This is an announcement for the paper “Not every infinite dimensional Banach space coarsely contains Hilbert space” by Florent Baudierhttps://arxiv.org/find/math/1/au:+Baudier_F/0/1/0/all/0/1, Gilles Lancienhttps://arxiv.org/find/math/1/au:+Lancien_G/0/1/0/all/0/1, Thomas Schlumprechthttps://arxiv.org/find/math/1/au:+Schlumprecht_T/0/1/0/all/0/1.

Abstract: In this article a new concentration inequality is proven for Lipschitz maps on the infinite Hamming graphs and taking values in Tsirelson's original space. This concentration inequality is then used to disprove the conjecture that the separable infinite dimensional Hilbert space coarsely embeds into every infinite dimensional Banach space. Some positive embeddability results are proven for the infinite Hamming graphs and the countably branching trees using the theory of spreading models. A purely metric characterization of finite dimensionality is also obtained, as well as a rigidity result pertaining to the spreading model set for Banach spaces coarsely embeddable into Tsirelson's original space.

The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1705.06797