Abstract
This thesis examines the C*-algebras associated to semigroups of partial isometries. There are many interesting examples of C*-algebras generated by families of partial isometries, for example the C*-algebras associated to directed graphs and the C*-algebras associated to inverse semigroups.
In 1992 Nica introduced a class of partially ordered groups called quasi-lattice ordered groups, and studied the C*-algebras generated by semigroups of isometries satisfying a covariance condition. We have adapted Nica's construction for semigroups of partial isometries associated to what we call doubly quasi-lattice ordered groups. For each doubly quasi-lattice ordered group we construct two algebras: a concretely defined reduced algebra, and a universal algebra generated by a covariant family of partial isometries. We examine when representations of the universal algebra are faithful, and this gives rise to a notion of amenability for doubly quasi-lattice ordered groups.
We prove several recognition theorems for amenability. In particular, we prove that the universal and reduced algebras are isomorphic if and only if the doubly quasi-lattice ordered group is amenable. Further, we prove that if there is an order preserving homomorphism from a doubly quasi-lattice ordered group to an amenable group, then the quasi-lattice ordered group is amenable and the associated universal algebra is nuclear.