Aspects of spectral theory for algebras of measurable operators
Doctoral Thesis
2008
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University of Cape Town
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Abstract
The spectral theory for bounded normal operators on a Hilbert space and the various functional calculi for such operators is closely related to the representation theory of commutative C*- and von Neumann algebras as algebras of bounded continuous or measurable functions. For unbounded operators the corresponding theory leads to algebras of unbounded densely defined operators. The thesis looks at aspects of spectral theory in the non-commutative generalisations of these algebras. Given a von Neumann algebra M, there are various notions of measurability for operators affiliated with M, and the measurable operators of a particular kind form an involutive algebra under the strong sum and product. Algebras of this kind can usually be equipped with a topology modelled on the topology of convergence in measure under which they become topological algebras. The emphasis in this thesis is on a semifinite von Neumann algebra M equipped with a semi-finite faithful normal trace τ and the corresponding algebra M~ of τ-measurable operators.
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Includes abstract.
Includes bibliographical references (p. 124-129).
Reference:
Tembo, I. 2008. Aspects of spectral theory for algebras of measurable operators. University of Cape Town.