On the universality of semigroup $C^*$-algebras

Reduced semigroup $C^*$-algebras given by regular representations of discrete semigroups are considered in the talk. Their characterizations are given as universal $C^*$-algebras defined by generating elements and relations. This approach allows us to study various properties of $C^*$-algebras, in particular, to obtain a semigroup $C^*$-algebra representation for the semidirect product of semigroups of integers $\mathbb{Z}\rtimes \mathbb{Z}^{\ times}$ as a crossed product of its $C^*$-subalgebra with a cyclic group of order two.