Representation of group isomorphisms. The compact case

Let $G$ be a discrete group and let $\mathcal A$ and $\mathcal B$ be two subgroups of $G$-valued continuous functions defined on two $0$-dimensional compact spaces $X$ and $Y$. A group isomorphism $H$ defined between $\mathcal A$ and $\mathcal B$ is called \textit{separating} when for each pair of maps $f,g\in \mathcal A$ satisfying that $f^{-1}(e_G)\cup g^{-1}(e_G)=X$, it holds that $Hf^{-1}(e_G)\cup Hg^{-1}(e_G)=Y$. We prove that under some mild conditions every separating isomorphism $H:\mathcal A\longrightarrow \mathcal B$ can be represented by means of a continuous function $h: Y\longrightarrow X$ 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.


Introduction
Let G be a discrete group and let X and Y be topological spaces. If A and B are groups of G-valued continuous maps, we say that A and B are equivalent when there is a homeomorphism h : Y → X and a continuous map ω : Y → Aut(G) satisfying that Hf (y) = ω[y](f (h(y))) for all y ∈ Y , where Aut(G) is equipped with the pointwise convergence topology. We say in this case that H is represented as a weighted composition operator. There are many results that are concerned with the representation of linear operators as weighted composition maps and the equivalence of specific groups of continuous functions in the literature, which is vast in this regard. We will only mention here the classic Banach-Stone Theorem that, when G is the field of real or complex numbers, establishes that if the Banach spaces of continuous functions C(X, G) and C(Y, G) are isometric, then they are equivalent and the isometry can be represented as a weighted composition map (cf. [1,2,8,9,11,13,14,15,17,26]). Another important example appears in coding theory, where the well known MacWilliams Equivalence Theorem asserts that, when G is a finite field and X and Y are finite sets, two codes (linear subspaces) A and B of G X and G Y , respectively, are equivalent when they are isometric for the Hamming metric (see [18,19,4]). This result has been generalized to convolutional codes in [12] and it also makes sense in other areas, as for example functional analysis and linear dynamical systems (cf [7,10,12,24,27]). The main motivation of this research has been to extend MacWilliams Equivalence Theorem to more general settings and explore the possible application of these methods to the study of convolutional codes or linear dynamical systems. However, throughout this paper, we will only deal with 0-dimensional compact spaces X and Y , and a discrete group G. We will look at the possible application of this abstract approach elsewhere.
There are many precedents in the study the representation of group homomorphisms for group-valued continuous functions. Among them, the following ones are relevant here (cf [3,6,16,20,21,23,28,29]). Most basic facts and notions related to topological properties may be found in [5].
Throughout this paper all spaces are assumed to be Hausdorff 0-dimensional and compact. If X is a topological space and G is a topological (discrete) group, we denote by C(X, G) the group of continuous functions from X to G. Let e G be the neutral and the zero of f is the set Z(f ) = X \ coz(f ). Since G is discrete coz(f ) and Z(f ) are both closed and open (clopen) subsets of X.
) and a function g ∈ A such that We now formulate our main results. (1) h is a homeomorphism of Y onto X.
(3) H is an isomorphism with respect to the pointwise convergence topology. We notice that some of the requirements we have imposed on the previous results could be relaxed in general. However, this would take us to a wider setting in general.
For instance, if we assume that A does not separate points in X, then there must be some point x ∈ X such that f (x) = e G for all f ∈ A, then we can replace X by the largest subspace X A ⊆ X where A separates points. This subspace X A is open but not necessarily closed in general. Thus, the study of subgroups that does not separate points lead us to consider locally compact spaces. We shall discuss these spaces in a subsequent paper.

Basic notions and facts
The following lemma is easily verified using a standard compactness argument. Recall that we are assuming that all spaces are compact and 0-dimensional. Next proposition shows that the notions of separating and strongly separating points are equivalent for controllable subgroups. Proposition 2.3. If A is a controllable subgroup of C(X, G) that separates the points of X, then A strongly separates the points of X.
Proof. Set D = Z(A) and take two distinct elements Applying that D is a subbase of closed subsets, and using a compactness argument, . Therefore x 2 ∈ coz(g 2 ) ⊆ U 2 , which yields coz(g 1 ) ∩ coz(g 2 ) = ∅. This completes the proof.
Definition 2.4. Let A be a subgroup of C(X, G) and let ϕ : A → G a group homo- Some basic properties of support subsets are shown in the next proposition. Observe that, since A ⊆ A X ⊆ Z(f ), we may assume WLOG that all support subsets are closed and therefore compact subsets of X.
Proposition 2.5. Let ϕ : A → G a non null group homomorphism. The following assertions hold : (1) X is a support for ϕ.
(2) If A is a support for ϕ then A = ∅.
(3) If A is a support for ϕ and A ⊆ B then B is a support for ϕ.
(4) Let A be a support for ϕ and f, g ∈ A such that f | A = g| A . Then ϕ(f ) = ϕ(g).
If, in addition, A is controllable and separates points in X, then we have: Proof. Assertions (1) − (4) are obvious.
Applying the controllability of A, we obtain U ∈ σ(coz(A)) and g ∈ A such that yields a contradiction as the evaluation of ϕ(g) shows. Indeed, since g(x) = f (x) for all x ∈ A, by item (2) it follows that ϕ(g) = ϕ(f ) = e G . On the other hand, we have that g(x) = e G for all x ∈ B, which imples ϕ(g) = e G . This contradiction completes the proof.
if for each pair of maps f, g ∈ A satisfying that f −1 (e G ) ∪ g −1 (e G ) = X, it holds that . In case H is bijective, the map H is said to be biseparating if both H and H −1 are separating. Remark that this definition makes sense and extends naturally to maps ϕ : A → G.
Next we will see that every non null separating group homomorphism ϕ : A → G, where A is controllable, has the smallest possible compact support set. For that purpose, set There is a canonical partial order that can be defined on S Proof. Let S be a minimal element of S, which is nonempty by Proposition 2.5. Suppose now that there are two different elements x 1 , x 2 that are contained in S. As X is Hausdorff, we can select two disjoint open subsets V 1 , V 2 in X such that x 1 ∈ V 1 and x 2 ∈ V 2 . Since S is minimal, the compact subset S \ V i is not a support for ϕ. Hence, Since ϕ is separating, it follows that A = coz(f 1 ) ∩ coz(f 2 ) is a nonempty compact subset of X.
We claim that S ∩ A = ∅. Otherwise, pick up an element a ∈ S ∩ A. If a ∈ V 1 then a ∈ S \ V 2 and a ∈ Z(f 2 ), which is a contradiction; but if a / ∈ V 1 then a ∈ S \ V 1 , which implies that a ∈ Z(f 1 ) and we get a contradiction again. Therefore S ∩ A = ∅.
By Lemma 2.1, we can take two disjoint sets D S , D A ∈ σ(Z(A)) such that S ⊆ D S and A ⊆ D A . Applying that A is controllable to D S , D A and f 1 , we obtain a set U ∈ σ(coz(A)) and a map g ∈ A such that S ⊆ D S ⊆ U ⊆ X \ D A ⊆ X \ A, g| S = f 1 | S and g| Z(f 1 )∪(X\U ) ≡ e G . Then U ∩ A = ∅, ϕ(g) = ϕ(f 1 ) = e G and A ⊆ Z(g). Since ϕ is separating the set B = coz(g) ∩ coz(f 2 ) = ∅. Take b ∈ B. Then b ∈ coz(f 2 ) and b ∈ coz(g) ⊆ coz(f 1 ), that is, b ∈ coz(f 1 ) ∩ coz(f 2 ) = A. As a consequence g(b) = e G , which is a contradiction. Therefore we have proved that |S| = 1. This completes the proof.

Proof of main results
Along this section A (resp. B) is a controllable subgroup of C(X, G) (resp. C(Y, G)) that separates points in X (resp. Y ).
Let H : A → B be a separating group homomorphism. The maps δ y • H are a separating group homomorphisms of A into G for all y ∈ Y . Furthermore, since A is controllable and separates points in X, we can apply Proposition 2.7, in order to obtain that each partial ordered set S y = {A ⊆ X : S is a compact support for δ y • H} has a minimum element, which is a singleton denoted by h(y). Therefore, by sending y ∈ Y to h(y) ∈ X for every y ∈ Y , we have defined the support map of Y into X that is associated to H. 1) h is continuous.

4) If H is one-to-one, then h is onto.
Moreover, when H is a bijection of A onto B, we have in addition:

5)
If H is biseparating, then h is a homeomorphism of Y onto X.
Proof. 1) Let (y d ) d∈D be a net in Y converging to y ∈ Y . By a standard compactness argument, we may assume WLOG that (h(y d )) d converges to x ∈ X. Reasoning by contradiction, suppose h(y) = x. Since X is Hausdorff, we can take two disjoint open neighborhoods V h(y) and V x of h(y) and x, respectively. Using convergence, there is As every support subset for δ y ′ • H contains h(y ′ ), for all y ′ ∈ Y , the subset X \ V h(y) may not be a support for δ y •H. Therefore there exists f ∈ A such that X \V h(y) ⊆ Z(f ) and and Hf 3 (y d 3 ) = e G . This means that y d 3 ∈ coz(Hf 3 ) ∩ coz(Hf ) and, since H is a separating map, it follows that coz( . This is a contradiction that completes the proof. 2) Let ∅ = A X be an open subset, f ∈ A and A ⊆ Z(f ). If we take y ∈ h −1 (A), then X \ A is a nonempty compact subset that is not a support for δ y • H. Then there is g ∈ A such that X \ A ⊆ Z(g) and Hg(y) = e G . Therefore coz(g) ⊆ A and coz(f ) ⊆ X \A. Since H is separating, we have that coz(Hg)∩coz(Hf ) = ∅. Therefore Hf (y) = e G .

4) Suppose h(Y ) = X and take
x ∈ X such that x / ∈ h(Y ). Since h is continuous and Y is compact, we have that h(Y ) is a proper compact subset of X. Applying Lemma 2.1, Moreover, as A separates points in X, there exists f ∈ A such that f (x) = e G . Again, by the controllability of A, we may take a subset U ⊆ σ(coz(A)) and a map g ∈ A such and H(f )(y) = e G for all y ∈ Y . Then Hf ≡ e G . Since H is an injective group homomorphism, this yields f ≡ e G , which is a contradiction. 5) Since X and Y are compact spaces, it will suffice to prove that h is one-to-one.
Suppose there are two elements Now, by the controllability of B, there are U 2 ∈ σ(coz(B)) and g 2 ∈ A such that Hence, since coz(Hg i ) ⊆ U i , U 1 ∩U 2 = ∅, and H is biseparating, it follows that coz(g 1 )∩coz(g 2 ) = ∅.
We have just seen how a separating group homomorphism H has associated a continuous map h that assigns to each point y ∈ Y the support subset of δ y • H. Our next goal now is to obtain a complete representation of H by means of the support map h.

Having this in mind, set
which is a subgroup of G for all y ∈ Y , and denote by Hom(G h(y) , G y ) the set of all group homomorphisms on G h(y) into G y . Consider now the set We can think of the elements of G as partial functions on G. That is, functions α : Dom(α) ⊆ G −→ G whose domain is a (not necessarily proper) subset of G. Since the group G is discrete, we can equip G with the product (or pointwise convergence) topology as follows: neighborhood of a map α ∈ G G . If now α is a partial map, we can restrict this basic neighborhood to G by letting [α; g 1 , . . . , g n ] be the set of all partial maps β : Dom(β) ⊆ G −→ G such that g 1 , . . . , g n ∈ Dom(β) and α(g i ) = β(g i ), 1 ≤ i ≤ n. It is easily verified that this procedure extends the pointwise convergence topology on G (cf. [22]).
With this notation, we define ω : Y → G by ω[y](f (h(y))) def = Hf (y) for each y ∈ Y . We shall see next that ω is well defined and continuous. (1) ω[y] is a well defined group homomorphism of G h(y) into G y for all y ∈ Y .
(2) ω is continuous when G is equipped with the pointwise convergence topology.
(2) Let (y d ) d∈D be a net converging to y in Y . If g is an arbitrary element in Thus g ∈ Dom(ω[y d ]) for all d ≥ d 1 (g). In like manner, as ω[y](g) = Hf (y) = g y ∈ G and Hf ∈ C(Y, G), we have that (Hf ) −1 (g y ) is a clopen neighborhood of y. As a consequence there is d 2 ≥ d 1 (g) such that Hf (y d ) = g y for all d ≥ Observe that, since G is discrete, the compact subsets in G are all finite. Therefore, we have also proved that ω is also continuous if we consider the compact open topology on G. We are in position now of establishing a main result in this paper. (1) For each y ∈ Y and every f ∈ A it holds Hf (y) = ω[y](f (h(y))).
(2) H is continuous with respect to the pointwise convergence topology.  (1) For each y ∈ Y and every f ∈ A it holds Hf (y) = ω[y](f (h(y))).
(2) H is continuous with respect to the pointwise convergence topology. We are in now position of establishing the results formulated at the Introduction.
Proof of Theorem 1.1. After Theorem 3.3 and Corollary 3.4, we only need to verify that ω[y] ∈ Aut(G) for all y ∈ Y . Applying Theorem 3.3 to H −1 , we obtain maps ρ : X → End(G) and k : X → Y such that for every x ∈ X and g ∈ B, we have H −1 g(x) = ρ[x](g(k(x))).
Thus, for every f ∈ A and x ∈ X, we have (Hf (k(x))) = ρ[x](ω[k(x)](f (h(k(x))))) which means that the support subset of δ x • (H −1 • H) is both x and h(k(x)). Since the support of a separating map is unique, this means that h • k = id X , which implies that k is a right inverse of h. Analogously, for every g ∈ B and y ∈ Y , we have Applying the former equality to x = h(y), it follows that ρ[h(y)] • ω[y] = id G for all y ∈ Y , and from the latter, we also have that ω[y] • ρ[h(y)] = id G . This means that ω[y] has left and right inverse and, therefore, it is an automorphism on G, which completes the proof.
Proof of Corollary 1.2. It follows directly from Theorem 1.1.