Group structures and their applications to categories of analysis and topological algebra

Given a covering space of a compact topological group, we consider the problem of whether the group structure can be lifted to the covering space in such a way that the respective covering map becomes a group homomorphism. We discuss some applications to polynomials over the Banach algebra of continuous functions on a compact abelian group. We also consider some properties of morphisms of topological groups and of related semigroup $C^*$-algebras.