Let A be a C∗-algebra of real-rank zero and B be a C∗-algebra with unit I . It is shown that the mapping : A −→ B which preserves arithmetic mean and satisfies (A∗ A) = (A) ∗ (A) + (A)(A) ∗ 2 , for all normal elements A ∈ A, is an R-linear continuous Jordan ∗-homomorphism provided that 0 ∈ Ran . Also, is the sum of a linear Jordan ∗-homomorphism and a conjugate-linear Jordan ∗-homomorphism. This result also presents an application of maps which preserve the square absolute value.
