Abstract:
In this thesis we study operational quantities characteristic of semi-Fredholm elements relative to a norm-closed ideal in a von Neumann algebra. In recent years, there have been various attempts at generalising various classical results relating to compact operators on a Banach space to the setting of a von Neumann algebra containing a closed ideal. In the study of operators and the quantities associated with them, the concept of stability under small perturbations is central. In this work we mainly concentrate on results relating quantities like the reduced minimum modulus and the lower bound of an element to the study of Fredholm theory. We answer an open question in the affirmative, namely that the reduced essential minimum modulus of an element in a von Neumann algebra relative to a closed ideal is equal to the reduced minimum modulus of the element perturbed by an element from the ideal. As a corollary, we extend some basic perturbation results on semi-Fredholm elements. We also find a complete characterisation of the points of continuity of the reduced essential minimum modulus in terms of Fredholm properties and study the asymptotic behaviour of this quantity. On the other hand it is known in the classical theory of operators on a Hilbert space that the lower bound and the essential lower bound of an operator measures the distance from the operator to the sets of unitary and more generally invertible operators. We study these bounds, by results, connecting the topological divisors of zero with the boundary of the group of invertible elements. We also find necessary and sufficient conditions for regular elements in a von Neumann algebra to be in the closure of the group of invertible elements.
