Ring isomorphisms in norm between Banach algebras of continuous complex-valued functions
T. Miura, T. Takahashi
TL;DR
The paper studies bijections $T:C(X)\to C(Y)$ between complex function algebras on compact Hausdorff spaces that preserve ring structure in norm, augmented by $T(\overline{f})=\overline{T(f)}$. The authors show that such a map is, after normalization, a real-linear isometry and is represented by a weighted composition operator determined by a homeomorphism $\tau:Y\to X$ and a clopen set $K\subset Y$, with a unimodular multiplier $u\in C_\mathbb{R}(Y)$ satisfying $u(Y)\subset\{\pm1\}$. The construction relies on converting the ring-preserving in-norm property into additivity, forming a unital map $S(f)=T(1_X)T(f)$, proving that $S$ is a bijective real-linear isometry, and applying Ellis's theorem to obtain the explicit representation $T(f)(y)=u(y)f(\tau(y))$ on $K$ and $u(y)\overline{f(\tau(y))}$ on $Y\setminus K$, thereby linking the algebraic and topological data via a weighted composition operator. This extends classical results by showing that norm-ring isomorphisms between complex function algebras are governed by underlying homeomorphisms with a dichotomy on a clopen partition.
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.