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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.
##### MSC:
 22A05 Structure of general topological groups 22C05 Compact groups
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