Arcs in the Pontryagin dual of a topological abelian group

Abstract

In this article we study the arcwise connected component in the Pontryagin dual of an abelian topological group. It is clear that the set of continuous characters that can be lifted over the reals is contained in the arcwise connected component of the dual group. We show that the converse is true if all arcs in the character group are equicontinuous sets. This property is present in Pontryagin duals of pseudocompact groups, of reflexive groups and of groups which are k-spaces as topological spaces. We study the meaning of such a property and its presence in groups and vector spaces endowed with weak topologies. We also characterize the image of the exponential mapping of a dual group as formed by those characters which can be lifted over the reals endowed with the Bohr topology.