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 \(C_0(L,{\mathbb {R}})\) 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 \(T:C_0^+(X,{\mathbb {R}})\rightarrow C_0^+(Y,{\mathbb {R}})\) between the positive cones of \(C_0(X,{\mathbb {R}})\) and \(C_0(Y,{\mathbb {R}})\) is a composition operator induced by a homeomorphism between X and Y. Furthermore, we show that any surjective phase-isometry \(T:C_0^+(X,{\mathbb {R}})\rightarrow C_0^+(Y,{\mathbb {R}})\) extends to a surjective linear isometry from \(C_0(X,{\mathbb {R}})\) onto \(C_0(Y,{\mathbb {R}})\).