Group structures and their applications

We will consider properties of coverings of compact groups, in particular, the $P$-adic solenoids. The theorem on a covering group and its applications will be discussed. It turns out that the coverings are closely related with the Weierstrass polynomials over Banach algebras of continuous functions. We will discuss $*$-automorphisms of semigroup $C^*$-algebras for semigroups in the character groups of the $P$-adic solenoids.