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On the “three-space” problem for locally quasi-convex topological groups. (English) Zbl 0955.22006
The main result of this paper is the analogue for topological groups of a theorem due to S. Dierolf which states that there exists a nonlocally convex twisted sum of the locally convex spaces $X$ and $Y$ if and only if there exists a nonlocally convex twisted sum of $X$ and $\Bbb R$. It reads as follows: Given two abelian locally quasi-convex groups $H$ and $G$ there exists a nonlocally quasi-convex extension of $H$ and $G$ if and only if there exists a nonlocally quasi-convex extension of the circle group $S^1$ and $G$. The proof relies on methods of homological algebra.

22B05General properties and structure of LCA groups
46B20Geometry and structure of normed linear spaces
46B28Spaces of operators; tensor products; approximation properties
46B08Ultraproduct techniques in Banach space theory
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