Development of the theory of Kumjian-Pask fibrations, their path groupoids, and their C*-algebras - de Wolf, Lydia (Kansas State University, 2023)
The higher-rank graphs, or k-graphs, of Kumjian and Pask [1], introduced in 2000 to generalize the construction of C*-algebras of directed graphs, are in fact examples of a notion in category theory called a discrete Conduché fibration. In 2017, Brown and Yetter developed a theory of C*-algebras associated to discrete Conduché fibrations, enforced additional conditions to define Kumjian-Pask fibrations, and investigated the generalized infinite path spaces and path groupoids arising from these fibrations [2]. This dissertation continues to explore this generalization, investigating the properties of k-graphs that may or may not hold in the general case, and providing examples of Kumjian-Pask fibrations beyond the motivating k-graph case, to justify and motivate further study. We generalize various constructions, properties, and techniques that are useful in the theory of graph algebras, as well as integrating recent studies to bring our general results up to date. After reformulating or introducing several examples, we cover two classes of Kumjian-Pask fibrations in detail which admit much more development and demonstrate opportunity for continuing research.
Keywords: Operator algebras; C*-algebras; Category theory