Title: Geometric properties of noncommutative symmetric spaces of measurable operators and unitary matrix ideals
Speaker: Anna Kaminska (University of Memphis)
Abstract: This is a survey lecture presenting a number of geometric properties of noncommutative symmetric spaces of measurable operators $E(\mathcal{M},\tau)$ and unitary matrix ideals $C_E$, where $\mathcal{M}$ is a von Neumann algebra with a semi-finite, faithful and normal trace $\tau$, and $E$ is a (quasi)Banach function and a sequence lattice, respectively. We provide auxiliary definitions, notions, examples and we discuss a number of properties that are most often used in studies of local and global geometry of (quasi) Banach spaces. We interpret the general spaces $E(\mathcal{M},\tau)$ in the case when $E=L_p$ obtaining $L_p(\mathcal{M},\tau)$ spaces, and in the case when $E$ is a sequence space we explain how the unitary matrix space $C_E$ can be in fact identified with the symmetric space of measurable operators $G(\mathcal{M},\tau)$ for some Banach function lattice $G$. We present the results on some of the following properties: (complex) extreme points, (complex) strict convexity, strong extreme points and midpoint local uniform convexity, $k$-extreme points and $k$-convexity, (complex or local) uniform convexity, smoothness and strong smoothness, (strongly) exposed points, (uniform) Kadec-Klee properties, Banach-Saks properties, Radon-Nikodym property and stability in the sense of Krivine-Maurey.
