On coarse geometry of separable dual Banach spaces

TL;DR

This paper investigates the coarse geometric properties of separable dual Banach spaces, establishing non-universality results and distinguishing coarse non-universality from non-equi-coarse embeddings, using novel ultrafilter techniques within ZFC.

Contribution

It introduces new methods using Ramsey ultrafilters to prove coarse non-universality of certain dual Banach spaces without relying on CH, and clarifies the distinction between coarse non-universality and embeddings.

Findings

Proves coarse non-universality for dual spaces with conditional spreading bases.

Establishes coarse non-universality for generalized James and James tree spaces.

Uses ultrafilter techniques to derive results within ZFC, avoiding additional set-theoretic assumptions.

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.