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: SX, SY will denote their unit spheres. Give a surjective mapping f: SX → SY satisfying the functional equation {||f(x) + f(y)||, ||f(x)− f(y)||} = {||x+y||, ||x−y||} (x, y ∈ SX ) We show that there exists a function ε: SX → {−1,1} such that ε f is an isometry. Moreover, this isometry is the restriction of a real linear isometry from X to Y.