THE COARSE GEOMETRY OF TSIRELSON'S SPACE AND APPLICATIONS

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21230%2F18%3A00323439" target="_blank" >RIV/68407700:21230/18:00323439 - isvavai.cz</a>

Výsledek na webu

<a href="http://dx.doi.org/10.1090/jams/899" target="_blank" >http://dx.doi.org/10.1090/jams/899</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1090/jams/899" target="_blank" >10.1090/jams/899</a>

Jazyk výsledku

angličtina

Název v původním jazyce

THE COARSE GEOMETRY OF TSIRELSON'S SPACE AND APPLICATIONS

Popis výsledku v původním jazyce

Abstract: The main result of this article is a rigidity result pertaining to the spreading model structure for Banach spaces coarsely embeddable into Tsirelson's original space $ T^*$. Every Banach space that is coarsely embeddable into $ T^*$ must be reflexive, and all of its spreading models must be isomorphic to $ c_0$. Several important consequences follow from our rigidity result. We obtain a coarse version of an influential theorem of Tsirelson: $ T^*$ coarsely contains neither $ c_0$ nor $ ell _p$ for $ pin [1,infty )$. We show that there is no infinite-dimensional Banach space that coarsely embeds into every infinite-dimensional Banach space. In particular, we disprove the conjecture that the separable infinite-dimensional Hilbert space coarsely embeds into every infinite-dimensional Banach space. The rigidity result follows from a new concentration inequality for Lipschitz maps on the infinite Hamming graphs that take values into $ T^*$, and from the embeddability of the infinite Hamming graphs into Banach spaces that admit spreading models not isomorphic to $ c_0$. Also, a purely metric characterization of finite dimensionality is obtained.

Název v anglickém jazyce

THE COARSE GEOMETRY OF TSIRELSON'S SPACE AND APPLICATIONS

Popis výsledku anglicky

Abstract: The main result of this article is a rigidity result pertaining to the spreading model structure for Banach spaces coarsely embeddable into Tsirelson's original space $ T^*$. Every Banach space that is coarsely embeddable into $ T^*$ must be reflexive, and all of its spreading models must be isomorphic to $ c_0$. Several important consequences follow from our rigidity result. We obtain a coarse version of an influential theorem of Tsirelson: $ T^*$ coarsely contains neither $ c_0$ nor $ ell _p$ for $ pin [1,infty )$. We show that there is no infinite-dimensional Banach space that coarsely embeds into every infinite-dimensional Banach space. In particular, we disprove the conjecture that the separable infinite-dimensional Hilbert space coarsely embeds into every infinite-dimensional Banach space. The rigidity result follows from a new concentration inequality for Lipschitz maps on the infinite Hamming graphs that take values into $ T^*$, and from the embeddability of the infinite Hamming graphs into Banach spaces that admit spreading models not isomorphic to $ c_0$. Also, a purely metric characterization of finite dimensionality is obtained.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2018

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY

ISSN

0894-0347

e-ISSN

1088-6834

Svazek periodika

31

Číslo periodika v rámci svazku

3

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

19

Strana od-do

699-717

Kód UT WoS článku

000430377800004

EID výsledku v databázi Scopus

2-s2.0-85045897954