Pure injective and $\ast$-pure injective LCA groups

Abstract

A proper short exact sequence $0\to A\to B\to C\to 0$ in the category $\mathcal L$ of locally compact abelian (LCA) groups is called $\ast$-pure if the induced sequence $0\to A\[n]\to B\[n]\to C\[n]\to 0$ is proper exact for all positive integers $n$. An LCA group is called $\ast$-pure injective in $\mathcal L$ if it has the injective property relative to all $\ast$-pure sequences in $\mathcal L$. In this paper, we give a complete description of the $\ast$-pure injectives in $\mathcal L$. They coincide with the injectives in $\mathcal L$ and therefore with the pure injectives in $\mathcal L$. Dually, we determine the topologically pure projectives in $\mathcal L$.