Continued fractions and representations of graphs

Linden, Christopher

Permalink

https://hdl.handle.net/2142/107874 Copy

Description

Title

Continued fractions and representations of graphs

Author(s)

Linden, Christopher

Issue Date

2020-04-10

Director of Research (if dissertation) or Advisor (if thesis)

Boca, Florin

Doctoral Committee Chair(s)

Junge, Marius

Committee Member(s)

• Dor-On, Adam
• Tyson, Jeremy
• Zaharescu, Alexandru

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• Continued fractions
• Cuntz algebras
• Non-self-adjoint operator algebras

Abstract

This thesis is concerned with results about continued fractions and the representation theory of operator algebras associated with graphs. First, we study analogues of Minkowski’s question mark function ?(x) for continued fractions with even or with odd partial quotients. We prove these functions are Hölder continuous with precise exponents, and that they linearize analogues of the Gauss and Farey maps. We also show that certain Bratteli diagrams which arise in the study of continued fractions all yield isomorphic approximately finite-dimensional C*-algebras. Second, we construct representations of the Cuntz algebra O_N from dynamical systems associated to slow continued fraction algorithms. We give their irreducible decomposition formulas in terms of the modular group action on real numbers, as a generalization of results by Kawamura, Hayashi and Lascu. Third, we consider free semigroupoid algebras associated to graphs. We show that every non-cycle finite transitive directed graph has a Cuntz-Krieger family whose WOT-closed algebra is B(H). As a consequence, we prove that finite disjoint unions of finite transitive directed graphs are exactly those finite graphs which admit self-adjoint free semigroupoid algebras. Finally, we prove two results about operator algebras constructed from stochastic matrices. We improve the classification result proved by Dor-On and Markiewicz with a new characterization of conditional probabilities in terms of (generalized) Doob transforms. We also characterize the non-commutative peak points of the associated operator algebra in a way that allows one to determine them from inspecting the graph. This leads to a concrete analogue of the maximum modulus principle for computing the norm of operators in the ampliated operator algebras.

Graduation Semester

2020-05

Type of Resource

Thesis

Permalink

http://hdl.handle.net/2142/107874

Copyright and License Information

Copyright 2020 Christopher Linden

Owning Collections