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

TL;DR

This paper introduces a new, functorial definition of semigroup C*-algebras for left cancellative semigroups using inverse semigroup theory, resolving previous construction issues and linking amenability to nuclearity.

Contribution

It provides a novel inverse semigroup-based construction of semigroup C*-algebras that is functorial and clarifies the relationship with group C*-algebras and crossed products.

Findings

The new construction is functorial and independent of ideal choices.

The group C*-algebra appears as a natural quotient.

Crossed products of semigroups are isomorphic under certain actions.

Abstract

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.