Ring isomorphisms in norm between Banach algebras of continuous functions

Abstract

Let $X$ and $Y$ be locally compact Hausdorff spaces and let $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$. We say that a bijection $T:C_0(X,\mathbb{K})\to C_0(Y,\mathbb{K})$ is a ring isomorphism in norm if \[ \|T(f+g)\|=\|T(f)+T(g)\|,\qquad \|T(fg)\|=\|T(f)T(g)\| \] for every $f,g\in C_0(X,\mathbb{K})$. We determine the form of such maps. When $\mathbb{K}=\mathbb{C}$, under the additional assumption that $\Vert T(\overline{f})\Vert =\Vert T(f)\Vert$ for every $f\in C_0(X,\mathbb{C})$, there exist a continuous function $w:Y\to \{\lambda \in \mathbb{C}:|\lambda |=1\}$, a homeomorphism $\phi :Y\to X$, and a closed and open subset $Y_0\subset Y$ such that \[ T(f)(y)= \begin{cases} w(y)f(φ(y)),& y\in Y_0,\\ w(y)\overline{f(φ(y))},& y\in Y\setminus Y_0, \end{cases} \] for every $f\in C_0(X,\mathbb{C})$ and $y\in Y$. When $\mathbb{K}=\mathbb{R}$, there exist a continuous function $w:Y\to \{\pm 1\}$ and a homeomorphism $\phi :Y\to X$ such that \[ T(f)(y)=w(y)f(φ(y)) \] for every $f\in C_0(X,\mathbb{R})$ and $y\in Y$. In particular, the real case extends the norm version of the Gelfand--Kolmogoroff theorem to the locally compact setting.