Ring isomorphisms in norm between Banach algebras of continuous complex-valued functions

Abstract

Let $X$ and $Y$ be compact Hausdorff spaces, and let $C(X)$ and $C(Y)$ denote the commutative Banach algebras of all continuous complex-valued functions on $X$ and $Y$, respectively. We study bijective maps $T$ from $C(X)$ onto $C(Y)$ which preserve the ring structure in the norm in the following sense: \[ \|T(f+g)\|=\|T(f)+T(g)\|,\quad \|T(fg)\|=\|T(f)T(g)\| \qquad(f,g\in C(X)). \] Our main objective is to clarify whether such maps must necessarily be induced by homeomorphisms between the underlying spaces. Under the additional assumption that $T(\overline{f})=\overline{T(f)}$ for $f\in C(X)$, we prove that $T$ is a real-linear isometry. As a consequence, we obtain a concrete representation of such maps as weighted composition operators.