## Abstract and Applied Analysis

### Extending Topological Abelian Groups by the Unit Circle

#### Abstract

A twisted sum in the category of topological Abelian groups is a short exact sequence $0\to Y\to X\to Z\to 0$ where all maps are assumed to be continuous and open onto their images. The twisted sum splits if it is equivalent to $0\to Y\to Y×Z\to Z\to 0$. We study the class ${S}_{\text{T}\text{G}}(\mathrm{\Bbb T})$ of topological groups G for which every twisted sum $0\to \mathrm{\Bbb T}\to X\to G\to 0$ splits. We prove that this class contains Hausdorff locally precompact groups, sequential direct limits of locally compact groups, and topological groups with ${\scr L}_{\infty }$ topologies. We also prove that it is closed by taking open and dense subgroups, quotients by dually embedded subgroups, and coproducts. As means to find further subclasses of ${S}_{\text{T}\text{G}}(\mathrm{\Bbb T})$, we use the connection between extensions of the form $0\to \mathrm{\Bbb T}\to X\to G\to 0$ and quasi-characters on G, as well as three-space problems for topological groups. The subject is inspired on some concepts known in the framework of topological vector spaces such as the notion of $\mathrm{\scr K}$-space, which were interpreted for topological groups by Cabello.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 590159, 9 pages.

Dates
First available in Project Euclid: 26 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393450162

Digital Object Identifier
doi:10.1155/2013/590159

Mathematical Reviews number (MathSciNet)
MR3121509

Zentralblatt MATH identifier
1295.22009

#### Citation

Bello, Hugo J.; Chasco, María Jesús; Domínguez, Xabier. Extending Topological Abelian Groups by the Unit Circle. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 590159, 9 pages. doi:10.1155/2013/590159. https://projecteuclid.org/euclid.aaa/1393450162

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