[Paper Review] Some results about the Schroeder-Bernstein Property for separable Banach spaces

Valentin Ferenczi, Elói Medina Galego|ArXiv.org|Jun 23, 2004

Advanced Banach Space Theory11 references4 citations

TL;DR

This paper constructs a continuum of mutually non-isomorphic separable Banach spaces that are complemented in each other, demonstrating that the Schroeder-Bernstein Index of each is $2^{\aleph_0}$. The construction builds on Gowers and Maurey's 1997 Banach space and applies descriptive set theory to refine results on the Schroeder-Bernstein Property for spaces with unconditional finite-dimensional Schauder decompositions.

ABSTRACT

We construct a continuum of mutually non-isomorphic separable Banach spaces which are complemented in each other. Consequently, the Schroeder-Bernstein Index of any of these spaces is $2^{\aleph_0}$. Our construction is based on a Banach space introduced by W. T. Gowers and B. Maurey in 1997. We also use classical descriptive set theory methods, as in some work of V. Ferenczi and C. Rosendal, to improve some results of P. G. Casazza and of N. J. Kalton on the Schroeder-Bernstein Property for spaces with an unconditional finite-dimensional Schauder decomposition.

Motivation & Objective

• To resolve the structure of complemented biembeddable families of separable Banach spaces by constructing a continuum of mutually non-isomorphic spaces that are complemented in each other.
• To determine the Schroeder-Bernstein Index (SBi) for certain Banach spaces, particularly showing it can reach $2^{\aleph_0}$.
• To improve existing results on the Schroeder-Bernstein Property (SBP) for Banach spaces with unconditional finite-dimensional Schauder decompositions (UFDD) using descriptive set theory.
• To extend and simplify prior results of Kalton and Casazza on $\kappa$-primary and primary Banach spaces with unconditional bases or UFDDs, achieving uniformity in isomorphism constants.
• To establish that non-perfectly decomposable Banach spaces with unconditional bases or $l_p$-sums of finite-dimensional spaces have the SBP under $\kappa < 2^{\aleph_0}$.

Proposed method

• Constructs a family of Banach spaces using a variant of the Gowers–Maurey space, which is isomorphic to its cube but not its square, to generate complemented biembeddable but non-isomorphic spaces.
• Applies classical descriptive set theory techniques, particularly those from Ferenczi and Rosendal, to analyze isomorphism classes and uniformity in embeddings.
• Uses the notion of $\kappa$-primary spaces and introduces the concept of 'perfectly decomposable' spaces to classify structural complexity in decompositions.
• Employs a family of functions $f_r(x) = \log_2(x+1)^r$ for $r \in [1/2,1]$ to model growth rates in embeddings and verify conditions for complemented biembeddability.
• Applies inequalities involving iterated logarithms and exponentials (e.g., $\log \log \log n \geq 4m^2$) to control growth and ensure the required embedding properties.
• Uses tangent line approximations and inequalities (e.g., $t(x) \geq x/(2(\log_2(x_0+1))^{r+1})$) to bound ratios of norms and verify embedding conditions.

Experimental results

Research questions

• RQ1 Can a continuum of mutually non-isomorphic separable Banach spaces be constructed such that each is complemented in the others?
• RQ2 What is the maximal possible value of the Schroeder-Bernstein Index (SBi) for a separable Banach space?
• RQ3 Does a $\kappa$-primary Banach space with an unconditional basis or UFDD have the SBP restricted to spaces with UFDD when $\kappa < 2^{\aleph_0}$?
• RQ4 Can descriptive set theory methods uniformly control isomorphism constants in the context of complemented biembeddability for UFDD spaces?
• RQ5 Under what conditions does a Banach space with an unconditional basis or $l_p$-sum of finite-dimensional spaces fail to be perfectly decomposable, and how does this affect the SBP?

Key findings

• The authors construct a family of $2^{\aleph_0}$ mutually non-isomorphic separable Banach spaces, each complemented in the others, proving that the Schroeder-Bernstein Index of each is $2^{\aleph_0}$.
• For any $\kappa < 2^{\aleph_0}$, a $\kappa$-primary Banach space with an unconditional basis has the SBP restricted to spaces with an unconditional finite-dimensional Schauder decomposition (UFDD).
• The paper improves on Casazza’s results by showing that an $l_p$-sum of finite-dimensional spaces with $\kappa < 2^{\aleph_0}$ or non-perfectly decomposable structure has the SBP.
• A primary Banach space with a UFDD has the SBP, and this result is extended via descriptive set theory to a broader class of spaces.
• The method yields uniform isomorphism constants for embeddings between spaces with UFDDs, strengthening prior results of Kalton and Casazza.
• The construction relies on a carefully chosen sequence $J = (j_n)$ satisfying $j_1 > 10^{10^{10^{400}}}$ and $j_{n+1} > 10^{10^{10^{4j_n^2}}}$, ensuring the required growth conditions for embedding inequalities.