Doctor of Philosophy
University of Wollongong. School of Mathematics and Applied Statistics
Webster, Samuel Brendon, Directed graphs and k-graphs: topology of the path space and how it manifests in the associated C*-algebra, Doctor of Philosophy thesis, University of Wollongong. School of Mathematics and Applied Statistics, University of Wollongong, 2010. https://ro.uow.edu.au/theses/3175
Directed graphs and their higher-rank analogues provide an intuitive frame-work for the analysis of a broad class of C*-algebras which we call graph algebras. Kumjian, Pask, Raeburn and Renault built a groupoid ςE from the infinite-path space of a locally finite directed graph E, and used the theory of groupoid C*-algebras to define the graph C*-algebra. Local finiteness was required so that ςE was locally compact and r-discrete, with unit space homeomorphic to the infinite path space of E. Similarly, the higher-rank graphs of Kumjian and Pask were initially studied with similar restrictive hypotheses in order to use groupoid based analysis of their higher-rank C*-algebras. In particular, the topology on the path space of a directed graph or higher-rank graph is crucial in the analysis of graph C*-algebras.
Drinen and Tomforde defined a process called desingularisation which can be used to extend many results about the C*-algebras of locally finite directed graphs to those of arbitrary directed graphs. Drinen and Tomforde construct from an arbitrary directed graph E a row-finite directed graph E^ with no sources such that C*(E) embeds in C*(E^) as a full corner. Subsequently, Farthing developed a partial desingularisation for higher-rank graphs, which constructs from a row-finite higher-rank graph Λ with sources a row-finite higher-rank graph Λ~ with no sources such that C*(Λ) embeds in C*(Λ~) as a full corner.
In Chapter 2, we construct a topology on the path space of an arbitrary directed graph E and prove that it is locally compact and Hausdorff. We show that there is a homeomorphism φ∞ from a subspace of the infinite-path space of the Drinen-Tomforde desingularisation E^ onto the boundary-path space ∂E of E. We then show that there is a commutative C*-subalgebra DE of C*(E) which is homeomorphic to the continuous functions on ∂E. Concluding our results on directed graphs, we show that the embedding of C*(E) in C*(E^) restricts to an embedding of DE in DE^ which implements φ∞. In Chapter 3, we develop a modifcation of Farthing's desingularisation procedure for row-finite higher-rank graphs which contains cleaner proofs of her results. We use this modification to prove analogues for higher-rank graphs of the results from Chapter 2.
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