Topics on the geometry and classification of Banach lattices

We examine topics related to the geometric structure of Banach lattices of various classes, their properties, and classification using tools from functional analysis and mathematical logic. This work can be roughly divided into four parts. The first part (Chapter 2) presents several geometric results on Banach lattice analogues of classical Banach space theorems which ground results from later sections. The second part (Chapters 3 and 4) presents various results on the descriptive complexity of classes of Banach lattices and determines the complexity of the lattice isomorphism and isometry equivalence relations. The focus of the third part (Chapter 5) is the construction of a lattice isometrically universal separable ”Gurarij” Banach lattice by combining properties of the geometric structure of Banach lattices with Fraısse machinery that was developed in the context of continuous logic. Finally, we return to geometric considerations in the fourth and last part of chapter 6, which describes a method of renorming AM spaces so that the only lattice isometry is the trivial isometry.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2021

Author dc:creator

• Gramcko-Tursi, Mary Angelica

Contributors dc:contributor

• Oikhberg, Timur
• Junge, Marius
• Kutzarova, Denka
• Tserunyan, Anush

• Banach lattices
• universal spaces
• standard Borel spaces
• Isomorphism equivalence relations
• Fraisse limits
• homogeneity
• renormings

Statement dc:rights

• Copyright 2021 Mary Gramcko-Tursi

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/110490

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/110490