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

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.

MSC:

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

Citations:

Zbl 0030.24102
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