Aspects of duality theory for spaces of measurable operators
Doctoral Thesis
1997
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University of Cape Town
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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.
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Bibliography: pages 98-101.
Reference:
Crowther, C. 1997. Aspects of duality theory for spaces of measurable operators. University of Cape Town.