Let H and K be two complex Hilbert spaces andAB(H) and BB(K) be two unital C-algebras. It is shown that if :A!B is an additive surjective mapping satisfying (jAj)¼j(A)j for every A2A and (I ) is a projection, then the restriction of mapping to both As and Ask is a Jordan -homomorphism onto corresponding set in B, where As and Ask 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 is a C-linear or C-antilinear -homomorphism.
