C*-algebras generated by semigroups of partial isometries

C*-algebras;partial isometries;operator algebras;quasi-lattice ordered groups;nuclear C*-algebras;amenability;semigroups

Tolich, Ilija ; an Huef, Astrid ; Raeburn, Iain

University of Otago

关键词: C*-algebras;    partial isometries;    operator algebras;    quasi-lattice ordered groups;    nuclear C*-algebras;    amenability;    semigroups;   

Others  :  https://ourarchive.otago.ac.nz/bitstream/10523/7277/3/TolichIlija2017PhD.pdf

来源: Otago University Research Archive

【 摘 要 】

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.