Semigroup C⁎-algebras and amenability of semigroups

We construct reduced and full semigroup C⁎-algebras for left cancellative semigroups. Our new construction covers particular cases already considered by A. Nica and also Toeplitz algebras attached to rings of integers in number fields due to J. Cuntz. Moreover, we show how (left) amenability of semigroups can be expressed in terms of these semigroup C⁎-algebras in analogy to the group case