On pure subgroups of locally compact abelian groups

In this note, we construct an example of a locally compact abelian groupG = C × D (where C is a compact group and Dis 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 suchthat $G in {frak K}$ 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 ofH is topologically pure in$hat{G}$. This result extends a theorem of Hartman and Hulanicki.Received: 4 April 2002