Representation of group isomorphisms: The compact case

Abstract

Let G be a discrete group and let A and 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 A and B is called separating when, for each pair of maps f, gA satisfying that f-1eG∪g-1eG=X, it holds that Hf-1eG∪Hg-1eG=Y. We prove that under some mild conditions every biseparating isomorphism H:A→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.