Publication Date

8-25-2016

Abstract

In this document we solve some local connectivity problems in matrix representations of the form C(T^N) -> M_n and C(T^N) -> M_n <- C([-1, 1]^N) using the so called toroidal matrix links, which can be interpreted as normal contractive matrix analogies of free homotopies in algebraic topology. In order to deal with the locality constraints, we have combined some techniques introduced in this document with several versions of the Basic Homotopy Lemma L.2.3.2, T.2.3.1 and C.2.3.1 obtained initially by Bratteli, Elliot, Evans and Kishimoto in [4] and generalized by Lin in [19] and [22]. We have also implemented some techniques from matrix geometry, combinatorial optimization and noncommutative topology developed by Loring [24, 27], Shulman [27], Bhatia [2], Chu [8], Brockett [5], Choi [7, 6], Effros [6], Exel [11], Eilers [11], Elsner [12], Pryde [31, 30], McIntosh [30] and Ricker [30].

Degree Name

Mathematics

Level of Degree

Doctoral

Department Name

Mathematics & Statistics

First Advisor

Loring, Terry A.

First Committee Member (Chair)

Buium, Alexandr

Second Committee Member

Boyer, Charles

Third Committee Member

Packer, Judith

Language

English

Keywords

Matrix homotopy, relative lifting problems, matrix representation, noncommutative semialgebraic sets, K-theory, amenable C*-algebra, joint spectrum.

Document Type

Dissertation

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