{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/101546"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/101546","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"On quantum Euclidean spaces: Continuous deformation and pseudo-differential operators","abstract":"Quantum Euclidean spaces are noncommutative deformations of Euclidean spaces. They are prototypes of locally compact noncommutative manifolds in Noncommutative Geometry. In this thesis, we study the continuous deformation and Pseudo-differential calculus of quantum Euclidean spaces. After reviewing the basic definitions and representation theory of quantum Euclidean spaces in Chapter 1, we prove in Chapter 2 a Lip^(1/2) continuous embedding of the family of quantum Euclidean spaces. This result is the locally compact analog of U. Haagerup and M. R\\o rdom's work on Lip^(1/2) continuous embedding for quantum 2-torus. As a corollary, we also obtained Lip^(1/2) embedding for quantum tori of all dimensions. In Chapter 3, we developed a Pseudo-differential calculus for noncommuting covariant derivatives satisfying the Canonical Commutation Relations. Based on some basic analysis on quantum Euclidean spaces, we introduce abstract symbol classs following the idea of abstract pseudo-differential operators introduced by A. Connes and H. Moscovici. We proved the two main ingredients pseudo-differential calculus ---the L2-boundedness of 0-order operators and the composition identity. We also identify the principal symbol map in our setting. Chapter 4 is devoted to application in the local index formula in noncommutative Geometry. We show that our setting with noncommuting covariant derivatives is an example of locally compact noncommutative manifold. After developed the Getzler super-symmetric symbol calculus, we calculate the local index formula for the a noncommutative analog of Bott projection.","abstract_html":"Quantum Euclidean spaces are noncommutative deformations of Euclidean spaces. They are prototypes of locally compact noncommutative manifolds in Noncommutative Geometry. In this thesis, we study the continuous deformation and Pseudo-differential calculus of quantum Euclidean spaces. After reviewing the basic definitions and representation theory of quantum Euclidean spaces in Chapter 1, we prove in Chapter 2 a Lip^(1/2) continuous embedding of the family of quantum Euclidean spaces. This result is the locally compact analog of U. Haagerup and M. R\\o rdom's work on Lip^(1/2) continuous embedding for quantum 2-torus. As a corollary, we also obtained Lip^(1/2) embedding for quantum tori of all dimensions. In Chapter 3, we developed a Pseudo-differential calculus for noncommuting covariant derivatives satisfying the Canonical Commutation Relations. Based on some basic analysis on quantum Euclidean spaces, we introduce abstract symbol classs following the idea of abstract pseudo-differential operators introduced by A. Connes and H. Moscovici. We proved the two main ingredients pseudo-differential calculus ---the L2-boundedness of 0-order operators and the composition identity. We also identify the principal symbol map in our setting. Chapter 4 is devoted to application in the local index formula in noncommutative Geometry. We show that our setting with noncommuting covariant derivatives is an example of locally compact noncommutative manifold. After developed the Getzler super-symmetric symbol calculus, we calculate the local index formula for the a noncommutative analog of Bott projection.","abstract_has_math":false,"creators":["Gao, Li"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Junge, Marius","Ruan, Zhong-Jin","Boca, Florin P.","Oikhberg, Timur"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2018,"date_issued":"2018-09-27T16:17:44Z","date_published":"2018-09-27T16:17:44Z","updated_at":"2026-07-22T22:24:40Z","subjects":["Noncommutative Euclidean spaces","Moyal Deformation","Pseudo-differential operators"],"languages":["en"],"rights":["Copyright 2018 by Li Gao"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/101546","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Junge, Marius","Ruan, Zhong-Jin","Boca, Florin P.","Oikhberg, Timur"]},{"key":"dc:creator","label":"Author","values":["Gao, Li"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2018-09-27T16:17:44Z","2018-07-10","2018-08"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"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":["Noncommutative Euclidean spaces","Moyal Deformation","Pseudo-differential operators"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2018 by Li Gao"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/101546"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Quantum Euclidean spaces are noncommutative deformations of Euclidean spaces. They are prototypes of locally compact noncommutative manifolds in Noncommutative Geometry. In this thesis, we study the continuous deformation and Pseudo-differential calculus of quantum Euclidean spaces. After reviewing the basic definitions and representation theory of quantum Euclidean spaces in Chapter 1, we prove in Chapter 2 a Lip^(1/2) continuous embedding of the family of quantum Euclidean spaces. This result is the locally compact analog of U. Haagerup and M. R\\o rdom's work on Lip^(1/2) continuous embedding for quantum 2-torus. As a corollary, we also obtained Lip^(1/2) embedding for quantum tori of all dimensions. In Chapter 3, we developed a Pseudo-differential calculus for noncommuting covariant derivatives satisfying the Canonical Commutation Relations. Based on some basic analysis on quantum Euclidean spaces, we introduce abstract symbol classs following the idea of abstract pseudo-differential operators introduced by A. Connes and H. Moscovici. We proved the two main ingredients pseudo-differential calculus ---the L2-boundedness of 0-order operators and the composition identity. We also identify the principal symbol map in our setting. Chapter 4 is devoted to application in the local index formula in noncommutative Geometry. We show that our setting with noncommuting covariant derivatives is an example of locally compact noncommutative manifold. After developed the Getzler super-symmetric symbol calculus, we calculate the local index formula for the a noncommutative analog of Bott projection.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2018-09-27 without embargo terms","The student, Li Gao, accepted the attached license on 2018-07-09 at 13:22.","The student, Li Gao, submitted this Dissertation for approval on 2018-07-09 at 13:37.","This Dissertation was approved for publication on 2018-07-10 at 08:36.","DSpace SAF Submission Ingestion Package generated from Vireo submission #12777 on 2018-09-27 at 10:47:26","Made available in DSpace on 2018-09-27T16:17:44Z (GMT). No. of bitstreams: 3 GAO-DISSERTATION-2018.pdf: 745083 bytes, checksum: 7a0f65dda33bda7890444e0cfc35a03b (MD5) LICENSE.txt: 4203 bytes, checksum: 5789b910b66f2cca97d3516d7fdc1b9d (MD5) PROQUEST_LICENSE.txt: 4549 bytes, checksum: 6ed0d0dc126ae16eeb56d703aa479169 (MD5) Previous issue date: 2018-07-10"]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["On quantum Euclidean spaces: Continuous deformation and pseudo-differential operators"]}]}],"canonical_facts":{"dc:contributor":["Junge, Marius","Ruan, Zhong-Jin","Boca, Florin P.","Oikhberg, Timur"],"dc:creator":["Gao, Li"],"dc:date":["2018-09-27T16:17:44Z","2018-07-10","2018-08"],"dc:description":["Quantum Euclidean spaces are noncommutative deformations of Euclidean spaces. They are prototypes of locally compact noncommutative manifolds in Noncommutative Geometry. In this thesis, we study the continuous deformation and Pseudo-differential calculus of quantum Euclidean spaces. After reviewing the basic definitions and representation theory of quantum Euclidean spaces in Chapter 1, we prove in Chapter 2 a Lip^(1/2) continuous embedding of the family of quantum Euclidean spaces. This result is the locally compact analog of U. Haagerup and M. R\\o rdom's work on Lip^(1/2) continuous embedding for quantum 2-torus. As a corollary, we also obtained Lip^(1/2) embedding for quantum tori of all dimensions. In Chapter 3, we developed a Pseudo-differential calculus for noncommuting covariant derivatives satisfying the Canonical Commutation Relations. Based on some basic analysis on quantum Euclidean spaces, we introduce abstract symbol classs following the idea of abstract pseudo-differential operators introduced by A. Connes and H. Moscovici. We proved the two main ingredients pseudo-differential calculus ---the L2-boundedness of 0-order operators and the composition identity. We also identify the principal symbol map in our setting. Chapter 4 is devoted to application in the local index formula in noncommutative Geometry. We show that our setting with noncommuting covariant derivatives is an example of locally compact noncommutative manifold. After developed the Getzler super-symmetric symbol calculus, we calculate the local index formula for the a noncommutative analog of Bott projection.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2018-09-27 without embargo terms","The student, Li Gao, accepted the attached license on 2018-07-09 at 13:22.","The student, Li Gao, submitted this Dissertation for approval on 2018-07-09 at 13:37.","This Dissertation was approved for publication on 2018-07-10 at 08:36.","DSpace SAF Submission Ingestion Package generated from Vireo submission #12777 on 2018-09-27 at 10:47:26","Made available in DSpace on 2018-09-27T16:17:44Z (GMT). No. of bitstreams: 3 GAO-DISSERTATION-2018.pdf: 745083 bytes, checksum: 7a0f65dda33bda7890444e0cfc35a03b (MD5) LICENSE.txt: 4203 bytes, checksum: 5789b910b66f2cca97d3516d7fdc1b9d (MD5) PROQUEST_LICENSE.txt: 4549 bytes, checksum: 6ed0d0dc126ae16eeb56d703aa479169 (MD5) Previous issue date: 2018-07-10"],"dc:format":["application/pdf"],"dc:identifier":["http://hdl.handle.net/2142/101546"],"dc:language":["en"],"dc:rights":["Copyright 2018 by Li Gao"],"dc:subject":["Noncommutative Euclidean spaces","Moyal Deformation","Pseudo-differential operators"],"dc:title":["On quantum Euclidean spaces: Continuous deformation and pseudo-differential operators"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:24:40Z"}