A new coarsely rigid class of Banach spaces
Abstract
We prove that the class of reflexive asymptotic- Banach spaces is coarsely rigid, meaning that if a Banach space coarsely embeds into a reflexive asymptotic- space , then is also reflexive and asymptotic-. 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- space, we show that this concentration inequality is not equivalent to the non equi-coarse embeddability of the Hamming graphs.
Keywords
Cite
@article{arxiv.1806.00702,
title = {A new coarsely rigid class of Banach spaces},
author = {Florent Baudier and Gilles Lancien and Pavlos Motakis and Thomas Schlumprecht},
journal= {arXiv preprint arXiv:1806.00702},
year = {2020}
}
Comments
v2 discussed 3 topics. The coarse rigidity results have been published in J. Inst. Math. Jussieu and form the content of v3 (now 17 pages). The material related to the geometry of Hamming-type metrics has been reworked and is now submission arXiv:2004.04805. The work on coarse universality has been considerably expanded with new results and is now the stand-alone submission arXiv:2004.04806