Extending topological abelian groups by the unit circle. (English) Zbl 1295.22009

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

Abstr. Appl. Anal. 2013, Article ID 590159, 9 p. (2013).

A twisted sum in the category of topological abelian groups is a short 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 is said to split if it is equivalent to \(0\to Y\to Y\times Z\to Z\to 0\). In the paper under review, the authors study the class \(S_{TG}(\mathbb{T})\) of topological abelian groups \(G\) for which every twisted sum \(0\to \mathbb{T}\to X\to G\to 0\) splits, where \(\mathbb{T}\) denotes the unit circle of the complex plane \(\mathbb{C}\). In the first part of the paper, the authors prove the following:

(1)

The class \(S_{TG}(\mathbb{T})\) is closed under forming open and dense subgroups (Corollaries 14 and 15), quotient by dually embedded subgroups (Theorem 21), and coproducts (Proposition 24).

(2)

The class \(S_{TG}(\mathbb{T})\) contains locally precompact Hausdorff abelian groups (Theorem 18).

Further, the authors show that the direct limit of a countable direct system of nuclear abelian groups in \(S_{TG}(\mathbb{T})\) is in \(S_{TG}(\mathbb{T})\) (Theorem 25). In particular, since every locally compact abelian group is nuclear, a sequential direct limit of locally compact abelian groups is in \(S_{TG}(\mathbb{T})\).
A topological group is called an \(\mathcal{L}_\infty\) group if its topology is the intersection of a decreasing sequence of Hausdorff locally compact group topologies. By combining the above results with a structure theorem of \(\mathcal{L}_\infty\) groups, which asserts that every abelian \(\mathcal{L}_\infty\) group has an open subgroup which is a strict inductive limit of a sequence of Hausdorff locally compact abelian groups [L. J. Sulley, J. Lond. Math. Soc., II. Ser. 5, 629–637 (1972; Zbl 0243.22002)], the authors prove (Corollary 27): every abelian \(\mathcal{L}_\infty\) group is in \(S_{TG}(\mathbb{T})\). At the end of the paper, the authors use the notion of approximable quasi-homomorphisms to obtain new examples of groups in the class \(S_{TG}(\mathbb{T})\).

Reviewer: Hatem Hamrouni (Sfax)

MSC:

22B05	General properties and structure of LCA groups
22A05	Structure of general topological groups

Keywords:

topological group; twisted sum; split twisted sum; character; quasi-homomorphism; approximable quasi-homomorphism; dual group; dually embedded subgroup; \(\mathcal{L}_\infty\) group; three space property

Citations:

Zbl 0243.22002

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