On the coarse geometry of James spaces
Gilles Lancien, Colin Petitjean, Anton\'in Proch\'azka

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 nor into its dual . 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 of Kalton. We conclude with a remark on the coarse geometry of the James tree space and of its predual.
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