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�, 𝑔� ∈ A satisfying that 𝑓�−1(𝑒� 𝐺� ) ∪ 𝑔�−1(𝑒� 𝐺� ) = 𝑋�, it holds that 𝐻�𝑓�−1(𝑒� 𝐺� ) ∪ 𝐻�𝑔�−1(𝑒� 𝐺� ) = 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.