Ideals in von Neumann algebras and in associated operator algebras

 

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dc.contributor.advisor Conradie, Jurie en_ZA
dc.contributor.author West, Graeme Philip en_ZA
dc.date.accessioned 2015-12-28T06:03:42Z
dc.date.available 2015-12-28T06:03:42Z
dc.date.issued 1993 en_ZA
dc.identifier.citation West, G. 1993. Ideals in von Neumann algebras and in associated operator algebras. University of Cape Town. en_ZA
dc.identifier.uri http://hdl.handle.net/11427/15968
dc.description Bibliography: pages 127-130. en_ZA
dc.description.abstract The compact operators on a Hilbert space are those operators for which the image of the unit ball is relatively compact in the norm topology. These operators form an ideal, in the algebra of all continuous linear operators on the Hilbert space, which is closed in the uniform norm. In the case that the underlying Hilbert space is separable this is the only such ideal, while for non-separable Hilbert spaces the norm-closed ideals are easily characterised by means of cardinal numbers. The algebra of all continuous linear operators on a Hilbert space is a specific example of a van Neumann algebra, and the theory of compact operators and the ideal they form admit certain generalisations to van Neumann algebras. One of the characterisations of the ideal of compact operators is that it is the closure of the ideal of finite rank operators, and hence the closed ideal generated by the finite dimensional projections. Kaftal has considered the ideal of so called algebraically compact operators, which is defined to be the closed ideal generated by the algebraically finite projections in the von Neumann algebra, and has shown that this ideal consists of those operators which map the unit ball to sets which have compact-like properties. This characterisation was generalised to arbitrary norm-closed ideals by Stroh. In this thesis we explore the extent to which norm-closed ideals in van Neumann algebras resemble the ideal of compact operators on a Hilbert space. We extend the theory developed by Kaftal and Stroh, and show that arbitrary ideals in van Neumann algebras can be characterised in terms of homologies and topologies. We also consider continuity characterisations of norm-closed ideals in von Neumann algebras, generalising the characterisation of the compact operators as being those that are continuous from the unit ball equipped with the weak topology, to the Hilbert space equipped with the norm topology. Furthermore we briefly consider sequential continuity characterisations as first analysed by Kaftal in the case of the algebraically compact ideal. Finally, in the case of a semifinite von Neumann algebra equipped with a faithful semifinite normal trace T, we generalise the characterisation of the compact operators given in terms of the singular value sequence, by showing that the ideal of T-measurable operators whose generalised singular function decreases to 0 possess many of the same properties as the ideal of compact operators. en_ZA
dc.language.iso eng en_ZA
dc.subject.other Mathematics and Applied Mathematics en_ZA
dc.title Ideals in von Neumann algebras and in associated operator algebras 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 West, G. P. (1993). <i>Ideals in von Neumann algebras and in associated operator algebras</i>. (Thesis). University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics. Retrieved from http://hdl.handle.net/11427/15968 en_ZA
dc.identifier.chicagocitation West, Graeme Philip. <i>"Ideals in von Neumann algebras and in associated operator algebras."</i> Thesis., University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics, 1993. http://hdl.handle.net/11427/15968 en_ZA
dc.identifier.vancouvercitation West GP. Ideals in von Neumann algebras and in associated operator algebras. [Thesis]. University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics, 1993 [cited yyyy month dd]. Available from: http://hdl.handle.net/11427/15968 en_ZA
dc.identifier.ris TY - Thesis / Dissertation AU - West, Graeme Philip AB - The compact operators on a Hilbert space are those operators for which the image of the unit ball is relatively compact in the norm topology. These operators form an ideal, in the algebra of all continuous linear operators on the Hilbert space, which is closed in the uniform norm. In the case that the underlying Hilbert space is separable this is the only such ideal, while for non-separable Hilbert spaces the norm-closed ideals are easily characterised by means of cardinal numbers. The algebra of all continuous linear operators on a Hilbert space is a specific example of a van Neumann algebra, and the theory of compact operators and the ideal they form admit certain generalisations to van Neumann algebras. One of the characterisations of the ideal of compact operators is that it is the closure of the ideal of finite rank operators, and hence the closed ideal generated by the finite dimensional projections. Kaftal has considered the ideal of so called algebraically compact operators, which is defined to be the closed ideal generated by the algebraically finite projections in the von Neumann algebra, and has shown that this ideal consists of those operators which map the unit ball to sets which have compact-like properties. This characterisation was generalised to arbitrary norm-closed ideals by Stroh. In this thesis we explore the extent to which norm-closed ideals in van Neumann algebras resemble the ideal of compact operators on a Hilbert space. We extend the theory developed by Kaftal and Stroh, and show that arbitrary ideals in van Neumann algebras can be characterised in terms of homologies and topologies. We also consider continuity characterisations of norm-closed ideals in von Neumann algebras, generalising the characterisation of the compact operators as being those that are continuous from the unit ball equipped with the weak topology, to the Hilbert space equipped with the norm topology. Furthermore we briefly consider sequential continuity characterisations as first analysed by Kaftal in the case of the algebraically compact ideal. Finally, in the case of a semifinite von Neumann algebra equipped with a faithful semifinite normal trace T, we generalise the characterisation of the compact operators given in terms of the singular value sequence, by showing that the ideal of T-measurable operators whose generalised singular function decreases to 0 possess many of the same properties as the ideal of compact operators. DA - 1993 DB - OpenUCT DP - University of Cape Town LK - https://open.uct.ac.za PB - University of Cape Town PY - 1993 T1 - Ideals in von Neumann algebras and in associated operator algebras TI - Ideals in von Neumann algebras and in associated operator algebras UR - http://hdl.handle.net/11427/15968 ER - en_ZA


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