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 $\mathcal{L}$.