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Analysis and Operator Theory Seminar - Anna Kaminska

Anna Kaminska
October 16, 2018
11:00AM - 11:55AM
Cockins Hall 240

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Add to Calendar 2018-10-16 11:00:00 2018-10-16 11:55:00 Analysis and Operator Theory Seminar - Anna Kaminska 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. Cockins Hall 240 Department of Mathematics math@osu.edu America/New_York public

Title: Geometric properties of noncommutative symmetric spaces of measurable operators and unitary matrix ideals

SpeakerAnna 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.

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