Nuclearity of semigroup C∗-algebras and the connection to amenability

We study C∗-algebras associated with subsemigroups of groups. For a large class of such semigroups including positive cones in quasi-lattice ordered groups and left Ore semigroups, we describe the corresponding semigroup C∗-algebras as C∗-algebras of inverse semigroups, groupoid C∗-algebras and full corners in associated group crossed products. These descriptions allow us to characterize nuclearity of semigroup C∗-algebras in terms of faithfulness of left regular representations and amenability of group actions. Moreover, we also determine when boundary quotients of semigroup C∗-algebras are UCT Kirchberg algebras. This leads to a unified approach to Cuntz algebras and ring C∗-algebras.