Additive and multiplicative maps in norm on the positive cone of continuous function algebras

Let $X$ and $Y$ be locally compact Hausdorff spaces. We denote by $C_0^{+}(X)$ the positive cone of all real-valued continuous functions on $X$ vanishing at infinity. In this paper, we consider a bijection $T\text{\hspace{0.17em}⁣}:C_0^{+}(X)\to C_0^{+}(Y)$ satisfying the following two norm conditions for all $f,g\in C_0^{+}(X)$: $$\mathrm{\parallel }T(f+g)\mathrm{\parallel }=\mathrm{\parallel }T(f)+T(g)\mathrm{\parallel },\mathrm{\parallel }T(f\cdot g)\mathrm{\parallel }=\mathrm{\parallel }T(f)\cdot T(g)\mathrm{\parallel }\mathrm{.}$$ The main result of this paper is that such a map $T$ is a composition operator of the form $T(f)=f\circ \tau$, induced by a homeomorphism $\tau \text{\hspace{0.17em}⁣}:Y\to X$.