Decompositions of the automorphism group of a locally compact abelian group

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.

💡 Research Summary

The paper investigates the structure of the automorphism group Aut(L) of a locally compact abelian (LCA) group L when Aut(L) is equipped with the so‑called g‑topology. The starting point is the classical decomposition theorem for LCA groups: every such group can be written as a topological direct sum
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