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

TL;DR

This paper investigates bijective maps between Banach algebras of continuous functions on compact spaces that preserve the ring structure in norm, showing they are induced by homeomorphisms and are essentially weighted composition operators.

Contribution

It characterizes norm-preserving ring isomorphisms between function algebras as weighted composition operators induced by homeomorphisms, under certain conjugation conditions.

Findings

Such maps are necessarily induced by homeomorphisms between the underlying spaces.

Under conjugation preservation, these maps are real-linear isometries.

They can be represented explicitly as weighted composition operators.

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.