  To each discrete left cancellative semigroup $S$ one may associate a certain
inverse semigroup $I_l(S)$, often called the left inverse hull of $S$. We show
how the full and the reduced C*-algebras of $I_l(S)$ are related to the full
and reduced semigroup C*-algebras for $S$ recently introduced by Xin Li, and
give conditions ensuring that these algebras are isomorphic. Our picture
provides an enhanced understanding of Li's algebras.
