Let A be a C*-algebra of real rank zero and B be a C*-algebra with unit I. It is shown that if φ : A –→B is an additive mapping which satisfies |φ(A)φ(B)| ≤ φ(|AB|) for every A, B ∈ A+ and φ(A) = I for some A ∈ As with A≤ 1, then the restriction of mapping φ to As is a Jordan homomorphism, where As denotes the set of all self-adjoint elements. We will also show that if φ is surjective preserving the product and an absolute value, then φ is a C-linear or C-antilinear *-homomorphism on A.
