×

zbMATH — the first resource for mathematics

One-parameter subgroups of topological abelian groups. (English) Zbl 1315.22003
Let \(G\) be an abelian topological group and denote by \(\alpha_G:G\longrightarrow G^{\wedge\wedge},\;x\longmapsto (\chi\mapsto \chi(x))\) the canonical mapping in the bidual group \(G^{\wedge\wedge}\). Further, let \(G^{\wedge\wedge}_{\text{lift}}=\{\exp(2\pi i f):\;f:G^\wedge\to \mathbb R \;\) a continuous homomorphism\(\}\) be the subgroup of all characters of \(G^\wedge\) which can be lifted over the reals. The main result of the paper is the following: If \(G\) is a Hausdorff locally quasi-convex abelian group such that \(\alpha_G\) is surjective, then \[ \text{im}\exp_G=G_a=\alpha_G^{-1}(G^{\wedge\wedge}_{\text{lift}}); \] i.e., the arc component \(G_a\) of \(G\) is the image of the exponential function \(\exp_G:{\mathcal L}(G)\to G,\;\lambda\mapsto \lambda(1)\) where \({\mathcal L}(G) \) is the Lie algebra of \(G\), that means the set of continuous functions \(\mathbb R\to G\). Further, the author shows that if \(G\) is a Polish reflexive group and \(G\) is arcwise or locally arcwise connected, then the exponential mapping is a quotient mapping.
MSC:
22A05 Structure of general topological groups
22C05 Compact groups
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Außenhofer, L.; Chasco, M. J.; Domínguez, X., Arcs in the Pontryagin dual of a topological abelian group, J. Math. Anal. Appl., 425, 1, 337-348, (2015) · Zbl 1332.22010
[2] Ardanza-Trevijano, S.; Chasco, M. J.; Domínguez, X., The role of real characters in the Pontryagin duality of topological abelian groups, J. Lie Theory, 18, 1, 193-203, (2008) · Zbl 1151.22003
[3] Außenhofer, L., Contributions to the duality theory of abelian topological groups and to the theory of nuclear groups, Diss. Math., 384, (1999) · Zbl 0953.22001
[4] Außenhofer, L., A survey on nuclear groups, (Nuclear Groups and Lie Groups, Madrid, 1999, Res. Exp. Math., vol. 24, (2001), Heldermann Lemgo), 1-30 · Zbl 1020.22001
[5] Außenhofer, L., The Lie algebra of a nuclear group, J. Lie Theory, 13, 1, 263-270, (2003) · Zbl 1014.22006
[6] Banaszczyk, W., Additive subgroups of topological vector spaces, Lecture Notes in Mathematics, vol. 1466, (1991), Springer Berlin · Zbl 0743.46002
[7] Chasco, M. J., Pontryagin duality for metrizable groups, Arch. Math. (Basel), 70, 1, 22-28, (1998) · Zbl 0899.22001
[8] Chasco, M. J.; Dikranjan, D.; Martín-Peinador, E., A survey on reflexivity of abelian topological groups, Topol. Appl., 159, 9, 2290-2309, (2012) · Zbl 1247.22001
[9] Chasco, M. J.; Martín-Peinador, E.; Tarieladze, V., A class of angelic sequential non-Fréchet-Urysohn topological groups, Topol. Appl., 154, 3, 741-748, (2007) · Zbl 1106.22002
[10] Gleason, A. M., Arcs in locally compact groups, Proc. Natl. Acad. Sci. USA, 36, 663-667, (1950) · Zbl 0040.15301
[11] Hewitt, E.; Ross, K. A., Abstract harmonic analysis. vol. I, Grundlehren Math. Wiss., vol. 115, (1979), Springer Berlin · Zbl 0115.10603
[12] Hofmann, K. H.; Morris, S. A., The structure of compact groups, De Gruyter Stud. Math., vol. 25, (2013), de Gruyter Berlin
[13] Hofmann, K. H.; Morris, S. A.; Poguntke, D., The exponential function of locally connected compact abelian groups, Forum Math., 16, 1, 1-16, (2004) · Zbl 1041.22005
[14] Kąkol, J.; López-Pellicer, M.; Martín-Peinador, E.; Tarieladze, V., Lindelöf spaces \(C(X)\) over topological groups, Forum Math., 20, 2, 201-212, (2008) · Zbl 1171.22001
[15] Nickolas, P., Reflexivity of topological groups, Proc. Am. Math. Soc., 65, 1, 137-141, (1977) · Zbl 0369.22002
[16] Rickert, N. W., Arcs in locally compact groups, Math. Ann., 172, 222-228, (1967) · Zbl 0239.22009
[17] Warner, S., The topology of compact convergence on continuous function spaces, Duke Math. J., 25, 265-282, (1958) · Zbl 0081.32802
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.