Nuclearity of semigroup C*-algebras

TL;DR

This paper establishes conditions under which semigroup C*-algebras of weakly quasi-lattice ordered groups are nuclear, using generalized length functions called controlled maps, with applications to one-relator semigroups and graph products.

Contribution

It introduces a new definition of controlled maps and applies it to prove nuclearity for classes of semigroup C*-algebras, including Baumslag--Solitar semigroups and graph products.

Findings

Nuclearity characterized by controlled maps into amenable groups.

Nuclearity proven for semigroup C*-algebras of certain one-relator semigroups.

Graph products of weak quasi-lattices are weak quasi-lattices with nuclear C*-algebras when groups are amenable.

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\cap 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.