Nuclearity of semigroup C*-algebras

Abstract

We study the semigroup C * -algebra of a positive cone P of a weakly quasi-lattice ordered group. That is, P is a subsemigroup of a discrete group G with P ∩ P −1 = {e} and such that any two elements of P with a common upper bound in P also have a least upper bound. We find sufficient conditions for the semigroup C * -algebra of P to be nuclear. These conditions involve the idea of a generalised length function, called a "controlled map", into an amenable group.Here we give a new definition of a controlled map and discuss examples from different sources. We apply our main result to establish nuclearity for semigroup C * -algebras of a class of one-relator semigroups, motivated by a recent work of Li, Omland and Spielberg. This includes all the Baumslag-Solitar semigroups. We also analyse semidirect products of weakly quasi-lattice ordered groups and use our theorem in examples to prove nuclearity of the semigroup C * -algebra. Moreover, we prove that the graph product of weak quasi-lattices is again a weak quasi-lattice, and show that the corresponding semigroup C * -algebra is nuclear when the underlying groups are amenable.