C(star)-algebras of higher-risk graphs: Desingularization and groupoid methods

Higher-rank graphs are higher dimensional analogues of directed graphs that were developed in 2000 by Kumjian and Pask. Originally, C*-algebras were associated to row-finite higher-rank graphs without sources. Extending the Cuntz-Krieger relations to define C*-algebras of a broader class of higher-rank graphs known as finitely aligned higher-rank graphs has been one focus of research in this area since the introduction of higher-rank graphs.; Two types of vertices prove to be problematic when defining the C*-algebras of higher-rank graphs: sources and infinite receivers. Sources are vertices that do not have any edges pointing toward them, while infinite receivers have infinitely many edges pointing toward them. Similar difficulties were previously encountered in defining C*-algebras of directed graphs. One solution that was developed to deal with the complications sources and infinite receivers create in directed graphs is a process known as desingularization. This thesis develops a process that removes sources from a higher-rank graph and that is analogous to the desingularization used for directed graphs. That is, given a row-finite higher-rank graph with sources, we build a row-finite higher-rank graph without sources whose C*-algebra is in the same Morita equivalence class as the original graph.; The C*-algebras of row-finite higher-rank graphs without sources were originally studied using groupoid methods. However, the C*-algebras of finitely aligned graphs were defined without groupoids. This thesis constructs a groupoid associated to a finitely-aligned higher-rank graph. We first define an inverse semigroup that encodes the structure of the higher-rank graph and an action of the inverse semigroup on the path space of the graph. The groupoid obtained is the groupoid of germs of this action. Using the groupoid, we are able to show that two different types of boundary paths used in the analysis of higher-rank graphs are, in some cases, essentially the same. We also prove a generalization of the Cuntz-Krieger Uniqueness Theorem that was previously established for row-finite higher-rank graphs without sources.