

Let A be a standard operator algebra on a Banach space of dimension > 1 and B be an arbitrary algebra over Q the field of rational numbers. Suppose that M : A → B and M* : B → A are surjective maps such that M(r(aM*(x) + M*(x)a)) = r(M(a)x + xM(a)), M(r(M(a)x + xM(a))) = r(aM*(x) + M*(x)a) for all a ∈ A,x ∈ B, where r is a fixed nonzero rational number. Then both M and M* are additive.