A new coarsely rigid class of Banach spaces

Florent Baudier; Gilles Lancien; Pavlos Motakis; Thomas Schlumprecht

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A new coarsely rigid class of Banach spaces

Florent Baudier, Gilles Lancien, Pavlos Motakis, Thomas Schlumprecht

Metric Geometry · Submitted 2 Jun 2018

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The authors' main claim 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-$c_0$.

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arXiv ID1806.00702 (version 3)

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CitationFlorent Baudier, Gilles Lancien, Pavlos Motakis, Thomas Schlumprecht. "A new coarsely rigid class of Banach spaces." arXiv:1806.00702 (2018). https://pith.science/paper/V2A5PP7U

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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.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	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-$c_0$.	result 0.94	0.95	0.95	asserted 1.00
s1	In order to achieve this result we provide a purely metric characterization of this class of Banach spaces.	method 0.83	0.94	0.04	asserted 0.98
s2	This metric characterization takes the form of a concentration inequality for Lipschitz maps on the Hamming graphs, which is rigid under coarse embeddings.	method 0.73	0.89	0.07	asserted 1.00
s3	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.	result 0.88	0.93	0.03	asserted 1.00

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Original 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.

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5 arXiv papers cite this in 23 sentences

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Papers citing it · 18 sentences without a confident label

Builds on it 1 paper

It follows, again from , [ref] that $T_p(T_p)$ has a uniformly unique $ℓ_p$ spreading model.

On the complete separation of asymptotic structures in Banach spaces · Uniqueness of asymptotic notions · 0.84

Uses its method or tool 2 papers

The approach used in [ref] can be used to show that the spaces $JT^ξ_r,p$ and $JT^ξ_p$ have the property that any joint spreading model generated by an array of weakly null sequences is isometrically equivalent to the unit vector basis of $ℓ_p$.

Non-asymptotic $\ell_1$ spaces with unique $\ell_1$ asymptotic model · More non-asymptotic $ℓ_p$ spaces with uniformly unique $ℓ_p$ joint spreading models · 0.93
As it is mentioned in [ref] , its proof follows the ideas of the proof of [ref] *Lemma 4.1.

On Kalton's interlaced graphs and nonlinear embeddings into dual Banach spaces · General properties of Lipschitz maps into a dual space · 0.81

Compares with it 1 paper

This was also observed in [ref] for a slightly different type of spaces.

Non-asymptotic $\ell_1$ spaces with unique $\ell_1$ asymptotic model · More non-asymptotic $ℓ_p$ spaces with uniformly unique $ℓ_p$ joint spreading models · 0.81

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