Representation of group isomorphisms. The compact case

TL;DR

This paper characterizes separating group isomorphisms between subgroups of continuous functions on 0-dimensional compact spaces, showing they can be represented as weighted composition operators under mild conditions.

Contribution

It provides a representation theorem for separating isomorphisms as weighted composition operators in the compact case, linking subgroup equivalence to biseparating isomorphisms.

Findings

Separating isomorphisms can be represented by continuous functions as weighted composition operators.

Equivalence of subgroups is characterized by the existence of a biseparating isomorphism.

Under mild conditions, all separating isomorphisms have a specific functional form.

Abstract

Let $G$ be a discrete group and let $\mathcal A$ and $\mathcal B$ be two subgroups of $G$-valued continuous functions defined on two $0$-dimensional compact spaces $X$ and $Y$. A group isomorphism $H$ defined between $\mathcal A$ and $\mathcal B$ is called \textit{separating} when for each pair of maps $f,g\in \mathcal A$ satisfying that $f^{-1}(e_G)\cup g^{-1}(e_G)=X$, it holds that $Hf^{-1}(e_G)\cup Hg^{-1}(e_G)=Y$. We prove that under some mild conditions every separating isomorphism $H:\mathcal A\longrightarrow \mathcal B$ can be represented by means of a continuous function $h: Y\longrightarrow X$ as a weighted composition operator. As a consequence we establish the equivalence of two subgroups of continuous functions if there is a biseparating isomorphism defined between them.