Dissertation
Rigidity for von Neumann algebras of graph product groups
University of Iowa
Doctor of Philosophy (PhD), University of Iowa
Spring 2024
DOI: 10.25820/etd.007396
Abstract
This thesis is the culmination of work done by Ionut Chifan, Daniel Drimbe and myself investigating the structure of von Neumann algebras arising from graph product groups. In particular, this thesis investigates various rigidity aspects of the von Neumann algebra L(Γ) where Γ is a graph product group whose underlying graph is a certain cycle of cliques and the vertex groups are wreath-like product property (T) groups. Using an approach that combines methods from Popa’s deformation/rigidity theory with new techniques pertaining to graph product algebras, a description of all symmetries of these von Neumann algebras and reduced C˚ -algebras is given by establishing formulas in the spirit of Genevois and Martin’s results on automorphisms of graph product groups.
The other main result of this work is the fact that these graph product groups Γ are entirely recognizable from the category of all von Neumann algebras arising from an arbitrary non-trivial graph product group with infinite vertex groups. A sharper C˚ -algebraic version of this statement is also obtained. Proving these results leads to an extension of the main W˚ -superrigidity result from [CIOS21] to direct products of property (T) wreath-like product groups.
Details
- Title: Subtitle
- Rigidity for von Neumann algebras of graph product groups
- Creators
- Michael Davis
- Contributors
- Ionut Chifan (Advisor)Ben Cooper (Committee Member)Raul Curto (Committee Member)Palle Jorgensen (Committee Member)
- Resource Type
- Dissertation
- Degree Awarded
- Doctor of Philosophy (PhD), University of Iowa
- Degree in
- Mathematics
- Date degree season
- Spring 2024
- Publisher
- University of Iowa
- DOI
- 10.25820/etd.007396
- Number of pages
- vi, 97 pages
- Copyright
- Copyright 2024 Michael Davis
- Language
- English
- Date submitted
- 04/21/2024
- Description illustrations
- illustrations
- Description bibliographic
- Includes bibliographical references (pages 92-97).
- Public Abstract (ETD)
- Von Neumann algebras were introduced by John von Neumann in the early half of the 20th century to provide a rigorous mathematical framework for quantum mechanics. In quantum physics, quantum states are represented as points in a Hilbert space and the observable quantities as linear operators acting on that Hilbert space. This framework has remained standard for modelling quantum systems, and recently von Neumann algebras have even been used in the theory of general relativity to model black hole phenomena. This thesis is concerned with the mathematical classification of von Neumann algebras, specifically what are called “group von Neumann algebras”, and is the culmination of work with Ionut Chifan and Daniel Drimbe investigating von Neumann algebras associated with “graph prod uct groups”, a kind of group arising from a given collection of groups, called vertex groups, and a finite graph. With this work we were able to obtain strong results related to the rigidity of these algebras, an important step towards classification. These groups are important in other areas of mathematics such as topology and geometric group theory, and studying them has opened up new approaches in all these areas.
- Academic Unit
- Mathematics
- Record Identifier
- 9984647647002771
Metrics
7 File views/ downloads
14 Record Views