A class of metrizable locally quasi-convex groups which are not Mackey

A topological group $(G,\mu)$ from a class $\mathcal G$ of MAP topological abelian groups will be called a {\it Mackey group} in $\mathcal G$ if it has the following property: if $\nu$ is a group topology in $G$ such that $(G,\nu)\in \mathcal G$ and $(G,\nu)$ has the same continuous characters, say $(G,\nu)^{\wedge}=(G,\mu)^{\wedge}$, then $\nu\le \mu$. If $\rm{LCS}$ is the class of Hausdorff topological abelian groups which admit a structure of a locally convex topological vector space over $\mathbb R$, it is well-known that every metrizable $(G,\mu) \in \rm{LCS}$ is a Mackey group in $\rm{LCS}$. For the class $\rm{LQC}$ of locally quasi-convex Hausdorff topological abelian groups, it was proved in 1999 that every {\bf complete} metrizable $(G,\mu)\in \rm{LQC}$ is a Mackey group in $\rm{LQC}$ (\cite{CMPT}). The completeness cannot be \NB dropped within the class $\rm{LQC}$ as we prove in this paper. In fact, we provide a large family of metrizable precompact \NB(noncompact) groups which {\bf are not} Mackey groups in \rm{LQC} (Theorem \ref{basth}). Those examples are constructed from groups of the form $c_0(X)$, whose elements are the null sequences of a topological abelian group $X$, and whose topology is the uniform topology. We first show that for a compact metrizable group $X\ne\{0\}$ the topological group $c_0(X)$ is a non-compact complete metrizable locally quasi-convex group, which has {\bf countable} topological dual iff $X$ is connected. Then we prove that for a connected compact metrizable group $X\ne\{0\}$ the group $c_0(X)$ endowed with the product topology induced from the product $X^{\N}$ is metrizable precompact but not a Mackey group in LQC.