  We give a new definition of the semigroup C*-algebra of a left cancellative
semigroup, which resolves problems of the construction by X. Li. Namely, the
new construction is functorial, and the independence of ideals in the semigroup
does not influence the independence of the generators. It has a group
C*-algebra as a natural quotient. The C*-algebra of the old construction is a
quotient of the new one. All this applies both to the full and reduced
C*-algebras. The construction is based on the universal inverse semigroup
generated by a left cancellative semigroup. We apply this approach to connect
amenability of a semigroup to nuclearity of its C*-algebra. Large classes of
actions of these semigroups are in one-to-one correspondence, and the crossed
products are isomorphic. A crossed product of a left Ore semigroup is
isomorphic to the partial crossed product of the generated group.
