{"id":{"repo_id":"qu-belfast","oai_identifier":"oai:pure.qub.ac.uk/portal:studenttheses/8b5752eb-db39-425e-a216-15803fd48d4b"},"canonical_url":"https://search.dev.ndltd.org/etd/qu-belfast/oai:pure.qub.ac.uk/portal:studenttheses/8b5752eb-db39-425e-a216-15803fd48d4b","repository":{"repo_id":"qu-belfast","name":"Queen's University Belfast","base_url":"https://pureadmin.qub.ac.uk/ws/oai"},"display":{"title":"Cohomological dimension for C*-algebras","abstract":"We define and explore invariants for C*-algebras that arise as cohomological dimensions for associated categories of operator space modules. The setting of exact categories provides us with a robust framework to utilise homological techniques.<br/><br/>We develop initial global dimension theorems for two of these categories. In the additive category of operator modules over a C*-algebra, equipped with the exact structure of all kernel-cokernel pairs, we show how an extension theorem of Wittstock and a representation theorem of Christensen-Effros-Sinclair can be used to build injective resolutions. From there, we establish a lower bound for the associated global dimension.<br/><br/>We also investigate the sub–exact structure of kernel-cokernel pairs that split as completely bounded linear maps. This provides a new context in which to discuss relative homological algebra for operator modules over a C*-algebra. We provide a proof that the cohomological dimension of this exact category is zero ifand only if the C*-algebra is classically semisimple.","abstract_html":"We define and explore invariants for C*-algebras that arise as cohomological dimensions for associated categories of operator space modules. The setting of exact categories provides us with a robust framework to utilise homological techniques.&lt;br/&gt;&lt;br/&gt;We develop initial global dimension theorems for two of these categories. In the additive category of operator modules over a C*-algebra, equipped with the exact structure of all kernel-cokernel pairs, we show how an extension theorem of Wittstock and a representation theorem of Christensen-Effros-Sinclair can be used to build injective resolutions. From there, we establish a lower bound for the associated global dimension.&lt;br/&gt;&lt;br/&gt;We also investigate the sub–exact structure of kernel-cokernel pairs that split as completely bounded linear maps. This provides a new context in which to discuss relative homological algebra for operator modules over a C*-algebra. We provide a proof that the cohomological dimension of this exact category is zero ifand only if the C*-algebra is classically semisimple.","abstract_has_math":false,"creators":["Rosbotham, Michael"],"institution":"Queen's University Belfast","degree_name":"Doctor of Philosophy","degree_level":"Doctoral Thesis","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["De Chiara, Gabriele"],"committee_chairs":[],"committee_members":[],"year":2021,"date_issued":"2021-12","date_published":"2021-12","updated_at":"2026-07-24T03:56:04Z","subjects":["Operator module","C*-algebra","exact category","dimension","homological algebra","cohomology","operator space"],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["oai:pure.qub.ac.uk/portal:studenttheses/8b5752eb-db39-425e-a216-15803fd48d4b"],"render_values":[{"text":"oai:pure.qub.ac.uk/portal:studenttheses/8b5752eb-db39-425e-a216-15803fd48d4b","href":null,"code":true}]}]},"links":{"outbound_url":"https://pure.qub.ac.uk/en/studentTheses/8b5752eb-db39-425e-a216-15803fd48d4b","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["De Chiara, Gabriele"]},{"key":"dc:contributor.sponsor","label":"Sponsor","values":["Northern Ireland Department for the Economy"]},{"key":"dc:creator","label":"Author","values":["Rosbotham, Michael"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2021-12"]},{"key":"dc:date.issued","label":"Date","values":["2021-12"]},{"key":"dc:publisher.department","label":"Dc Publisher Department","values":["School of Mathematics and Physics"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["Queen's University Belfast"]},{"key":"dc:relation.isreferencedby","label":"Dc Relation Isreferencedby","values":["https://pure.qub.ac.uk/en/studentTheses/8b5752eb-db39-425e-a216-15803fd48d4b"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["Doctoral Thesis"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["Doctor of Philosophy"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Operator module","C*-algebra","exact category","dimension","homological algebra","cohomology","operator space"]}]},{"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":["oai:pure.qub.ac.uk/portal:studenttheses/8b5752eb-db39-425e-a216-15803fd48d4b","https://pure.qub.ac.uk/en/studentTheses/8b5752eb-db39-425e-a216-15803fd48d4b"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://pure.qub.ac.uk/files/250429789/Cohomological_dimension_for_C_star_algebras.pdf"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["We define and explore invariants for C*-algebras that arise as cohomological dimensions for associated categories of operator space modules. The setting of exact categories provides us with a robust framework to utilise homological techniques.<br/><br/>We develop initial global dimension theorems for two of these categories. In the additive category of operator modules over a C*-algebra, equipped with the exact structure of all kernel-cokernel pairs, we show how an extension theorem of Wittstock and a representation theorem of Christensen-Effros-Sinclair can be used to build injective resolutions. From there, we establish a lower bound for the associated global dimension.<br/><br/>We also investigate the sub–exact structure of kernel-cokernel pairs that split as completely bounded linear maps. This provides a new context in which to discuss relative homological algebra for operator modules over a C*-algebra. We provide a proof that the cohomological dimension of this exact category is zero ifand only if the C*-algebra is classically semisimple."]},{"key":"dc:title","label":"Title","values":["Cohomological dimension for C*-algebras"]}]}],"canonical_facts":{"dc:contributor.advisor":["De Chiara, Gabriele"],"dc:contributor.sponsor":["Northern Ireland Department for the Economy"],"dc:creator":["Rosbotham, Michael"],"dc:date":["2021-12"],"dc:date.issued":["2021-12"],"dc:description.abstract":["We define and explore invariants for C*-algebras that arise as cohomological dimensions for associated categories of operator space modules. The setting of exact categories provides us with a robust framework to utilise homological techniques.<br/><br/>We develop initial global dimension theorems for two of these categories. In the additive category of operator modules over a C*-algebra, equipped with the exact structure of all kernel-cokernel pairs, we show how an extension theorem of Wittstock and a representation theorem of Christensen-Effros-Sinclair can be used to build injective resolutions. From there, we establish a lower bound for the associated global dimension.<br/><br/>We also investigate the sub–exact structure of kernel-cokernel pairs that split as completely bounded linear maps. This provides a new context in which to discuss relative homological algebra for operator modules over a C*-algebra. We provide a proof that the cohomological dimension of this exact category is zero ifand only if the C*-algebra is classically semisimple."],"dc:identifier":["oai:pure.qub.ac.uk/portal:studenttheses/8b5752eb-db39-425e-a216-15803fd48d4b","https://pure.qub.ac.uk/en/studentTheses/8b5752eb-db39-425e-a216-15803fd48d4b"],"dc:identifier.uri":["https://pure.qub.ac.uk/files/250429789/Cohomological_dimension_for_C_star_algebras.pdf"],"dc:language":["eng"],"dc:publisher.department":["School of Mathematics and Physics"],"dc:publisher.institution":["Queen's University Belfast"],"dc:relation.isreferencedby":["https://pure.qub.ac.uk/en/studentTheses/8b5752eb-db39-425e-a216-15803fd48d4b"],"dc:subject":["Operator module","C*-algebra","exact category","dimension","homological algebra","cohomology","operator space"],"dc:title":["Cohomological dimension for C*-algebras"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["Doctoral Thesis"],"dc:type.qualificationname":["Doctor of Philosophy"]},"updated_at":"2026-07-24T03:56:04Z"}