Abstract and Applied Analysis

Extending Topological Abelian Groups by the Unit Circle

Hugo J. Bello, María Jesús Chasco, and Xabier Domínguez

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Abstract

A twisted sum in the category of topological Abelian groups is a short exact sequence 0 Y X Z 0 where all maps are assumed to be continuous and open onto their images. The twisted sum splits if it is equivalent to 0 Y Y × Z Z 0 . We study the class S T G 𝕋 of topological groups G for which every twisted sum 0 𝕋 X G 0 splits. We prove that this class contains Hausdorff locally precompact groups, sequential direct limits of locally compact groups, and topological groups with 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 T G 𝕋 , we use the connection between extensions of the form 0 𝕋 X G 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 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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