Some new results on the Chu duality of discrete groups

This paper deals mainly with the Chu duality of discrete groups. Among other results, we give sufficient conditions for an $FC$ group to satisfy Chu duality and characterize when the Chu quasi-dual and the Takahashi quasi-dual of a group $G$ coincide. As a consequence, it follows that when $G$ is a weak sum of a family of finite simple groups, if the exponent of the groups in the family is bounded then $G$ satisfies Chu duality; on the other hand, if the exponent of the groups goes to infinite then the Chu quasi-dual of $G$ coincide with its Takahashi quasi-dual. We also present examples of discrete groups whose Chu quasi-duals are not locally compact and examples of discrete Chu reflexive groups which contain non-trivial sequences converging in the Bohr topology of the groups. Our results systematize some previous work and answer some open questions in the subject.