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.
Reference:
West, G. 1993. Ideals in von Neumann algebras and in associated operator algebras. University of Cape Town.
Bibliography: pages 127-130.