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

Abstract

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 $T:C_0^{+}(X)\to C_0^{+}(Y)$ satisfying the following two norm conditions for all $f,g\in C_0^{+}(X)$: \[ \|T(f+g)\| = \|T(f)+T(g)\|,\qquad \|T(f \cdot g)\| = \|T(f) \cdot T(g)\|. \] 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 $\tau :Y\to X$.