  The notion of locally quasi-convex abelian group, introduce by Vilenkin, is
extended to maximally almost-periodic non-necessarily abelian groups. For that
purpose, we look at certain bornologies that can be defined on the set
$\hbox{rep}(G)$ of all finite dimensional continuous representations on a
topological group $G$ in order to associate well behaved group topologies (dual
topologies) to them. As a consequence, the lattice of all Hausdorff totally
bounded group topologies on a group $G$ is shown to be isomorphic to the
lattice of certain special subsets of $\hbox{rep}(G_d)$. Moreover, generalizing
some ideas of Namioka, we relate the structural properties of the dual
topological groups to topological properties of the bounded subsets belonging
to the associate bornology. In like manner, certain type of bornologies that
can be defined on a group $G$ allow one to define canonically associate
uniformities on the dual object $\hat G$. As an application, we prove that if
for every dense subgroup $H$ of a compact group $G$ we have that $\hat H$ and
$\hat G$ are uniformly isomorphic, then $G$ is metrizable. Thereby, we extend
to non-abelian groups some results previously considered for abelian
topological groups.
