[Paper Review] Universal objects and associations between classes of Banach spaces and classes of compact spaces

Piotr Koszmider|arXiv (Cornell University)|Sep 19, 2012

Advanced Banach Space Theory43 references5 citations

TL;DR

This paper investigates the existence and nonexistence of universal Banach spaces and compact spaces within various classes, using duality between compact Hausdorff spaces and Banach spaces via C(K) and dual unit balls. It establishes that the nonexistence of universal compact spaces is often easier to prove than for Banach spaces, and shows that under certain set-theoretic assumptions, $\ell_\infty/c_0$ may not contain all Banach spaces of density $2^\omega$ isomorphically, even when some universal objects exist consistently.

ABSTRACT

In the context of classical associations between classes of Banach spaces and classes of compact Hausdorff spaces we survey known results and open questions concerning the existence and nonexistence of universal Banach spaces and of universal compact spaces in various classes. This gives quite a complex network of interrelations which quite often depend on additional set-theoretic assumptions.

Motivation & Objective

• To analyze the interplay between classes of Banach spaces and compact Hausdorff spaces through duality, particularly via $C(K)$ and dual unit balls.
• To clarify the relationships between different notions of universality (isomorphic, isometric, surjective, injective) in these classes.
• To investigate the consistency of universal objects in classes such as Eberlein, Corson, and Asplund-generated spaces under various set-theoretic assumptions.
• To determine when $\ell_\infty/c_0$ serves as a universal space for Banach spaces of density $2^\omega$, and when it fails.
• To identify open problems concerning the existence of universal compact spaces and their implications for universal Banach spaces.

Proposed method

• Utilizes Stone duality between Boolean algebras and totally disconnected compact spaces as a foundational analogy for the duality between compact spaces and Banach spaces.
• Defines and compares notions of association between classes of compact spaces and Banach spaces: $K$-associated, $B$-associated, strongly associated, and associated.
• Applies functional-analytic techniques, including the study of $C(K)$ spaces and dual unit balls $B_{X^*}$ with the weak* topology.
• Employs set-theoretic methods, particularly forcing models, to establish consistency results about embeddings into $\ell_\infty/c_0$.
• Uses Kunen cardinals and $n$-Kunen cardinals to characterize when $C(K)$ spaces fail to embed isomorphically into $\ell_\infty/c_0$, based on combinatorial properties of $\kappa = 2^\omega$.
• Applies Holsztyński’s theorem and results on measure algebras to analyze isometric and isomorphic embeddings into $\ell_\infty/c_0$.

Experimental results

Research questions

• RQ1 Is it consistent that $\ell_\infty/c_0$ is universal for the class of Banach spaces of density $2^\omega$, even if $[0,2^\omega]$ is not a continuous image of $\mathbb{N}^*$?
• RQ2 Can $C([0,2^\omega])$ isomorphically embed into $\ell_\infty/c_0$ without $[0,2^\omega]$ being a continuous image of $\mathbb{N}^*$?
• RQ3 Is it consistent that $\mathbb{N}^*$ is not universal for the class of compact spaces of density $2^\omega$, yet $\ell_\infty/c_0$ is universal for the corresponding class of Banach spaces?
• RQ4 Under what conditions does $C(K)$ fail to embed isomorphically into $\ell_\infty/c_0$ for Corson or uniform Eberlein compacta $K$?
• RQ5 When does the measure algebra’s Stone space $K$ fail to be a continuous image of $\mathbb{N}^*$, yet $C(K) \cong L_\infty([0,1])$ still embeds into $\ell_\infty/c_0$?

Key findings

• It is consistent that there exist universal Banach spaces for the class of spaces of density $2^\omega$, but $\ell_\infty/c_0$ is not among them.
• The nonexistence of universal compact spaces is generally easier to prove than the nonexistence of universal Banach spaces, due to the stronger topological constraints.
• If $2^\omega$ is not an $n$-Kunen cardinal for any $n$, then there exists a uniform Eberlein compact space $K$ such that $C(K)$ does not isomorphically embed into $\ell_\infty/c_0$, even though such embeddings are possible in some models.
• There exists a zero-dimensional compact $K$ such that $[0,2^\omega]$ is not a continuous image of $K$, yet $C([0,2^\omega])$ isomorphically embeds into $C(K)$, showing a gap between topological and Banach space universality.
• The Stone space of the measure algebra (isometric to $L_\infty([0,1])$) may not be a continuous image of $\mathbb{N}^*$, but $C(K)$ still embeds isomorphically into $\ell_\infty/c_0$ via Pe{ł}czy{\i}ski's theorem.
• It is consistent that $\ell_\infty/c_0$ is universal for the class of Banach spaces of density $2^\omega$, even if $[0,2^\omega]$ is not a continuous image of $\mathbb{N}^*$, suggesting that $\ell_\infty/c_0$ can capture universal properties without topological universality.