{"id":{"repo_id":"lancaster","oai_identifier":"oai:eprints.lancs.ac.uk:79356"},"canonical_url":"https://search.dev.ndltd.org/etd/lancaster/oai:eprints.lancs.ac.uk:79356","repository":{"repo_id":"lancaster","name":"Lancaster University","base_url":"https://eprints.lancs.ac.uk/cgi/oai2"},"display":{"title":"Discontinuous homomorphisms from Banach algebras of operators","abstract":"The relationship between a Banach space X and its Banach algebra of bounded operators B(X) is rich and complex; this is especially so for non-classical Banach spaces. In this thesis we consider questions of the following form: does there exist a Banach space X such that B(X) has a particular (Banach algebra) property? If not, is there a quotient of B(X) with the property? The first of these is the uniqueness-of-norm problem for Calkin algebras: does there exist a Banach space whose Calkin algebra lacks a unique complete norm? We show that there does indeed exist such a space, answering a classical open question [101]. Secondly, we turn our attention to splittings of extensions of Banach algebras. Work of Bade, Dales and Lykova [12] inspired the problem of whether there exist a Banach space X and an extension of B(X) which splits algebraically but not strongly; this asks for a special type of discontinuous homomorphism from B(X). Using the categorical notion of a pullback we obtain, jointly with N. J. Laustsen [71], new general results about extensions and prove that such a space exists. The same space is used to answer our third question, which goes back to Helemskii, in the positive: is there a Banach space X such that B(X) has homological bidimension at least two? The proof uses techniques developed (with N. J. Laustsen [71]) during the solution to the second question. We use two main Banach spaces to answer our questions. One is due to Read [90], the other to Argyros and Motakis [8]; the former plays a much more prominent role. Together with Laustsen [72], we prove a major original result about Read’s space which allows for the new applications. The conclusion of the thesis examines a class of operators on Banach spaces which have previously received little attention; these are a weak analogue of inessential operators.","abstract_html":"The relationship between a Banach space X and its Banach algebra of bounded operators B(X) is rich and complex; this is especially so for non-classical Banach spaces. In this thesis we consider questions of the following form: does there exist a Banach space X such that B(X) has a particular (Banach algebra) property? If not, is there a quotient of B(X) with the property? The first of these is the uniqueness-of-norm problem for Calkin algebras: does there exist a Banach space whose Calkin algebra lacks a unique complete norm? We show that there does indeed exist such a space, answering a classical open question [101]. Secondly, we turn our attention to splittings of extensions of Banach algebras. Work of Bade, Dales and Lykova [12] inspired the problem of whether there exist a Banach space X and an extension of B(X) which splits algebraically but not strongly; this asks for a special type of discontinuous homomorphism from B(X). Using the categorical notion of a pullback we obtain, jointly with N. J. Laustsen [71], new general results about extensions and prove that such a space exists. The same space is used to answer our third question, which goes back to Helemskii, in the positive: is there a Banach space X such that B(X) has homological bidimension at least two? The proof uses techniques developed (with N. J. Laustsen [71]) during the solution to the second question. We use two main Banach spaces to answer our questions. One is due to Read [90], the other to Argyros and Motakis [8]; the former plays a much more prominent role. Together with Laustsen [72], we prove a major original result about Read’s space which allows for the new applications. The conclusion of the thesis examines a class of operators on Banach spaces which have previously received little attention; these are a weak analogue of inessential operators.","abstract_has_math":false,"creators":["Skillicorn, Richard","Laustsen, Niels"],"institution":"Lancaster University","degree_name":"Ph.D.","degree_level":"doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2016,"date_issued":"2016","date_published":"2016","updated_at":"2026-07-24T02:48:27Z","subjects":[],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":null,"outbound_label":null,"outbound_source":null},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Skillicorn, Richard","Laustsen, Niels"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2016"]},{"key":"dc:date.issued","label":"Date","values":["2016"]},{"key":"dc:publisher.commercial","label":"Dc Publisher Commercial","values":["Lancaster University"]},{"key":"dc:publisher.department","label":"Dc Publisher Department","values":["Mathematics and Statistics"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["Lancaster University"]},{"key":"dc:relation.isreferencedby","label":"Dc Relation Isreferencedby","values":["https://eprints.lancs.ac.uk/id/eprint/79356/"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["doctoral"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["Ph.D."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://eprints.lancs.ac.uk/id/eprint/79356/1/2016skillicornphd.pdf"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The relationship between a Banach space X and its Banach algebra of bounded operators B(X) is rich and complex; this is especially so for non-classical Banach spaces. In this thesis we consider questions of the following form: does there exist a Banach space X such that B(X) has a particular (Banach algebra) property? If not, is there a quotient of B(X) with the property? The first of these is the uniqueness-of-norm problem for Calkin algebras: does there exist a Banach space whose Calkin algebra lacks a unique complete norm? We show that there does indeed exist such a space, answering a classical open question [101]. Secondly, we turn our attention to splittings of extensions of Banach algebras. Work of Bade, Dales and Lykova [12] inspired the problem of whether there exist a Banach space X and an extension of B(X) which splits algebraically but not strongly; this asks for a special type of discontinuous homomorphism from B(X). Using the categorical notion of a pullback we obtain, jointly with N. J. Laustsen [71], new general results about extensions and prove that such a space exists. The same space is used to answer our third question, which goes back to Helemskii, in the positive: is there a Banach space X such that B(X) has homological bidimension at least two? The proof uses techniques developed (with N. J. Laustsen [71]) during the solution to the second question. We use two main Banach spaces to answer our questions. One is due to Read [90], the other to Argyros and Motakis [8]; the former plays a much more prominent role. Together with Laustsen [72], we prove a major original result about Read’s space which allows for the new applications. The conclusion of the thesis examines a class of operators on Banach spaces which have previously received little attention; these are a weak analogue of inessential operators."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Discontinuous homomorphisms from Banach algebras of operators"]}]}],"canonical_facts":{"dc:creator":["Skillicorn, Richard","Laustsen, Niels"],"dc:date":["2016"],"dc:date.issued":["2016"],"dc:description.abstract":["The relationship between a Banach space X and its Banach algebra of bounded operators B(X) is rich and complex; this is especially so for non-classical Banach spaces. In this thesis we consider questions of the following form: does there exist a Banach space X such that B(X) has a particular (Banach algebra) property? If not, is there a quotient of B(X) with the property? The first of these is the uniqueness-of-norm problem for Calkin algebras: does there exist a Banach space whose Calkin algebra lacks a unique complete norm? We show that there does indeed exist such a space, answering a classical open question [101]. Secondly, we turn our attention to splittings of extensions of Banach algebras. Work of Bade, Dales and Lykova [12] inspired the problem of whether there exist a Banach space X and an extension of B(X) which splits algebraically but not strongly; this asks for a special type of discontinuous homomorphism from B(X). Using the categorical notion of a pullback we obtain, jointly with N. J. Laustsen [71], new general results about extensions and prove that such a space exists. The same space is used to answer our third question, which goes back to Helemskii, in the positive: is there a Banach space X such that B(X) has homological bidimension at least two? The proof uses techniques developed (with N. J. Laustsen [71]) during the solution to the second question. We use two main Banach spaces to answer our questions. One is due to Read [90], the other to Argyros and Motakis [8]; the former plays a much more prominent role. Together with Laustsen [72], we prove a major original result about Read’s space which allows for the new applications. The conclusion of the thesis examines a class of operators on Banach spaces which have previously received little attention; these are a weak analogue of inessential operators."],"dc:format":["application/pdf"],"dc:identifier.uri":["https://eprints.lancs.ac.uk/id/eprint/79356/1/2016skillicornphd.pdf"],"dc:publisher.commercial":["Lancaster University"],"dc:publisher.department":["Mathematics and Statistics"],"dc:publisher.institution":["Lancaster University"],"dc:relation.isreferencedby":["https://eprints.lancs.ac.uk/id/eprint/79356/"],"dc:title":["Discontinuous homomorphisms from Banach algebras of operators"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["doctoral"],"dc:type.qualificationname":["Ph.D."]},"updated_at":"2026-07-24T02:48:27Z"}