The group Z(N) does not have a Mackey topology

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