  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.
