[Paper Review] Geometric properties of noncommutative symmetric spaces of measurable operators and unitary matrix ideals

Małgorzata Czerwiñska, Anna Kamińska|arXiv (Cornell University)|Apr 6, 2017

Advanced Operator Algebra Research58 references4 citations

TL;DR

This paper investigates geometric properties of noncommutative symmetric spaces of measurable operators $E(\mathcal{M},\tau)$ and their unitary matrix analogues $C_E$, establishing connections between properties of the underlying symmetric space $E$ and the corresponding operator space. The key contribution is the characterization of stability in the sense of Krivine-Maurey for $C_E$ in terms of the stability of $E$, proving that $E$ is stable if and only if $C_E$ is stable, extending earlier results on $L_p$ spaces and providing a foundational link between sequence space and operator space geometry.

ABSTRACT

This is a survey article of geometric properties of noncommutative symmetric spaces of measurable operators $E(\mathcal{M},τ)$, where $\mathcal{M}$ is a semifinite von Neumann algebra with a faithful, normal, semifinite trace $τ$, and $E$ is a symmetric function space. If $E\subset c_0$ is a symmetric sequence space then the analogous properties in the unitary matrix ideals $C_E$ are also presented. In the preliminaries we provide basic definitions and concepts illustrated by some examples and occasional proofs. In particular we list and discuss the properties of general singular value function, submajorization in the sense of Hardy, Littlewood and Pólya, Köthe duality, the spaces $L_p(\mathcal{M},τ)$, $1\le p

Motivation & Objective

• To establish a comprehensive framework for analyzing geometric properties of noncommutative symmetric spaces of measurable operators $E(\mathcal{M},\tau)$ and unitary matrix ideals $C_E$.
• To clarify the relationship between geometric and structural properties of the symmetric space $E$ and the induced properties in $E(\mathcal{M},\tau)$ and $C_E$.
• To resolve open problems concerning stability, Kadec-Klee properties, and the Radon-Nikodým property in noncommutative settings.
• To extend classical results from $L_p$ and $\ell_p$ spaces to general symmetric spaces and their noncommutative analogues.

Proposed method

• Utilizes the theory of singular value functions and submajorization in the sense of Hardy, Littlewood, and Pólya to define and analyze the structure of $E(\mathcal{M},\tau)$ and $C_E$.
• Applies Köthe duality and the identification of $C_E$ with a generalized noncommutative $L_p$-type space $G(B(H),\mathrm{tr})$ to transfer properties from function spaces to operator spaces.
• Employs trace-preserving $*$-isomorphisms and the removal of non-atomicity assumptions to generalize results to arbitrary semifinite von Neumann algebras.
• Applies ultrafilter-based definitions and criteria for Krivine-Maurey stability, particularly using the condition $\inf_{n>m}\|x_n + x_m\| \leq \sup_{n<m}\|x_n + x_m\|$, to characterize stability in $C_E$.
• Leverages results from Arazy and Raynaud on basic sequences in $C_E$, showing that every basic sequence has a subsequence equivalent to one in $\ell_2 \oplus E$, enabling reduction of complex properties to the sequence space level.
• Uses duality, monotonicity, and local uniform convexity techniques to analyze complex and real convexity, smoothness, and exposed point structures in $E(\mathcal{M},\tau)$ and $C_E$.

Experimental results

Research questions

• RQ1 Under what conditions on the symmetric space $E$ is the unitary matrix ideal $C_E$ stable in the sense of Krivine-Maurey?
• RQ2 How do geometric properties such as strict convexity, uniform convexity, and the Kadec-Klee property transfer from $E$ to $C_E$?
• RQ3 What is the relationship between the Radon-Nikodým property and stability in noncommutative symmetric spaces $E(\mathcal{M},\tau)$?
• RQ4 When does $E(\mathcal{M},\tau)$ inherit the Banach-Saks property from $E$?
• RQ5 Is the stability of $E$ sufficient for the stability of $E(\mathcal{M},\tau)$ when $\mathcal{M}$ is of type I?

Key findings

• The space $C_E$ is stable in the sense of Krivine-Maurey if and only if the symmetric sequence space $E$ is stable, establishing a fundamental equivalence between the stability of the sequence space and its unitary matrix ideal.
• If $E$ is stable, then $C_E$ does not contain a subspace isomorphic to $c_0$, and its shell decomposition is boundedly complete.
• The space $E(\mathcal{M},\tau)$ inherits the Radon-Nikodým property from $E$ if $E$ has the RNP, and this property is preserved under the $L_p$-type construction.
• For $\mathcal{M}$ of type I, $L_p(\mathcal{M},\tau)$ is stable for all $1 \leq p < \infty$, and this stability is equivalent to $\mathcal{M}$ being of type I.
• The Banach-Saks property holds in $C_E$ if and only if it holds in $E$, and this is equivalent to $E$ being $p$-convex for some $p > 1$.
• The space $E(\mathcal{M},\tau)$ satisfies the uniform Kadec-Klee property if and only if $E$ satisfies the same property, and this is linked to the local uniform convexity of $E$.