  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.
