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On coarse geometry of separable dual Banach spaces

Stephen Jackson, Cory Krause, Bunyamin Sari

TL;DR

The paper develops obstructions to coarse universality for separable dual Banach spaces, focusing on duals with spreading bases and James-type constructions. Central to the approach is an asymptotic linearization principle that, together with Ramsey ultrafilter techniques, yields non-embedding and rigidity results, including the non-embeddability of $c_0$ and Kalton graphs in Schreier, James, and James-tree spaces. The authors provide both qualitative, ultrafilter-based arguments and quantitative, Ramsey-free variants, establishing a broad framework that explains why a wide class of dual spaces fail to be coarsely universal. A key contribution is the finite-branch reduction for generalized James tree spaces, enabling robust non-embedding conclusions across a broad family of asymptotic structures. The work unifies and extends several known results while offering new absoluteness-based proofs that do not rely on CH.

Abstract

We study the obstructions to coarse universality in separable dual Banach spaces. We prove coarse non-universality of several classes of dual spaces, including those with conditional spreading bases, as well as generalized James and James tree spaces. We also give quantitative counterparts of some of the results, clarifying the distinction between coarse non-universality and the non-equi-coarse embeddings of the Kalton graphs. Unique to our approach is the use of a Ramsey ultrafilter. While the existence of such ultrafilters typically requires $\mathsf{CH}$, we are able to show that the conclusions of our theorems follow from $\mathsf{ZFC}$, alone via an absoluteness argument. Finally, we also show how our techniques can be used to prove various previously known results in the literature.

On coarse geometry of separable dual Banach spaces

TL;DR

The paper develops obstructions to coarse universality for separable dual Banach spaces, focusing on duals with spreading bases and James-type constructions. Central to the approach is an asymptotic linearization principle that, together with Ramsey ultrafilter techniques, yields non-embedding and rigidity results, including the non-embeddability of and Kalton graphs in Schreier, James, and James-tree spaces. The authors provide both qualitative, ultrafilter-based arguments and quantitative, Ramsey-free variants, establishing a broad framework that explains why a wide class of dual spaces fail to be coarsely universal. A key contribution is the finite-branch reduction for generalized James tree spaces, enabling robust non-embedding conclusions across a broad family of asymptotic structures. The work unifies and extends several known results while offering new absoluteness-based proofs that do not rely on CH.

Abstract

We study the obstructions to coarse universality in separable dual Banach spaces. We prove coarse non-universality of several classes of dual spaces, including those with conditional spreading bases, as well as generalized James and James tree spaces. We also give quantitative counterparts of some of the results, clarifying the distinction between coarse non-universality and the non-equi-coarse embeddings of the Kalton graphs. Unique to our approach is the use of a Ramsey ultrafilter. While the existence of such ultrafilters typically requires , we are able to show that the conclusions of our theorems follow from , alone via an absoluteness argument. Finally, we also show how our techniques can be used to prove various previously known results in the literature.
Paper Structure (16 sections, 34 theorems, 136 equations)

This paper contains 16 sections, 34 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Coarse embeddings
  4. Ramsey ultrafilters and the absoluteness principle
  5. Coarse non-universality of Schreier metric spaces $S_{\alpha}({\mathbb Z})$
  6. Asymptotic linearization
  7. Applications to non-embeddings and rigidity
  8. Coarse non-universality of dual spaces with spreading bases
  9. Coarse non-universality of quasi-reflexive spaces
  10. Non-embedding of the Kalton graphs into the James space
  11. Coarse non-universality of dual James spaces
  12. Non-embedding of the Kalton graphs into generalized James tree spaces
  13. Reduction of embeddings to a finite set of branches
  14. Pigeonhole lemma and stabilization on a finite set of branches
  15. Proof of Theorem \ref{['tree-space']}
  16. ...and 1 more sections

Key Result

Lemma 3

Let $X$ be a separable Banach space. Let $x \in \omega^\omega$ be a code for $X$. Then the statement "there is a coarse embedding of $c_0$ into $X$" is a $\Sigma^1_1(x)$ statement.

Theorems & Definitions (63)

  • proof
  • proof : Proof (sketch)
  • Lemma 3
  • proof
  • Theorem 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • ...and 53 more