On pure subgroups of locally compact abelian groups

Abstract.

In this note, we construct an example of a locally compact abelian group G = C × D (where C is a compact group and D is a discrete group) and a closed pure subgroup of G having nonpure annihilator in the Pontrjagin dual $\hat{G}$, answering a question raised by Hartman and Hulanicki. A simple proof of the following result is given: Suppose ${\frak K}$ is a class of locally compact abelian groups such that $G \in {\frak K}$ implies that $\hat{G} \in {\frak K}$ and nG is closed in G for each positive integer n. If H is a closed subgroup of a group $G \in {\frak K}$, then H is topologically pure in G exactly if the annihilator of H is topologically pure in $\hat{G}$. This result extends a theorem of Hartman and Hulanicki.

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Correspondence to P. Loth.

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Received: 4 April 2002

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Loth, P. On pure subgroups of locally compact abelian groups. Arch. Math. 81, 255–257 (2003). https://doi.org/10.1007/s00013-003-0823-z

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Mathematics Subject Classification (1991):

  • Primary 20K27, 22B05
  • Secondary 20K45, 22D35