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On the Coarse Geometry of James Spaces
- Part of
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- Journal:
- Canadian Mathematical Bulletin / Volume 63 / Issue 1 / March 2020
- Published online by Cambridge University Press:
- 16 December 2019, pp. 77-93
- Print publication:
- March 2020
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- Article
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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 
${\mathcal{J}}^{\ast }$. 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 structure. 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.
