Representation of Group Isomorphisms I

Let G be a metric group and let A u t ( 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 → A u t ( G ) satisfying H f ( 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 ( e G ) ∪ g − 1 ( e G ) = X , it holds that ( H f ) − 1 ( e G ) ∪ ( H g ) − 1 ( e G ) = 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.