On Pure Subgroups of Locally Compact Abelian Groups
Document Type
Peer-Reviewed Article
Publication Date
9-2003
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.
DOI
10.1007/s00013-003-0823-z
Recommended Citation
Loth, P. (2003). On pure subgroups of locally compact abelian groups. Archiv der Mathematik 81(3), 255-257. doi: 10.1007/s00013-003-0823-z
