Extension of phase-isometries between the unit spheres of complex lp(Γ)-spaces (p>1)

Let Γ, ∆ be nonempty index sets. For p ∈ (1, ∞), we prove that every surjective mapping f: S lp(Γ) → S lp(∆) satisfying the functional equation {||f(x) + f(y)||, ||f(x)− f(y)||} = {||x+y||, ||x−y||} (x, y ∈ S lp(Γ) ), its positive homogeneous extension is a phase-isometry which is phase equivalent a real linear isometry.