Symmetric spaces of measurable operators E([special characters omitted], τ), known also as noncommutative symmetric spaces, were introduced first by Ovčinnikov in 1970 [58, 59]. They emerged as a generalization of the theory of unitary matrix spaces introduced by Schatten in sixties [71], as well as the theory of noncommutative L_p spaces introduced by Segal and Dixmier in the early fifties [20, 66]. Their study provides a unified approach to the theory of ideals of compact operators in Hilbert space due to Schatten [71], and to the classical theory of rearrangement invariant Banach function spaces [4, 51]. With the development of noncommutative theory, it was natural to expect the space E([special characters omitted], τ) to reflect the properties of the corresponding symmetric function space E. Establishing those lifting-type results from the space E to E([special characters omitted], 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([special characters omitted], 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([special characters omitted], τ).