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