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A NEW COARSELY RIGID CLASS OF BANACH SPACES

Part of: Special aspects of infinite or finite groups Graph theory Normed linear spaces and Banach spaces; Banach lattices Nonlinear functional analysis

Published online by Cambridge University Press:  13 January 2020

F. Baudier ,
G. Lancien Open the ORCID record for G. Lancien [Opens in a new window] ,
P. Motakis  and
Th. Schlumprecht Open the ORCID record for Th. Schlumprecht [Opens in a new window]
F. Baudier
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX77843, USA (florent@math.tamu.edu)
G. Lancien
Affiliation:
Laboratoire de Mathématiques de Besançon, Université Bourgogne Franche-Comté, 16 route de Gray, 25030Besançon Cédex, Besançon, France (gilles.lancien@univ-fcomte.fr)
P. Motakis
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL61801, USA (pmotakis@illinois.edu)
Th. Schlumprecht
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX77843-3368, USA Faculty of Electrical Engineering, Czech Technical University in Prague, Zikova 4, 16627, Prague, Czech Republic (schlump@math.tamu.edu)
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Abstract

We prove that the class of reflexive asymptotic-$c_{0}$ Banach spaces is coarsely rigid, meaning that if a Banach space $X$ coarsely embeds into a reflexive asymptotic-$c_{0}$ space $Y$, then $X$ is also reflexive and asymptotic-$c_{0}$. In order to achieve this result, we provide a purely metric characterization of this class of Banach spaces. This metric characterization takes the form of a concentration inequality for Lipschitz maps on the Hamming graphs, which is rigid under coarse embeddings. Using an example of a quasi-reflexive asymptotic-$c_{0}$ space, we show that this concentration inequality is not equivalent to the non-equi-coarse embeddability of the Hamming graphs.

Keywords

Coarse embeddingscoarsely rigid classes of banach spacesasymptotic-c0 spacesHamming graphs

MSC classification

Primary: 46B06: Asymptotic theory of Banach spaces 46B20: Geometry and structure of normed linear spaces 46B85: Embeddings of discrete metric spaces into Banach spaces; applications in topology and computer science
Secondary: 46T99: None of the above, but in this section 05C63: Infinite graphs 20F65: Geometric group theory
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 20 , Issue 5 , September 2021 , pp. 1729 - 1747
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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Footnotes

F. Baudier was supported by the National Science Foundation under Grant Number DMS-1800322. G. Lancien was supported by the French ‘Investissements d’Avenir’ program, project ISITE-BFC (contract ANR-15-IDEX-03). P. Motakis was supported by the National Science Foundation under Grant Numbers DMS-1600600 and DMS-1912897. Th. Schlumprecht was supported by the National Science Foundation under Grant Numbers DMS-1464713 and DMS-1711076.

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