R ESEARCH T ERM
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M ETRIC S PACES
http:// www.icmat.es/ RT/ AGMS2015/ seminars.php
Junior Seminar 13
Wednesday, June 10th, 2015
16:30 h., Aula Naranja (ICMat, Campus de Cantoblanco)
Jorge López Abad
ICMAT
The Ramsey property of finite
dimensional normed spaces
Abstract:
It is well known that the approximate structural Ramsey property characterizes the fixed point
property of the automorphism group of “Rich” metric structures. Recall that a topological group
G has the fixed point property (extremely amenable) when every continuous action of G on a
compactum has a fixed point. The group of linear isometries of the separable infinite dimensional
Hilbert space or the group of the isometries of the universal Urysohn space, with their pointwise
topologies, are extremely amenable. This is a consequence of the approximate Ramsey property
(ARP) of those structures. A family F of metric structures of the same sort has the ARP when for every
X, Y ∈ F and every ε > 0 there is Z ∈ C such that for every real-valued 1-Lipschitz mapping f defined
on the set of linear isometric embeddings Emb(X, Z) from X into Z there exists γ ∈ Emb(Y, Z) such
that
Osc(f, γ ◦ Emb(X, Y )) < ε.
In a joint work with D. Bartosova and B. Mbombo (U. Sao Paulo) we prove
Theorem
The finite dimensional normed spaces have the approximate Ramsey property.
It follows that the group of linear surjective isometries of the universal Gurarij space is extremely
amenable, and that the universal minimal flow of the Poulsen simplex is the Poulsen simplex itself with
the natural action. Finally, we prove the following
Theorem For every integers d and m and every , K > 0 there exists n such that for every K-Lipschitz
mapping f defined on the grassmannian Gr(d, Rn ) there exists a cd · K-Lipschitz mapping g defined
on the Banach-Mazur compactum Bd and there exists a subspace V of Rn isometric to `m
∞ such that
|f (W ) − gτ (W )| < ε
for every W ∈ Gr(d, V ), where τ (W ) ∈ Bd is the isometric type of the normed space (W, k · k∞ ).