Residual finite-dimensionality and realizations of approximate rigidity phenomena for operator algebras
| dc.contributor.author | Thompson, Ian | |
| dc.contributor.examiningcommittee | Farenick, Douglas (University of Regina) | |
| dc.contributor.examiningcommittee | Martin, Robert (Mathematics) | |
| dc.contributor.examiningcommittee | Zorboska, Nina (Mathematics) | |
| dc.contributor.supervisor | Clouâtre, Raphaël | |
| dc.date.accessioned | 2024-07-17T15:16:20Z | |
| dc.date.available | 2024-07-17T15:16:20Z | |
| dc.date.issued | 2024-07-11 | |
| dc.date.submitted | 2024-07-11T16:42:33Z | en_US |
| dc.date.submitted | 2024-07-17T14:01:58Z | en_US |
| dc.degree.discipline | Mathematics | |
| dc.degree.level | Doctor of Philosophy (Ph.D.) | |
| dc.description.abstract | Representations on a Hilbert space are a common tool for understanding both C*-algebras and (non self-adjoint) operator algebras. For operator algebras, a major theme has been to capitalize on the structural theory of an enveloping C*-algebra (or C*-cover) to better understand the underlying subalgebra. This philosophy has spanned several advancements, yet there is a subtlety involved in the process: the C*-algebra can vary greatly depending on the choice of representation of the operator algebra. Thus, it becomes advantageous to leverage information from several C*-covers to better understand the underlying operator algebra. Here, we focus on interactions between an operator algebra and its C*-covers, with an emphasis on sufficiency and maximality conditions for two fundamental classes of representations: the finite-dimensional representations and the non-commutative Choquet boundary, respectively. Both families offer separate advantages because the former class is more tractable, yet the latter induces a minimal representation due to work of Arveson. In Chapter 3, we analyze the residual finite-dimensionality of the maximal C*-cover of an operator algebra. To this end, we consider several C*-covers that are formed through families of finite-dimensional representations and compare these C*-covers with the maximal C*-cover. Along the way, we generalize Hadwin's characterization of separable residually finite-dimensional C*-algebras. In Chapter 4, we study maximality conditions on the non-commutative Choquet boundary. A conjecture of Arveson asserts that maximality conditions imply a rigidity property that is, a priori, much stronger. In this chapter, we uncover a significant localization procedure that generalizes several past attempts at Arveson's conjecture. This localization procedure is also applied to another conjecture of Arveson that concerns quotient modules of the Drury-Arveson space. Arveson's rigidity conjecture is originally inspired by a development in approximation theory due to Šaškin. In Chapter 5, we achieve one non-commutative analogue to Šaškin's theorem. In the setting of classical function theory, this encodes a maximality condition for the Choquet boundary with a rigidity property. We find that a similar phenomenon is still true for a large class of C*-algebras. | |
| dc.description.note | October 2024 | |
| dc.description.sponsorship | Dr. Murray Gordon Bell Graduate Scholarship (2023) University of Manitoba Graduate Scholarship (2021) NSERC CGS-M (2020) University of Manitoba Graduate Scholarship (2019) Dr. Narrain D. Gupta Scholarship in Mathematics (2019) Graduate Enhancement of Tri-Agency Stipends Program (2021-2024) | |
| dc.identifier.uri | http://hdl.handle.net/1993/38325 | |
| dc.language.iso | eng | |
| dc.subject | Residually finite-dimensional | |
| dc.subject | Korovkin approximation | |
| dc.subject | Operator algebra | |
| dc.subject | Non self-adjoint | |
| dc.subject | C*-algebra | |
| dc.subject | Operator space | |
| dc.subject | Hyperrigidity | |
| dc.subject | Approximate unitary equivalence | |
| dc.title | Residual finite-dimensionality and realizations of approximate rigidity phenomena for operator algebras | |
| local.subject.manitoba | yes | |
| oaire.awardNumber | 466477489 | |
| oaire.awardTitle | CGS-D | |
| oaire.awardURI | https://www.nserc-crsng.gc.ca/Students-Etudiants/PG-CS/CGSD-BESCD_eng.asp | |
| project.funder.identifier | https://doi.org/10.13039/501100000038 | |
| project.funder.name | Natural Sciences and Engineering Research Council of Canada |