Representation of group isomorphisms I

Abstract

Let G be a metric group and let Aut(G) denote the automorphism group of G. If A and B are groups of G-valued maps defined on the sets X and Y, respectively, we say that A and B are equivalent if there is a group isomorphism H:A→B such that there is a bijective map h:Y→X and a map w:Y→Aut(G) satisfying Hf(y)=w[y](f(h(y))) for all y∈Y and f∈A. In this case, we say that H is represented as a weighted composition operator. 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(eG)∪g−1(eG)=X, it holds that (Hf)−1(eG)∪(Hg)−1(eG)=Y. Our main result establishes that under some mild conditions, every separating group isomorphism can be represented as a weighted composition operator. As a consequence we establish the equivalence of two function groups if there is a biseparating isomorphism defined between them.