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Aspects of duality theory for spaces of measurable operators
Bibliography: pages 98-101.
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| Format: | Thesis |
| Language: | English |
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Department of Mathematics and Applied Mathematics
2016
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| _version_ | 1867614444108906496 |
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| access_status_str | Open Access |
| author | Crowther, Charlotte Louise |
| author2 | Conradie, Jurie J |
| author_browse | Conradie, Jurie J Crowther, Charlotte Louise |
| author_facet | Conradie, Jurie J Crowther, Charlotte Louise |
| author_sort | Crowther, Charlotte Louise |
| collection | Thesis |
| description | Bibliography: pages 98-101. |
| format | Thesis |
| id | oai:open.uct.ac.za:11427/16107 |
| institution | University of Cape Town (South Africa) |
| language | eng |
| last_indexed | 2026-06-10T12:52:08.296Z |
| license_str | Not specified — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from UCTD — University of Cape Town Open Access Repository |
| publishDate | 2016 |
| publishDateRange | 2016 |
| publishDateSort | 2016 |
| publisher | Department of Mathematics and Applied Mathematics |
| publisherStr | Department of Mathematics and Applied Mathematics |
| record_format | dspace |
| source_str | UCTD — University of Cape Town Open Access Repository |
| spelling | oai:open.uct.ac.za:11427/16107 Aspects of duality theory for spaces of measurable operators Crowther, Charlotte Louise Conradie, Jurie J Mathematics and Applied Mathematics Bibliography: pages 98-101. It is well known that a commutative von Neumann algebra can be represented as a space of essentially bounded functions over a localizable measure space. In non-commutative integration theory, a von Neumann algebra takes over the role of the space of essentially bounded measurable functions. If the von Neumann algebra is semifinite, then there exists a faithful semifinite normal trace on it. Equipped with such a trace, a topology can be defined on the algebra, which in the commutative case is the familiar topology of convergence in measure. The completion of the algebra with respect to this topology yields an algebra of unbounded operators, the algebra of so-called measurable operators. In the first part of this thesis, the relationship between the nature of the lattice of projections of the von Neumann algebra and the properties of this topology, in particular its local convexity, is investigated. In the duality theory for commutative Banach function spaces, one distinguishes between normal functionals and singular functionals. The study of the former leads to Kothe duality theory. A non-commutative Kothe duality theory already exists and a second aim of this thesis is to initiate a theory for singular functionals in the noncommutative setting. As a preparation for this, singular functionals are characterised in several ways in the commutative case and one of these is used as definition for singular functionals on Banach spaces of measurable operators. The known association between singular functionals and the subspace of elements with order continuous norm in a Banach function space is extended to the non-commutative setting. Finally, duality for the space of measurable operators equipped with the measure topology is investigated. Its Kothe dual is first characterised, and then singular functionals on this space are investigated. In certain cases a full characterisation of the continuous dual is given. 2016-01-02T04:38:49Z 2016-01-02T04:38:49Z 1997 Doctoral Thesis Doctoral PhD http://hdl.handle.net/11427/16107 eng application/pdf Department of Mathematics and Applied Mathematics Faculty of Science University of Cape Town |
| spellingShingle | Mathematics and Applied Mathematics Crowther, Charlotte Louise Aspects of duality theory for spaces of measurable operators |
| thesis_degree_str | Doctoral |
| title | Aspects of duality theory for spaces of measurable operators |
| title_full | Aspects of duality theory for spaces of measurable operators |
| title_fullStr | Aspects of duality theory for spaces of measurable operators |
| title_full_unstemmed | Aspects of duality theory for spaces of measurable operators |
| title_short | Aspects of duality theory for spaces of measurable operators |
| title_sort | aspects of duality theory for spaces of measurable operators |
| topic | Mathematics and Applied Mathematics |
| url | http://hdl.handle.net/11427/16107 |
| work_keys_str_mv | AT crowthercharlottelouise aspectsofdualitytheoryforspacesofmeasurableoperators |