Research Doctorate - Doctor of Philosophy (PhD) / There is a strong connection between directed graphs and the shifts of finite type which are an important family of dynamical systems. Higher-rank graphs (or k-graphs) and their C*-algebras were introduced by Kumjian and Pask to generalise directed graphs and their C*-algebras. Kumjian and Pask showed how higher-dimensional shifts of finite type can be associated to k-graphs, but did not discuss how one might associate k-graphs to k-dimensional shifts of finite type. In this thesis we construct a family of 2-graphs A arising from a certain type of algebraic two-dimensional shift of finite type studied by Schmidt, and analyse the structure of their C*-algebras. Graph algebras and k-graph algebras provide a rich source of examples for the classication of simple, purely infinite, nuclear C*-algebras. We give criteria which ensure that the C*-algebra C*(A) is simple, purely infinite, nuclear, and satisfies the hypotheses of the Kirchberg-Phillips Classification Theorem. We perform K-theory calculations for a wide range of our 2-graphs A using the Magma computational algebra system. The results of our calculations lead us to conjecture that the K-groups of C*(A) are finite cyclic groups of the same order. We are able to prove under mild hypotheses that the K-groups have the same order, but we have only numerical evidence to suggest that they are cyclic. In particular, we find several examples for which K1(C*(A)) is nonzero and has torsion, hence these are examples of 2-graph C*-algebras which do not arise as the C*-algebras of directed graphs. Finally, we consider a subfamily of 2-graphs with interesting combinatorial connections. We identify the nonsimple C*-algebras of these 2-graphs and calculate their K-theory.
| Identifer | oai:union.ndltd.org:ADTP/245837 |
| Date | January 2009 |
| Creators | Weaver, Natasha |
| Source Sets | Australiasian Digital Theses Program |
| Language | English |
| Detected Language | English |
| Rights | Copyright 2009 Natasha Weaver |
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