#### Year

2010

#### Degree Name

Doctor of Philosophy

#### Department

University of Wollongong. School of Mathematics and Applied Statistics

#### Recommended Citation

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. http://ro.uow.edu.au/theses/3175

#### Abstract

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 ς_{E} 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 ς_{E} 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

*D*of

_{E}*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

^{^})*D*in

_{E}*D*which implements

_{E^}*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.

_{∞}02Whole.pdf (630 kB)