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C*-algebras associated to higher-rank graphs

Research Doctorate - Doctor of Philosophy (PhD) / Directed graphs are combinatorial objects used to model networks like fluid-flow systems in which the direction of movement through the network is important. In 1980, Enomoto and Watatani used finite directed graphs to provide an intuitive framework for the Cuntz-Krieger algebras introduced by Cuntz and Krieger earlier in the same year. The theory of the C*-algebras of directed graphs has since been extended to include infinite graphs, and there is an elegant relationship between connectivity and loops in a graph and the structure theory of the associated C*-algebra. Higher-rank graphs are a higher-dimensional analogue of directed graphs introduced by Kumjian and Pask in 2000 as a model for the higher-rank Cuntz-Krieger algebras introduced by Robertson and Steger in 1999. The theory of the Cuntz-Krieger algebras of higher-rank graphs is relatively new, and a number of questions which have been answered for directed graphs remain open in the higher-rank setting. In particular, for a large class of higher-rank graphs, the gauge-invariant ideal structure of the associated C*-algebra has not yet been identified. This thesis addresses the question of the gauge-invariant ideal structure of the Cuntz-Krieger algebras of higher-rank graphs. To do so, we introduce and analyse the collections of relative Cuntz-Krieger algebras associated to higher-rank graphs. The first two main results of the thesis are versions of the gauge-invariant uniqueness theorem and the Cuntz-Krieger uniqueness theorem which apply to relative Cuntz-Krieger algebras. Using these theorems, we are able to achieve our main goal, producing a classification of the gauge-invariant ideals in the Cuntz-Krieger algebra of a higher-rank graph analogous to that developed for directed graphs by Bates, Hong, Raeburn and SzymaƱski in 2002. We also demonstrate that relative Cuntz-Krieger algebras associated to higher-rank graphs are always nuclear, and produce conditions on a higher-rank graph under which the associated Cuntz-Krieger algebra is simple and purely infinite.

Identiferoai:union.ndltd.org:ADTP/222120
Date January 2003
CreatorsSims, Aidan Dominic
Source SetsAustraliasian Digital Theses Program
LanguageEnglish
Detected LanguageEnglish
RightsCopyright 2003 Aidan Dominic Sims

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