Consider a topological group $G$ and its unitary representation $\rho$. Consider the closure of $\rho(G)$ in the weak operator topology. Then we get a compact semigroup with separately continuous multiplication. For semisimple Lie groups with finite center this semigroup is simply the one-point compactification (R.Howe and C.Moore).
However, for Abelian groups (for instance for $\mathbb Z$) and for infinite-dimensional groups this leads to highely nontrivial (but handable) objects. The purpose of talk is an introduction to problems of this type (we do not suppose any preliminary knowledge of representation theory).