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- Creators: Debussy, Claude, 1862-1918
- Creators: Ravel, Maurice, 1875-1937
Description
In this thesis, I investigate the C*-algebras and related constructions that arise from combinatorial structures such as directed graphs and their generalizations. I give a complete characterization of the C*-correspondences associated to directed graphs as well as results about obstructions to a similar characterization of these objects for generalizations of directed graphs. Viewing the higher-dimensional analogues of directed graphs through the lens of product systems, I give a rigorous proof that topological k-graphs are essentially product systems over N^k of topological graphs. I introduce a "compactly aligned" condition for such product systems of graphs and show that this coincides with the similarly-named conditions for topological k-graphs and for the associated product systems over N^k of C*-correspondences. Finally I consider the constructions arising from topological dynamical systems consisting of a locally compact Hausdorff space and k commuting local homeomorphisms. I show that in this case, the associated topological k-graph correspondence is isomorphic to the product system over N^k of C*-correspondences arising from a related Exel-Larsen system. Moreover, I show that the topological k-graph C*-algebra has a crossed product structure in the sense of Larsen.
ContributorsPatani, Nura (Author) / Kaliszewski, Steven (Thesis advisor) / Quigg, John (Thesis advisor) / Bremner, Andrew (Committee member) / Kawski, Matthias (Committee member) / Spielberg, John (Committee member) / Arizona State University (Publisher)
Created2011
ContributorsDebussy, Claude, 1862-1918 (Composer)
ContributorsDebussy, Claude, 1862-1918 (Composer)
ContributorsDebussy, Claude, 1862-1918 (Composer)
ContributorsDebussy, Claude, 1862-1918 (Composer)
ContributorsDebussy, Claude, 1862-1918 (Composer)
Description
The purpose of this senior thesis is to explore the abstract ideas that give rise to the well-known Fourier series and transforms. More specifically, finite group representations are used to study the structure of Hilbert spaces to determine under what conditions an element of the space can be expanded as a sum. The Peter-Weyl theorem is the result that shows why integrable functions can be expressed in terms of trigonometric functions. Although some theorems will not be proved, the results that can be derived from them will be briefly discussed. For instance, the Pontryagin Duality theorem states that there is a canonical isomorphism between a group and the second dual of the group, and it can be used to prove $Plancherel$ theorem which essentially says that the Fourier transform is itself a unitary isomorphism.
ContributorsReyna De la Torre, Luis E (Author) / Kaliszewski, Steven (Thesis director) / Rainone, Timothy (Committee member) / School of Mathematical and Statistical Sciences (Contributor) / Department of Physics (Contributor) / Barrett, The Honors College (Contributor)
Created2019-05
Description
The theory of frames for Hilbert spaces has become foundational in the study of wavelet analysis and has far-reaching applications in signal and image-processing. Originally, frames were first introduced in the early 1950's within the context of nonharmonic Fourier analysis by Duffin and Schaeffer. It was then in 2000, when M. Frank and D. R. Larson extended the concept of frames to the setting of Hilbert C*-modules, it was in that same paper where they asked for which C*-algebras does every Hilbert C*-module admit a frame. Since then there have been a few direct answers to this question, one being that every Hilbert A-module over a C*-algebra, A, that has faithful representation into the C*-algebra of compact operators admits a frame. Another direct answer by Hanfeng Li given in 2010, is that any C*-algebra, A, such that every Hilbert C*-module admits a frame is necessarily finite dimensional. In this thesis we give an overview of the general theory of frames for Hilbert C*-modules and results answering the frame admittance property. We begin by giving an overview of the existing classical theory of frames in Hilbert spaces as well as some of the preliminary theory of Hilbert C*-modules such as Morita equivalence and certain tensor product constructions of C*-algebras. We then show how some results of frames can be extended to the case of standard frames in countably generated Hilbert C*-modules over unital C*-algebras, namely the frame decomposition property and existence of the frame transform operator. We conclude by going through some proofs/constructions that answer the question of frame admittance for certain Hilbert C*-modules.
ContributorsJaime, Arturo (Author) / Kaliszewski, Steven (Thesis director) / Spielberg, Jack (Committee member) / Aguilar, Konrad (Committee member) / School of Mathematical and Statistical Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2019-05
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
ContributorsDebussy, Claude, 1862-1918 (Composer)