{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/86956"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/86956","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Local Structure of Operator Algebras","abstract":"In this thesis some aspects of a local theory for operator algebras are explored. The main purpose is to provide some tools for studying locally compact quantum groups. We first consider inverse limits of $C\\sp*$-algebras (pro-$C\\sp*$-algebras); among them are the multipliers of the Pedersen ideal of a $C\\sp*$-algebra. We distinguish these as locally compact pro-$C\\sp*$-algebras and give a characterization of all locally compact $\\sigma$-$C\\sp*$-algebras. We show that in the commutative case, the locally compact $\\sigma$-$C\\sp*$-algebras are exactly those which correspond to locally compact Hausdorff topological spaces. Also we characterize these multipliers among the elements affiliated with the corresponding $C\\sp*$-algebra. As an application, we prove a version of the generalized Stone's theorem, and apply it to show that certain differential operators are affiliated with the group $C\\sp*$-algebras of Lie groups. Then we turn to inverse limits of $W\\sp*$-algebras and use the techniques of non-commutative topology to study the local structure of Kac algebras. Also we study inverse limits of Kac algebras.","abstract_html":"In this thesis some aspects of a local theory for operator algebras are explored. The main purpose is to provide some tools for studying locally compact quantum groups. We first consider inverse limits of $C\\sp*$-algebras (pro-$C\\sp*$-algebras); among them are the multipliers of the Pedersen ideal of a $C\\sp*$-algebra. We distinguish these as locally compact pro-$C\\sp*$-algebras and give a characterization of all locally compact <span class=\"etd-inline-math\">&sigma;</span>-$C\\sp*$-algebras. We show that in the commutative case, the locally compact <span class=\"etd-inline-math\">&sigma;</span>-$C\\sp*$-algebras are exactly those which correspond to locally compact Hausdorff topological spaces. Also we characterize these multipliers among the elements affiliated with the corresponding $C\\sp*$-algebra. As an application, we prove a version of the generalized Stone&#x27;s theorem, and apply it to show that certain differential operators are affiliated with the group $C\\sp*$-algebras of Lie groups. Then we turn to inverse limits of $W\\sp*$-algebras and use the techniques of non-commutative topology to study the local structure of Kac algebras. Also we study inverse limits of Kac algebras.","abstract_has_math":true,"creators":["Amini, Massoud"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Ruan, Zhong-Jin"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2015,"date_issued":"2015-09-28T15:20:20Z","date_published":"2015-09-28T15:20:20Z","updated_at":"2026-07-22T22:26:28Z","subjects":["Mathematics"],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(MiAaPQ)AAI9834650"],"render_values":[{"text":"(MiAaPQ)AAI9834650","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/86956","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Ruan, Zhong-Jin"]},{"key":"dc:creator","label":"Author","values":["Amini, Massoud"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2015-09-28T15:20:20Z","10000-01-01","1998"]},{"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":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/86956","(MiAaPQ)AAI9834650"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["In this thesis some aspects of a local theory for operator algebras are explored. The main purpose is to provide some tools for studying locally compact quantum groups. We first consider inverse limits of $C\\sp*$-algebras (pro-$C\\sp*$-algebras); among them are the multipliers of the Pedersen ideal of a $C\\sp*$-algebra. We distinguish these as locally compact pro-$C\\sp*$-algebras and give a characterization of all locally compact $\\sigma$-$C\\sp*$-algebras. We show that in the commutative case, the locally compact $\\sigma$-$C\\sp*$-algebras are exactly those which correspond to locally compact Hausdorff topological spaces. Also we characterize these multipliers among the elements affiliated with the corresponding $C\\sp*$-algebra. As an application, we prove a version of the generalized Stone's theorem, and apply it to show that certain differential operators are affiliated with the group $C\\sp*$-algebras of Lie groups. Then we turn to inverse limits of $W\\sp*$-algebras and use the techniques of non-commutative topology to study the local structure of Kac algebras. Also we study inverse limits of Kac algebras.","Made available in DSpace on 2015-09-28T15:20:20Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 9834650.pdf: 4546823 bytes, checksum: 3f5f1e27ad7e223013cab57098aee517 (MD5) Previous issue date: 1998","Embargo set by: Seth Robbins for item 88237 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","86 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1998."]},{"key":"dc:title","label":"Title","values":["Local Structure of Operator Algebras"]}]}],"canonical_facts":{"dc:contributor":["Ruan, Zhong-Jin"],"dc:creator":["Amini, Massoud"],"dc:date":["2015-09-28T15:20:20Z","10000-01-01","1998"],"dc:description":["In this thesis some aspects of a local theory for operator algebras are explored. The main purpose is to provide some tools for studying locally compact quantum groups. We first consider inverse limits of $C\\sp*$-algebras (pro-$C\\sp*$-algebras); among them are the multipliers of the Pedersen ideal of a $C\\sp*$-algebra. We distinguish these as locally compact pro-$C\\sp*$-algebras and give a characterization of all locally compact $\\sigma$-$C\\sp*$-algebras. We show that in the commutative case, the locally compact $\\sigma$-$C\\sp*$-algebras are exactly those which correspond to locally compact Hausdorff topological spaces. Also we characterize these multipliers among the elements affiliated with the corresponding $C\\sp*$-algebra. As an application, we prove a version of the generalized Stone's theorem, and apply it to show that certain differential operators are affiliated with the group $C\\sp*$-algebras of Lie groups. Then we turn to inverse limits of $W\\sp*$-algebras and use the techniques of non-commutative topology to study the local structure of Kac algebras. Also we study inverse limits of Kac algebras.","Made available in DSpace on 2015-09-28T15:20:20Z (GMT). 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