Doctor of Philosophy
School of Mathematics and Applied Statistics
Lewin, Peter, The structure and ergodic theory of higher-rank graph algebras, Doctor of Philosophy thesis, School of Mathematics and Applied Statistics, University of Wollongong, 2011. https://ro.uow.edu.au/theses/3500
A higher-rank graph is a higher-dimensional analogue of a directed graph. These were introduced by Kumjian and Pask in 2000 as a combinatorial model for the higher-rank Cuntz-Krieger algebras studied by Robertson and Steger in 1999. Since their inception, studying the properties of higher-rank graph algebras has been a fourishing research area. The most general class of higher-rank graphs currently under investigation is called nitely aligned. Raeburn, Sims and Yeend showed in 2004 how to associate to each nitely aligned higher-rank graph Λ a universal C*-algebra C*(Λ). These algebras share many important properties with the Cuntz- Krieger algebras, in particular they are highly tractable. Hence higher-rank graphs provide an excellent mechanism for constructing examples of tractable C*-algebras, which accounts for the interest in the area in recent years.
Recently, Shotwell has formulated two conditions - aperiodicity and cofinality - in a fnitely aligned higher-rank graph Λ which characterise simplicity of C*(Λ). Working independently of Shotwell, we give an alternate formulation of aperiodicity and cofnality in which characterises simplicity of C*(Λ). The key point of difference is that our formulations of aperiodicity and cofnality are given only in terms of nite paths. Thus our conditions are more easily veri ed in examples. Moreover, all of our proofs are direct and employ new techniques for studying higher-rank graph algebras. We also show how our cofinality condition simpli es in a number of special cases occurring in the literature; in these cases our results are also new.
One of they key dynamical invariants used in ergodic theory is topological en- tropy. Here we investigate topological entropy of an endomorphism, known as the noncommutative shift map, of the core of a higher-rank graph algebra. We con- sider the class of locally nite higher-rank graphs with no sources, a subclass of all nitely aligned higher-rank graphs. Recently, Jeong and Park provided estimates for topological entropy of the noncommutative shift map for C*-algebras associated to infnite directed graphs. In addition, Skalski and Zacharias have computed the topological entropy of the noncommutative shift map for C*-algebras associated to nite higher-rank graphs. We provide a common generalisation of the work of Jeong and Park, and of Skalski and Zacharias. That is, we provide estimates for topological entropy of the noncommutative shift map for C*-algebras associated to in nite higher-rank graphs.