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A NEW COARSELY RIGID CLASS OF BANACH SPACES
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 20 / Issue 5 / September 2021
- Published online by Cambridge University Press:
- 13 January 2020, pp. 1729-1747
- Print publication:
- September 2021
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- Article
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- You have access
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We prove that the class of reflexive asymptotic-
$c_{0}$ Banach spaces is coarsely rigid, meaning that if a Banach space 
$X$ coarsely embeds into a reflexive asymptotic-
$c_{0}$ space 
$Y$, then 
$X$ is also reflexive and asymptotic-
$c_{0}$. In order to achieve this result, we provide a purely metric characterization of this class of Banach spaces. This metric characterization takes the form of a concentration inequality for Lipschitz maps on the Hamming graphs, which is rigid under coarse embeddings. Using an example of a quasi-reflexive asymptotic-
$c_{0}$ space, we show that this concentration inequality is not equivalent to the non-equi-coarse embeddability of the Hamming graphs.
