Aspects of duality theory for spaces of measurable operators

 

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dc.contributor.advisor Conradie, Jurie J en_ZA
dc.contributor.author Crowther, Charlotte Louise en_ZA
dc.date.accessioned 2016-01-02T04:38:49Z
dc.date.available 2016-01-02T04:38:49Z
dc.date.issued 1997 en_ZA
dc.identifier.citation Crowther, C. 1997. Aspects of duality theory for spaces of measurable operators. University of Cape Town. en_ZA
dc.identifier.uri http://hdl.handle.net/11427/16107
dc.description Bibliography: pages 98-101. en_ZA
dc.description.abstract 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. en_ZA
dc.language.iso eng en_ZA
dc.subject.other Mathematics and Applied Mathematics en_ZA
dc.title Aspects of duality theory for spaces of measurable operators en_ZA
dc.type Doctoral Thesis
uct.type.publication Research en_ZA
uct.type.resource Thesis en_ZA
dc.publisher.institution University of Cape Town
dc.publisher.faculty Faculty of Science en_ZA
dc.publisher.department Department of Mathematics and Applied Mathematics en_ZA
dc.type.qualificationlevel Doctoral
dc.type.qualificationname PhD en_ZA
uct.type.filetype Text
uct.type.filetype Image
dc.identifier.apacitation Crowther, C. L. (1997). <i>Aspects of duality theory for spaces of measurable operators</i>. (Thesis). University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics. Retrieved from http://hdl.handle.net/11427/16107 en_ZA
dc.identifier.chicagocitation Crowther, Charlotte Louise. <i>"Aspects of duality theory for spaces of measurable operators."</i> Thesis., University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics, 1997. http://hdl.handle.net/11427/16107 en_ZA
dc.identifier.vancouvercitation Crowther CL. Aspects of duality theory for spaces of measurable operators. [Thesis]. University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics, 1997 [cited yyyy month dd]. Available from: http://hdl.handle.net/11427/16107 en_ZA
dc.identifier.ris TY - Thesis / Dissertation AU - Crowther, Charlotte Louise AB - 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. DA - 1997 DB - OpenUCT DP - University of Cape Town LK - https://open.uct.ac.za PB - University of Cape Town PY - 1997 T1 - Aspects of duality theory for spaces of measurable operators TI - Aspects of duality theory for spaces of measurable operators UR - http://hdl.handle.net/11427/16107 ER - en_ZA


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