Let A and B be two unital C∗-algebras. For an arbitrary bijective map φ : A → B that satisfies φ A∗B + BA∗ 2 = φ(A) ∗ φ(B) + φ(B)φ(A) ∗ 2 for every A, B ∈ A, it is shown that if A is prime, then φ is additive. Furthermore, if B is prime, then φ is either a ∗-isomorphism or a ∗-anti-isomorphism.
