A new coarsely rigid class of Banach spaces

Abstract

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.

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