Matching Items (3)
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- All Subjects: Theoretical mathematics
- Creators: Quigg, John
Description
Higher-rank graphs, or k-graphs, are higher-dimensional analogues of directed graphs, and as with ordinary directed graphs, there are various C*-algebraic objects that can be associated with them. This thesis adopts a functorial approach to study the relationship between k-graphs and their associated C*-algebras. In particular, two functors are given between appropriate categories of higher-rank graphs and the category of C*-algebras, one for Toeplitz algebras and one for Cuntz-Krieger algebras. Additionally, the Cayley graphs of finitely generated groups are used to define a class of k-graphs, and a functor is then given from a category of finitely generated groups to the category of C*-algebras. Finally, functoriality is investigated for product systems of C*-correspondences associated to k-graphs. Additional results concerning the structural consequences of functoriality, properties of the functors, and combinatorial aspects of k-graphs are also included throughout.
ContributorsEikenberry, Keenan (Author) / Quigg, John (Thesis advisor) / Kaliszewski, Steven (Thesis advisor) / Spielberg, John (Committee member) / Arizona State University (Publisher)
Created2016
Description
The author employs bundle theory to investigate dynamics on C*- algebras. Using methods old and new to define dynamics on topological spaces (often with additional structure), implications of the dynamics are investigated in the non-commutative setting, and in suitable situations the dynamics are classified. As a result, new Morita equivalence results are derived and new settings introduced in the study of crossed products, whether by group coactions or by actions of groups and groupoids.
ContributorsHall, Lucas (Author) / Quigg, John (Thesis advisor) / Kaliszewski, S. (Committee member) / Spielberg, Jack (Committee member) / Paupert, Julien (Committee member) / Kotschwar, Brett (Committee member) / Arizona State University (Publisher)
Created2022
Description
C*-algebras of categories of paths were introduced by Spielberg in 2014 and generalize C*-algebras of higher rank graphs. An approximately finite dimensional (AF) C*-algebra is one which is isomorphic to an inductive limit of finite dimensional C*-algebras. In 2012, D.G. Evans and A. Sims proposed an analogue of a cycle for higher rank graphs and show that the lack of such an object is necessary for the associated C*-algebra to be AF. Here, I give a class of examples of categories of paths whose associated C*-algebras are Morita equivalent to a large number of periodic continued fraction AF algebras, first described by Effros and Shen in 1980. I then provide two examples which show that the analogue of cycles proposed by Evans and Sims is neither a necessary nor a sufficient condition for the C*-algebra of a category of paths to be AF.
ContributorsMitscher, Ian (Author) / Spielberg, John (Thesis advisor) / Bremner, Andrew (Committee member) / Kalizsewski, Steven (Committee member) / Kawski, Matthias (Committee member) / Quigg, John (Committee member) / Arizona State University (Publisher)
Created2020