Nuclearity of semigroup C?-algebras

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