Date of Award
Doctor of Philosophy
Symmetric spaces of measurable operators E(M,τ), known also as noncommutative symmetric spaces, were introduced first by Ovcinnikov in 1970. They emerged as a generalization of the theory of unitary matrix spaces introduced by Schatten in sixties, as well as the theory of noncommutative Lpspaces introduced by Segal and Dixmier in the early fifties. Their study provides a unified approach to the theory of ideals of compact operators in Hilbert spaces due to Schatten, and to the classical theory of rearrangement invariant Banach function spaces. With the development of noncommutative theory, it was natural to expect the space E(M,τ) to reflect the properties of the corresponding symmetric function space E. Establishing those lifting-type results from the space E to E(M,τ) effectively reduces the study on geometric structures in noncommutative settings, to the corresponding questions in symmetric spaces of measurable functions.In this dissertation we explore strongly extreme points, complex extreme points, points of complex local uniform rotundity, smooth points, strongly smooth points of the unit ball in E(M,τ) and their global counterparts, midpoint local uniform rotundity, complex rotundity, complex local uniform rotundity, smoothness, Fréchet smoothness, respectively. Moreover, we investigate exposed and strongly exposed points in E(M,τ).
Dissertation or thesis originally submitted to the local University of Memphis Electronic Theses & dissertation (ETD) Repository.
Czerwinska, Malgorzata Marta, "Geometric Properties of Symmetric Spaces of Measurable Operators." (2011). Electronic Theses and Dissertations. 193.