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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.
22A05 Structure of general topological groups
22C05 Compact groups
Full Text: DOI arXiv
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