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

Abstract

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