Thomas Schlumprecht, Texas A&M Univesity, College Station, USA: On the coarse embeddability of Hilbert space and the metric characterization of asymptotic properties
Sprache des Titels:
A new concentration inequality is proven for Lipschitz maps on the infinite Hamming graphs taking values into Tsirelson's original space. This concentration inequality is then used to disprove the conjecture, originating in the context of the Coarse Novikov 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. Using part of the proof we also obtain a metric characterization of the property that a Banach space is reflexive and asymptotically $c_0$.
This is joint work with Florent Baudier, Gilles Lancien, and Pavlos Motakis.