On the continuity of the evaluation mapping associated with a group and its character group

Abstract:

For an abelian Hausdorff group $G$, let $G^{\ast }$ denote the character group endowed with the compact-open topology and let $\alpha _{G}:G\rightarrow G^{\ast \ast }$ denote the canonical homomorphism. We show that the evaluation mapping from $G^{\ast }\times G$ into the torus is continuous if and only if $G^{\ast }$ is locally compact and $\alpha _{G}$ is continuous. If $\alpha _{G}$ is injective and open, then the evaluation mapping is continuous if and only if $G$ is locally compact. Several examples and counterexamples are given.