{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/107874"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/107874","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Continued fractions and representations of graphs","abstract":"Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2020-08-25 without embargo terms","abstract_html":"Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2020-08-25 without embargo terms","abstract_has_math":false,"creators":["Linden, Christopher"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Boca, Florin","Junge, Marius","Dor-On, Adam","Tyson, Jeremy","Zaharescu, Alexandru"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2020,"date_issued":"2020-08-26T21:54:17Z","date_published":"2020-08-26T21:54:17Z","updated_at":"2026-07-22T22:24:47Z","subjects":["Continued fractions","Cuntz algebras","Non-self-adjoint operator algebras"],"languages":["en"],"rights":["Copyright 2020 Christopher Linden"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/107874","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Boca, Florin","Junge, Marius","Dor-On, Adam","Tyson, Jeremy","Zaharescu, Alexandru"]},{"key":"dc:creator","label":"Author","values":["Linden, Christopher"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2020-08-26T21:54:17Z","2020-04-10","2020-05"]},{"key":"dc:type","label":"Dc Type","values":["text","Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Continued fractions","Cuntz algebras","Non-self-adjoint operator algebras"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2020 Christopher Linden"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/107874"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2020-08-25 without embargo terms","The student, Christopher Linden, accepted the attached license on 2020-04-09 at 12:01.","The student, Christopher Linden, submitted this Dissertation for approval on 2020-04-09 at 12:10.","This Dissertation was approved for publication on 2020-04-10 at 15:24.","DSpace SAF Submission Ingestion Package generated from Vireo submission #14949 on 2020-08-25 at 17:06:41","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.","Made available in DSpace on 2020-08-26T21:54:17Z (GMT). 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The submission was exported from vireo on 2020-08-25 without embargo terms","The student, Christopher Linden, accepted the attached license on 2020-04-09 at 12:01.","The student, Christopher Linden, submitted this Dissertation for approval on 2020-04-09 at 12:10.","This Dissertation was approved for publication on 2020-04-10 at 15:24.","DSpace SAF Submission Ingestion Package generated from Vireo submission #14949 on 2020-08-25 at 17:06:41","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.","Made available in DSpace on 2020-08-26T21:54:17Z (GMT). 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