Additive mappings onC*-algebras preserving absolute values

Let H and K be two complex Hilbert spaces and 𝒜 ⊂ B(H) and ℬ ⊂ B(K) be two unital C*-algebras. It is shown that if φ : 𝒜 → ℬ is an additive surjective mapping satisfying φ(|A|) = |φ(A)| for every A ∈ 𝒜 and φ(I) is a projection, then the restriction of mapping φ 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 φ is a ℂ-linear or ℂ-antilinear *-homomorphism.