A note on radical and radical free topological groups. (English) Zbl 0744.22002

Rangan, G.

Publ. Math. Debr. 38, No. 3-4, 273-277 (1991).

The notion of the radical of a topological Abelian group can be found in N. Bourbaki [Eléments de mathématique, Livre III, Topologie générale (1947; Zbl 0030.24102), §2, Ex. 28]. A real character of a topological Abelian group is a continuous homomorphism from it into the group of all real numbers. It is shown that a connected topological Abelian group has sufficiently many real characters if and only if it is radical-free, i.e. its radical is equal to zero. For a locally compact Abelian group \(G\) it is proved that \(G\) is connected (totally disconnected) if and only if its dual group \(G^*\) is radical-free (radical). It is proved that the radical of a locally compact connected group is a topological direct summand of it.

Reviewer: M.I.Ursul (Kishinev)

MSC:

22A05	Structure of general topological groups
22A25	Representations of general topological groups and semigroups
22B05	General properties and structure of LCA groups

Keywords:

topological Abelian group; connected topological Abelian group; real characters; locally compact Abelian group; radical-free; radical; topological direct summand

Citations:

Zbl 0030.24102