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 $T\text{\hspace{0.17em}⁣}:C_0^{+}(X,\mathbb{R})\to C_0^{+}(Y,\mathbb{R})$ 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 $T\text{\hspace{0.17em}⁣}:C_0^{+}(X,\mathbb{R})\to C_0^{+}(Y,\mathbb{R})$ extends to a surjective linear isometry from $C_0(X,\mathbb{R})$ onto $C_0(Y,\mathbb{R})$.