Extension of phase-isometries between the unit spheres of complex L∞(Γ) spaces

Abstract

Let Γ be nonempty index set, and X, Y are complex L^∞(Γ)-type spaces. f: S_X, S_Y will denote their unit spheres. Give a surjective mapping f: S_X → S_Y satisfying the functional equation

{||f(x) + f(y)||, ||f(x)− f(y)||} = {||x+y||, ||x−y||} (x, y ∈ S_X )

We show that there exists a function ε: S_X → {−1,1} such that ε f is an isometry. Moreover, this isometry is the restriction of a real linear isometry from X to Y.