Linear topologies on \(\mathbb Z\) are not Mackey topologies (Q456821)

The Mackey topology is an important topic in the theory of topological vector spaces. The analogue of the Mackey topology for locally quasi-convex (LQC) abelian groups was introduced by \textit{M. J. Chasco}, \textit{E. Martín-Peinador} and \textit{V. Tarieladze} [Stud. Math. 132, No. 3, 257--284 (1999; Zbl 0930.46006)], where they proved in particular that the topology of a complete metrizable LQC abelian group is the Mackey topology. Recall that a LCQ topological group \((G,\tau)\) is called Mackey or LCQ-Mackey if it has the following property: if \(\nu\) is a group LCQ topology in \(G\) such that \((G,\nu)\) has the same character group as \((G,\tau)\), then \(\nu\leq\tau\). It is not known whether every locally quasi-convex non-complete metrizable group admits the Mackey topology. Moreover, even the following natural question posed by D. Dikranjan was open. Are the \(p\)-adic topologies on \(\mathbb{Z}\) Mackey topologies? The authors prove that, for every linear non-discrete topology \(\tau\) on the integers, there exists a strictly finer metrizable LCQ group topology \(\nu\) on \(\mathbb{Z}\) such that \((\mathbb{Z}, \tau)\) and \((\mathbb{Z}, \nu)\) have the same dual group. As a corollary of this result, they give a negative answer to Dikranjan's question.