Phase-isometries between the positive cones of the Banach space of continuous real-valued functions

For a locally compact Hausdorff space L, we denote by C₀(L,R) the Banach space of all continuous real-valued functions on L vanishing at infinity equipped with the supremum norm. We prove that every surjective phase-isometry T C₀⁺(X,R) → C₀⁺(Y,R) between the positive cones of C₀(X,R) and C₀(Y,R) is a composition operator induced by a homeomorphism between X and Y. Furthermore, we show that any surjective phase-isometry T C₀⁺(X,R) → C₀⁺(Y,R) extends to a surjective linear isometry from C₀(X,R) onto C₀(Y,R).