On pure subgroups of locally compact abelian groups

In this note, we construct an example of a locally compact abelian group G = C x 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 (G) over cap, answering a question raised by Hartman and Hulanicki. A simple proof of the following result is given: Suppose A is a class of locally compact abelian groups such that G is an element of k implies that (G) over cap is an element of k and nG is closed in G for each positive integer n. If H is a closed subgroup of a group G is an element of k, then H is topologically pure in G exactly if the annihilator of H is topologically pure in (G) over cap. This result extends a theorem of Hartman and Hulanicki.