Geometric Properties of Noncommutative Symmetric Spaces of Measurable Operators and Unitary Matrix Ideals

Start: 02/07/2018 - 4:15pm
End  : 02/07/2018 - 5:15pm


Anna Kaminska (U of Memphis)


This is a survey lecture presenting a number of geometric properties of  noncommutative symmetric spaces of measurable operators $E\Mtau$ and unitary matrix ideals $C_E$, where $\M$ 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 $E\Mtau$  in the case when $E=L_p$ obtaining $L_p\Mtau$ 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 $G\Mtau$ for some Banach function lattice $G$.  We present the results on (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-Nikod\'ym property and stability in the sense of Krivine-Maurey. We also present some open problems.

Freeberg Forum, LC 62, Kravis Center, CMC

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