2013/05/20 by Hugo J. Bello, Bello, Hugo J., María Jesús Chasco +4
Mathematics · #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topology and Set Theory #FOS: Mathematics #General Topology (math.GN) #math.GN
paper · pdf · doi:10.48550/arxiv.1305.4515
arxiv created 2013/05/20 · openalex publication_date 2013/05/20 · arxiv updated 2013/05/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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. \par We study the class \ST of topological groups G for which every twisted sum 0→ \T→ X → G→ 0 splits. We prove that this class contains locally precompact groups, sequential direct limits of locally compact groups and topological groups with L_∞ topologies. We also prove that it is closed by taking open and dense subgroups, quotients by dually embedded subgroups and coproducts. As a technique to find further examples of groups in \ST we use the relation of this class with the existence of quasi-characters on G and with 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.