Aspects of duality theory for spaces of measurable operators

Simple item page
dc.contributor.advisorConradie, Jurie Jen_ZA
dc.contributor.authorCrowther, Charlotte Louiseen_ZA
dc.date.accessioned2016-01-02T04:38:49Z
dc.date.available2016-01-02T04:38:49Z
dc.date.issued1997en_ZA
dc.descriptionBibliography: pages 98-101.en_ZA
dc.description.abstractIt 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.identifier.apacitationCrowther, 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/11427/16107en_ZA
dc.identifier.chicagocitationCrowther, 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/11427/16107en_ZA
dc.identifier.citationCrowther, C. 1997. Aspects of duality theory for spaces of measurable operators. University of Cape Town.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
dc.identifier.urihttp://hdl.handle.net/11427/16107
dc.identifier.vancouvercitationCrowther 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/11427/16107en_ZA
dc.language.isoengen_ZA
dc.publisher.departmentDepartment of Mathematics and Applied Mathematicsen_ZA
dc.publisher.facultyFaculty of Scienceen_ZA
dc.publisher.institutionUniversity of Cape Town
dc.subject.otherMathematics and Applied Mathematicsen_ZA
dc.titleAspects of duality theory for spaces of measurable operatorsen_ZA
dc.typeDoctoral Thesis
dc.type.qualificationlevelDoctoral
dc.type.qualificationnamePhDen_ZA
uct.type.filetypeText
uct.type.filetypeImage
uct.type.publicationResearchen_ZA
uct.type.resourceThesisen_ZA
Files
Original bundle
Now showing 1 - 1 of 1
Thumbnail Image
Name:
thesis_sci_1997_crowther_charlotte_louise.pdf
Size:
1.9 MB
Format:
Adobe Portable Document Format
Description:
Download
Collections
PhD / Doctoral