Representation of Group Isomorphisms: The Compact Case

Abstract EN

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 andY.

A group isomorphism H defined between A and B is called separating when,for each pair of maps f, g∈A satisfying that f-1eG∪g-1eG=X, it holds thatHf-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 establishthe equivalence of two subgroups of continuous functions if there is a biseparatingisomorphism defined between them.