Representation of group isomorphisms. The compact case

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 $H{f}^{-1}(e_G)\cup H{g}^{-1}(e_G)=Y$. We prove that under some mild conditions every separating isomorphism $H:\mathcal{A}⟶\mathcal{B}$ can be represented by means of a continuous function $h:Y⟶X$ 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.