It is well known that every locally compact abelian group L can be decomposed as L_1 \oplus R^n, where L_1 contains a compact-open subgroup. In this paper, we use this decomposition to study the topological group Aut(L) of automorphisms of L, equipped with the g-topology. We show that Aut(L) is topologically isomorphic to a matrix group with entries from Aut(L_1), Hom(L_1, R^n), Hom(R^n, L_1), and GL_n(R), respectively. It is also shown that the algebraic portion of the decomposition is not specific to locally compact abelian groups, but is also true for objects with a well-behaved decomposition in an additive category with kernels.
Given a collection of mathematical objects with a notion of isomorphism, it is often of interest to study the self-isomorphisms, or automorphisms, of those objects. In particular, the set of all such automorphisms is a group under composition, and there is an interplay between the structure of this group of automorphisms and the underlying object. Classical examples include permutation groups of sets, which encompasses the whole of group theory, automorphism groups of fields in the context of Galois theory, and groups of diffeomorphisms of smooth manifolds. In the setting of topological spaces, where automorphisms are self-homeomorphisms of a space X, it is natural to consider endowing this automorphism group Homeo(X) with a topology related to that of X. If X is locally compact, then Homeo(X) and its subgroups can be made into topological groups, via the so-called g-topology, or Birkhoff topology, generated by the subbasis consisting of sets of the form (C, U ) = {f ∈ Homeo(X) : f (C) ⊆ U and f -1 (C) ⊆ U }, where C is a compact subset of X, and U an open subset of X [1]. This is the coarsest refinement of the compact-open topology wherein both composition and inversion are continuous.
When L is a Hausdorff locally compact group, denote by Aut(L) the group of topological automorphisms of L, a closed subgroup of Homeo(L), endowed with the g-topology. In general, Aut(L) is not locally compact, even in the case where L is a compact abelian group [9, 26.18 (k)], which has led many to study conditions under which local compactness holds. For example, if L is compact, totally disconnected, and nilpotent, then local compactness, and in fact, compactness, of Aut(L) are equivalent to all Sylow subgroups having finitely many topological generators [16]. Recent work of Caprace and Monod has shown that if L is totally disconnected, compactly generated and locally finitely generated, then Aut(L) is locally compact [5, I.6]. It is also known that Aut(L) is a Lie group provided L is connected and finite dimensional [12]. It has been shown that automorphism groups of compact abelian groups are universal for the class of non-archimedean groups in the sense that every non-archimedean group embeds as a topological subgroup of Aut(K), for some compact abelian K; see [15] and [14].
In the case where L is a locally compact abelian (LCA) group, Levin [10] gave criterion for local compactness of Aut(L), provided L contained a discrete subgroup such that the quotient was compact. Levin’s analysis utilizes the additional structure of LCA groups afforded to us by their duality theory, and in particular, the following canonical decomposition of such groups. The main result of this paper is a structural decomposition of the automorphism group of an LCA group, using the decomposition in Theorem 1.1:
Then, as topological groups,
where the latter is equipped with the product topology.
The algebraic portion of Theorem A can be extracted and established in a more general setting.
Theorem B. Let C be an additive category with kernels, and A = B⊕C an object in C such that: (I) δ ∈ End(C) is an automorphism of C if and only if the zero morphism 0 is a kernel of δ; and (II) For every pair of morphisms γ : B → C and β : C → B, one has that γβ = 0. Then, as groups,
The paper is structured as follows: In §2, we provide topological preliminaries regarding the compact-open and g-topologies. §3 is a discussion of an abstract categoral setting wherein we prove Theorem B. In §4, we present the proof of Theorem A.
Throughout this paper, all spaces are assumed to be Hausdorff, and in particular, all topological groups are Tychonoff [11, 1.21]. Recall that if X and Y are topological spaces and F a collection of continuous functions from X to Y , the compact-open topology on F is the topology generated by the subbasis consisting of sets of the form
where C is a compact subset of X, and U an open subset of Y (see [7], [18, §43]). For locally compact X, composition of maps is continuous in Homeo(X) when endowed with the compact-open topology, a consequence of the following property: However, inversion may fail to be continuous in Homeo(X) with respect to the compact-open topology [4, p. 57-58]; this shortcoming is remedied by the g-topology. The two topologies coincide when X is compact, discrete, or locally connected, but not in general [1]. One can characterize convergence in the g-topology in terms of the compact-open topology as in the following proposition. –→ f -1 .
Given an LCA group L, Aut(L) is a closed subgroup of Homeo(L), endowed with the g-topology. Theorem 1.1 implies a decomposition of End(L), the (additive) group of topological endomorphisms of L, endowed with the compact-open topology, into a topological ring of 2×2 matrices. In particular, every element of Aut(L) can be algebraically represented in this way, but we caution that since Aut(L) carries the g-topology, it is not a subspace of End(L). We note for future reference that
This content is AI-processed based on open access ArXiv data.
