# Kurosh radical theory for topological groups?

Does anyone know if there has been much work done on radical and semisimple classes in the sense of Kurosh within the category of topological groups (or subcategories thereof)? For instance, for a radical class $\mathcal{R}$ in the usual sense, the union of an ascending chain of normal $\mathcal{R}$-subgroups of $G$ is also an $\mathcal{R}$-group; but in the topological context, we really want to know about the closure of the union, which may not be in $\mathcal{R}$ any more.

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I don't have Kurosh at hand: by radical, do you mean an idempotent subfunctor of the identity, $r : T \rightarrow T$, satisfying $r(T/r(T)) = 0$?

My undergrad thesis 'Radicals and torsion theories in locally compact abelian (LCA) groups' (Amherst, 1972+1/2) studied these and related notions in the category of LCA groups. There are two dual constructions, $\cap ker(f)$ for $f \in Hom(-,G)$, G in some class C, or $\overline{\Sigma im(f)}$, $f \in Hom(G,-)$, G in some class C, and every idempotent radical on LCA groups can be written as either. The classes C give rise to torsion theories, pairs $(T,F)$ of classes of groups, maximal w.r.t. the fact that there are no nontrivial homs from groups in $T$ to groups in $F$. There is a 1-1 correspondence between torsion theories and idempotent radicals.

Pontrjagin duality makes this an especially nice case. My thesis looks at the general theory and then considers 18 different pre-radicals and pre-coradicals (intersections and sums as above, not necessarily idempotent radicals) and studies them in detail.

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Yes, this is the kind of thing I mean. I found a survey talk by B.J.Gardner (here: aragorn.pb.bialystok.pl/~piotrgr/BanachCenter/meeting.html), but it made no mention of topology. Is your undergrad thesis online anywhere? –  Colin Reid Sep 23 '10 at 13:54
I have scanned my thesis and placed it at math.wayne.edu/~rrb/papers/UG_thesis.pdf FWIW, Bob –  Robert Bruner Sep 27 '10 at 16:22