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- All Subjects: Geometric analysis
- Creators: Kaliszewski, Steven
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
This thesis explores several questions concerning the preservation of geometric structure under the Ricci flow, an evolution equation for Riemannian metrics. Within the class of complete solutions with bounded curvature, short-time existence and uniqueness of solutions guarantee that symmetries and many other geometric features are preserved along the flow. However, much less is known about the analytic and geometric properties of solutions of potentially unbounded curvature. The first part of this thesis contains a proof that the full holonomy group is preserved, up to isomorphism, forward and backward in time. The argument reduces the problem to the preservation of reduced holonomy via an analysis of the equation satisfied by parallel translation around a loop with respect to the evolving metric. The subsequent chapter examines solutions satisfying a certain instantaneous, but nonuniform, curvature bound, and shows that when such solutions split as a product initially, they will continue to split for all time. This problem is encoded as one of uniqueness for an auxiliary system constructed from a family of time-dependent, orthogonal distributions of the tangent bundle. The final section presents some details of an ongoing project concerning the uniqueness of asymptotically product gradient shrinking Ricci solitons, including the construction of a certain system of mixed differential inequalities which measures the extent to which such a soliton fails to split.
ContributorsCook, Mary (Author) / Kotschwar, Brett (Thesis advisor) / Paupert, Julien (Committee member) / Kawski, Matthias (Committee member) / Kaliszewski, Steven (Committee member) / Fishel, Susanna (Committee member) / Arizona State University (Publisher)
Created2021