Additive mappings on C*-algebras preserving absolute values

Let H and K be two complex Hilbert spaces and A subset of B(H) and B subset of B(K) be two unital C*-algebras. It is shown that if phi : A -> B is an additive surjective mapping satisfying phi(vertical bar A vertical bar) = vertical bar phi(A)vertical bar for every A is an element of A and phi(I) is a projection, then the restriction of mapping phi to both A(s) and A(sk) is a Jordan *-homomorphism onto corresponding set in B, where A(s) and A(sk) denote the set of all self-adjoint and skew-self-adjoint elements, respectively. Furthermore, if B is a C*-algebra of real-rank zero then phi is a C-linear or C-antilinear *-homomorphism.