On the coarse geometry of James spaces

TL;DR

This paper demonstrates that Kalton interlaced graphs cannot be embedded into James spaces or their duals in a coarse manner, revealing a new coarse invariant related to the space's asymptotic structure.

Contribution

It establishes the non equire coarse embeddability of Kalton graphs into James spaces and introduces a new coarse invariant for Banach spaces based on this property.

Findings

Kalton interlaced graphs do not embed into James spaces or their duals.

A general non-embeddability result for quasi-reflexive spaces with specific asymptotic structures.

Introduction of a new coarse invariant related to graph embeddability.

Abstract

In this note we prove that the Kalton interlaced graphs do not equi-coarsely embed into the James space $\mathcal J$ nor into its dual $\mathcal J^*$. It is a particular case of a more general result on the non equi-coarse embeddability of the Kalton graphs into quasi-reflexive spaces with a special asymptotic stucture. This allows us to exhibit a coarse invariant for Banach spaces, namely the non equi-coarse embeddability of this family of graphs, which is very close to but different from the celebrated property $\mathcal Q$ of Kalton. We conclude with a remark on the coarse geometry of the James tree space $\mathcal J \mathcal T$ and of its predual.