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

An inverse semigroup approach to the C*-algebras and crossed products of cancellative semigroups