Not every infinite dimensional Banach space coarsely contains Hilbert space

Florent Baudier, Gilles Lancien, Thomas Schlumprecht

Published 2017-05-18Version 1

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.