The group [formula omitted] does not have a Mackey topology

Abstract

We prove that for a strictly increasing sequence (mn) of natural numbers and d∈N,d≥2 the subgroup G=∑n∈NZ(dmn) endowed with the topology induced by the product ∏n∈NZ(dmn) has no Mackey topology. In other words the supremum of all locally quasi–convex group topologies on G having as character group ∑n∈NZ(dmn)∧ has a strictly larger character group. As a consequence we obtain that also Z(N) with the topology induced by the product ZN has no Mackey topology.