Coarse embeddings into $c_0(\Gamma)$

TL;DR

This paper investigates the conditions under which large Banach spaces can be coarsely embedded into $c_0( ext{cardinality})$, revealing limitations related to cotype and isomorphism properties.

Contribution

It establishes that large Banach spaces admitting coarse embeddings into $c_0( ext{cardinality})$ lack nontrivial cotype and are linearly isomorphic to $c_0( ext{density})$ if they have a symmetric basis.

Findings

Spaces with large density fail to have nontrivial cotype.

Such spaces contain $ ext{ extlangle} ext{infty}^n ext{ extrangle}$ uniformly.

Spaces with symmetric bases are linearly isomorphic to $c_0( ext{density})$.

Abstract

Let $\lambda$ be a large enough cardinal number (assuming GCH it suffices to let $\lambda=\aleph_\omega$). If $X$ is a Banach space with $\text{dens}(X)\ge\lambda$, which admits a coarse (or uniform) embedding into any $c_0(\Gamma)$, then $X$ fails to have nontrivial cotype, i.e. $X$ contains $\ell_\infty^n$ $C$-uniformly for every $C>1$. In the special case when $X$ has a symmetric basis, we may even conclude that it is linearly isomorphic with $c_0(\text{dens}X)$.

Equations

$$\lambda =\underset{n\to \infty }{\lim }{\lambda }_n.$$

$$w_f(t)$$

$${\rho }_f(t)$$

$$\text{Lip}(f)=\underset{x\ne y}{\sup }\frac{d_2(f(x),f(y))}{d_1(x,y)}<\infty ,$$

$$n\in \mathbb{N}$$

$$(F_j)$$

$$i,j\in \mathbb{N}$$

$$F\in [\lambda {]}^{n-1}$$

$$H(a,i)=\big{\{}(b_{1},b_{2},\ldots,b_{i-1},a(i),b_{i+1},\ldots,\alpha_{n}):b_{j}\in\Gamma_{j},\text{ for $j\in\{1,2\ldots n\}\setminus\{i\}$}\big{\}},$$

$$L(a,i)=\big{\{}(a(1),\ldots,a(i-1),b_{i},a(i+1),\ldots a(n)):b_{i}\in\Gamma_{i}\big{\}},$$

$$\exp_{+}(\beta,n+1)=\big{(}2^{\exp_{+}(\beta,n)^{+}}\big{)}^{+}.$$

$$\lambda =\underset{n\to \infty }{\lim }{\exp }_{+}({\omega }_0,n).$$

$$\sigma :\Gamma :=\prod _{i=1}^n{\Gamma }_i\to \mathcal{C},$$

$$(a^{(j)}{)}_{j=1}^{\infty }$$

$$\sigma (a^{(i)})=\sigma (a^{(j)})$$

$$L\subset \Gamma$$

$$\sigma :\Gamma :=\prod _{i=1}^n{\Gamma }_i\to \mathcal{C},$$

$$\tilde{\sigma }:\Gamma :=\prod _{i=1}^n{\Gamma }_i\to \tilde{\mathcal{C}},$$

$$c\in \mathcal{C}$$

$${\sigma }^{-1}(\{c\})\subset \bigcup _{j=1}^{m(c)}\bigcup _{i=1}^{n}H(a^{(c,j)},i).$$

$$\tilde{\mathcal{C}}=\bigcup _{c\in \mathcal{C}}\{1,2,\dots ,m(c)\}\times \{1,2\dots n\}\times \{c\},$$

$$\tilde{\sigma }:\Gamma \to \tilde{\mathcal{C}},b\mapsto (c,j,i),\text{ where }$$

$$\displaystyle c=\sigma(b),\quad j=\min\Big{\{}j^{\prime}:b\in\bigcup_{i^{\prime}=1}^{n}H(a^{(c,j^{\prime})},i^{\prime})\Big{\}},\text{ and }i=\min\{i^{\prime}:b\in H(a^{(c,j)},i)\}.$$

$$\{b\in \Gamma :\sigma (b)=c\}=\bigcup _{j=1}^{m(c)}\bigcup _{i=1}^n\{b\in \Gamma :\tilde{\sigma }(b)=(c,j,i)\},$$

$$\tilde{\sigma }(L)\subset \{(j,i,c_k),k\le p,j\le m(c_k),\text{i}\le n\},$$

$$\forall (a_1,a_2,\dots ,a_{i-1},a_{i+1},$$

$$\displaystyle\big{|}\{a\in\Gamma_{i}^{\prime}:i(a_{1},a_{2},\ldots a_{i-1},a,a_{i+1},\ldots a_{n})=i\}\big{|}\leq K_{i}.$$

$$\displaystyle\big{|}\tilde{\Gamma}^{(0)}_{j}\big{|}$$

$$>2^{{\beta }_{n-j}}$$

$$=2^{|{\tilde{\Gamma }}_{j+1}^{(0)}\times {\tilde{\Gamma }}_{j+1}^{(0)}\times \dots \times {\tilde{\Gamma }}_n^{(0)}|}$$

$$\displaystyle=\big{|}\{f:\tilde{\Gamma}_{j+1}^{(0)}\times\tilde{\Gamma}_{j+1}^{(0)}\times\ldots\times\tilde{\Gamma}^{(0)}_{n}\to\mathbb{N}\}\big{|}.$$

$${\mathcal{F}}_j=\{f:{\tilde{\Gamma }}_j^{(0)}\times {\tilde{\Gamma }}_{j+1}^{(0)}\times \dots \times {\tilde{\Gamma }}_n^{(0)}\to \mathbb{N}\}$$

$$\phi_{1}:\prod_{j=1}^{n-1}\tilde{\Gamma}_{j}^{(0)}\to\mathbb{N},\quad(a_{1},a_{2},\ldots,a_{n-1})\mapsto\big{|}\{a\!\in\!\tilde{\Gamma}_{n}^{(0)}:i(a_{1},a_{2},\ldots,a_{n-1},a)\!=\!n\}\big{|}.$$

$${\varphi }_1(a_1,a_2,\dots ,a_{n-1})={\varphi }_2(a_2,\dots ,a_{n-1})=\dots ={\varphi }_{n-1}(a_{n-1}).$$

$$\big{|}\{a\in\Lambda_{i}:i(a_{1},a_{2},\ldots,a_{i-1},a,a_{i+1},\ldots a_{n})=i\}\big{|}\leq K_{i}.$$

$$\prod _{j=1}{l}_j$$

Full text

1 Introduction

The classical result of Aharoni states that every separable metric space (in particular every separable Banach space) can be bi-Lipschitz embedded (the definition is given below) into $c_{0}$.

The natural problem of embeddings of metric spaces into $c_{0}(\Gamma)$, for an arbitrary set $\Gamma$, has been treated by several authors, in particular Pelant and Swift. The characterizations that they obtained, and which play a crucial role in our argument, are described below.

Our main interest, motivated by some problems posed in [3], lies in the case of embeddings of Banach spaces into $c_{0}(\Gamma)$.

We now state the main results of this paper. We first define the following cardinal numbers inductively. We put $\lambda_{0}=\omega_{0}$, and, assuming for $n\in\mathbb{N}_{0}$, $\lambda_{n}$ has been defined, we put $\lambda_{n+1}=2^{\lambda_{n}}$. Then we let

[TABLE]

It is clear that assuming the generalized continuum hypothesis (GCH) $\lambda=\aleph_{\omega}$.

Theorem A**.**

If $X$ is a Banach space with density $\text{dens}(X)\geq\lambda$, which admits a coarse (or uniform) embedding into any $c_{0}(\Gamma)$, then $X$ fails to have nontrivial cotype, i.e. $X$ contains $\ell_{\infty}^{n}$ $C$-uniformly for some $C>1$ (equivalently, every $C>1$).

Our method of proof gives a much stronger result for Banach spaces with a symmetric basis. Namely, under the assumptions of Theorem A, such spaces are linearly isomorphic with $c_{0}(\Gamma)$ (Theorem (4.2)).

Theorem A will follow from the following combinatorial result which is of independent interest.

Theorem B**.**

Assume that $\Lambda$ is a set whose cardinality is at least $\lambda$, $n\in\mathbb{N}$, and $\sigma:[\Lambda]^{n}\to\mathcal{C}$ is a map into an arbitrary set $\mathcal{C}$. Then (at least) one of the following conditions holds:

There is a sequence $(F_{j})_{j=1}^{\infty}$ of pairwise disjoint elements of $[\Lambda]^{n}$, so that $\sigma(F_{i})=\sigma(F_{j})$, for all $i,j\in\mathbb{N}$. 2. 2.

There is an $F\in[\Lambda]^{n-1}$ so that \sigma\big{(}\big{\{}F\cup\{\gamma\}:\gamma\in\Lambda\big{\}}\big{)} is infinite.

The above Theorem B was previously deduced in [6] from a combinatorial result of Baumgartner, provided $\Lambda$ is a weakly compact cardinal number (whose existence is not provable in ZFC, as it is inaccessible [5] p. 325, p. 52). The authors in [6] pose a question whether assuming that $\Gamma$ is uncountable is sufficient in Theorem B.

Theorem B is used in order to obtain a scattered compact set $K$ of height $\omega_{0}$, such that $C(K)$ does not uniformly embed into $c_{0}(\Gamma)$. It is easy to check that our version of Theorem B implies a ZFC example of such a $C(K)$ space. It is further shown in [6] that the space $C[0,\omega_{1}]$ does not uniformly embed into any $c_{0}(\Gamma)$.

Let us point out that a special case of Theorem A was obtained by Pelant and Rodl [8], namely it was shown there that $\ell_{p}(\lambda),1\leq p<\infty$, spaces (which are well known to have nontrivial cotype) do not uniformly embed into any $c_{0}(\Gamma)$.

The paper is organized as follows. In Section 2 we recall Pelant’s [7, 6] amd Swift’s [10] conditions for Lipschitz, uniform, and coarse embeddability into $c_{0}(\Gamma)$. In Section 3 we provide a proof for Theorem B. Finally, in Section 4, we provide a proof of Theorem A as well as the symmetric version of the result.

All set theoretic concepts and results used in our note can be found in [5], whereas for facts concerning nonseparable Banach spaces [4] can be consulted.

2 Pelant’s and Swift’s criteria for Lipschitz, uniform, and coarse embeddability into $c_{0}(\Gamma)$

In this section we recall some of the notions and results by Pelant [7, 6] and Swift [10] about embeddings into $c_{0}(\Gamma)$.

For a metric space $(M,d)$ a cover is a set $\mathcal{U}$ of subsets of $M$ such that $M=\bigcup_{U\in\mathcal{U}}U$. A cover $\mathcal{U}$ of $M$ is called uniform if there is an $r>0$ so that for all $x\in M$ there is a $U\in\mathcal{U}$, so that $B_{r}(x)=\{x^{\prime}\in M:d(x^{\prime},x)<r\}\subset U$. It is called uniformly bounded if the diameters of the $U\in\mathcal{U}$ are uniformly bounded, and it is called point finite if every $x\in M$ lies in only finitely many $U\in\mathcal{U}$. A cover $\mathcal{V}$ of $M$ is a *refinement * of a cover $\mathcal{U}$, if for every $V\in\mathcal{V}$ there is a $U\in\mathcal{U}$, for which $V\subset U$.

Definition 2.1**.**

[6] A metric space $(M,d)$ is said to have the Uniform Stone Property if every uniform cover $\mathcal{U}$ of $M$ has a point finite uniform refinement.

Definition 2.2**.**

[10] A metric space $(M,d)$ is said to have the Coarse Stone Property if every bounded cover is the refinement of a point finite uniformly bounded cover.

Definition 2.3**.**

Let $(M_{1},d_{1})$ and $(M_{2},d_{2})$ be two metric spaces. For a map $f:M_{1}\to M_{2}$ we define the modulus of uniform continutiy $w_{f}:[0,\infty)\to[0,\infty]$, and the modulus of expansion $\rho:[0,\infty)\to[0,\infty]$ as follows

[TABLE]

The map $f$ is called *uniform continuous * if $\lim_{t\to 0}w_{f}(t)=0$, and it is called a uniform embedding if moreover $\rho_{f}(t)>0$ for every $t>0$. It is called coarse if $w_{f}(t)<\infty$, for all $0<t<\infty$ and it is called a coarse embedding, if $\lim_{t\to\infty}\rho_{f}(t)=\infty$. The map $f$ is called Lischitz continuous if

[TABLE]

and a bi-Lipschitz embedding, if $f$ is injective and $\text{{\rm Lip}}(f^{-1})$ is also finite.

The following result recalls results from [6](for (i)$\iff$(ii)) and [10] (for (ii)$\iff$(iii)$\iff$(iv)$\iff$(v)).

Theorem 2.4**.**

For a Banach space $X$ the following properties are equivalent.

1. (i)

$X$* has the uniform Stone Property.* 2. (ii)

$X$* is uniformly embeddable into $c_{0}(\Gamma)$, for some set $\Gamma$.* 3. (iii)

$X$* has the coarse Stone Property.* 4. (iv)

$X$* is coarsely embeddable into $c_{0}(\Gamma)$, for some set $\Gamma$.* 5. (v)

$X$* is bi-Lipschitzly embeddable into $c_{0}(\Gamma)$, for some set $\Gamma$.*

It is easy to see, and was noted in [6, 10], that the uniform Stone property and the coarse Stone property are inherited by subspaces. The equivalence (i)$\iff$(ii) was used in [6] to show that $C[0,\omega_{1}]$ does not uniformly embed in any $c_{0}(\Gamma)$. It was also used to prove that certain other $C(K)$-spaces do not uniformly embed into $c_{0}(\Gamma)$: Let $\Lambda$ be any set and denote for $n\in\mathbb{N}$ by $[\Lambda]^{\leq n}$ and $[\Lambda]^{n}$ the subsets of $\Lambda$ which have cardinality at most $n$ and exactly $n$, respectively. Endow $[\Lambda]^{\leq n}$ with the restriction of the product topology on $\{0,1\}^{\Lambda}$ (by identifying each set with its characteristic function). Then define $K_{\Lambda}$ to be the one-point Alexandroff compactification of the topological sum of the spaces $[\Lambda]^{\leq n}$, $n\in\mathbb{N}$. It was shown in [6] that if $\Lambda$ satisfies Theorem B then $C(K_{\Lambda})$ is not uniformly Stone and thus does not embed uniformly into any $c_{0}(\Gamma)$.

3 A combinatorial argument

We start by introducing property $P(\alpha)$ for a cardinal $\alpha$ as follows.

[TABLE]

As remarked in Section 2, if $\kappa$ is an uncountable weakly compact cardinal number, then $P(\kappa)$ holds. But the existence of weakly compact cardinal numbers requires further set theoretic axioms, beyond ZFC [5]. In [6, Question 3] the authors ask if $P(\omega_{1})$ is true.

Theorem 3.1**.**

For $\lambda$ defined by (1), $P(\lambda)$ holds.

For our proof of Theorem 3.1 it will be more convenient to reformulate it into a statement about $n$-tuples, instead of sets of cardinality $n$. We will first introduce some notation.

Let $n\in\mathbb{N}$ and $\Gamma_{1}$, $\Gamma_{2},\ldots\Gamma_{n}$ be sets of infinite cardinality, and put $\Gamma=\prod_{i=1}^{n}\Gamma_{i}$. For $a\in\Gamma$ and $1\leq i\leq n$ we denote the $i$-the coordinate of $a$ by $a(i)$. We say that two points $a$ and $b$ in $\Gamma$ are diagonal, if $a(i)\not=b(i)$, for all $i\in\{1,2,\ldots,n\}$.

Let $a\in\Gamma_{i}$ for $i\in\mathbb{N}$. For $i\in\{1,2\ldots,n\}$ we call the set

[TABLE]

the Hyperplane through the point $a$ orthogonal to $i$. We call the set

[TABLE]

the Line through the point $a$ in direction of $i$.

For a cardinal number $\beta$, we define recursively the following sequence of cardinal numbers \big{(}\exp_{+}(\beta,n):n\!\in\!\mathbb{N}_{0}): $\exp_{+}(\beta,0)=\beta$, and, assuming $\exp_{+}(\beta,n)$ has been defined for some $n\in\mathbb{N}_{0}$, we put

[TABLE]

Here $\gamma^{+}$ denotes the successor cardinal, for a cardinal $\gamma$, i.e., the smallest cardinal number $\gamma^{\prime}$ with $\gamma^{\prime}>\gamma$. Note that since $\exp_{+}(\gamma,1)\leq 2^{2^{2^{\gamma}}}$, it follows for the above defined cardinal number $\lambda$, that

[TABLE]

Secondly, successor cardinals are regular [5], and thus every set of cardinality $\gamma$, with $\gamma$ being a successor cardinal, can be partitioned for $n\in\mathbb{N}$ into $n$ disjoint sets $\Gamma_{1}$, $\Gamma_{2},\ldots,\Gamma_{n}$, all of them having also cardinality $\gamma$, and the map \Gamma_{1}\times\Gamma_{2}\times\ldots\times\Gamma_{n}\to\big{[}\bigcup_{i=1}^{n}\Gamma_{i}\big{]}^{n}, $(a_{1},a_{2},\ldots,a_{n})\mapsto\{a_{1},a_{2},\ldots a_{n}\}$, is injective. We therefore deduce that the following statement implies Theorem 3.1.

Theorem 3.2**.**

Let $n\in\mathbb{N}$ and a assume that the sets $\Gamma_{1}$, $\Gamma_{2},\ldots\Gamma_{n}$ have cardinality at least $\exp_{+}(\omega_{1},n^{2})$. For any function

[TABLE]

where $\mathcal{C}$ is an arbitrary set, at least one of the following two conditions hold

[TABLE]

We will make the following observation before proving Theorem 3.2.

Lemma 3.3**.**

Let $n\in\mathbb{N}$ and $\Gamma_{1}$, $\Gamma_{2},\ldots\Gamma_{n}$ be non empty sets. Let

[TABLE]

be a function that fails both conditions (4) and (5).

Then there is a set $\tilde{\mathcal{C}}$ and a function

[TABLE]

that fails both (4) and (5) and moreover has the property that

[TABLE]

Proof.

We may assume without loss of generality that $\sigma$ is surjective. Since (4) is not satisfied for each $c\in\mathcal{C}$ there exists an $m(c)\in\mathbb{N}$ and a (finite) sequence $(a^{(c,j)})_{j=1}^{m(c)}\subset\sigma^{-1}(\{c\})$, which is pairwise diagonal, and maximal, with this property. Hence

[TABLE]

Indeed, from the maximality of $(a^{(c,j)})_{j=1}^{m(c)}\subset\sigma^{-1}(\{c\})$, it follows that each $b\in\sigma^{-1}(\{b\})$ must have at least one coordinate in common with at least one element of $(a^{(c,j)})_{j=1}^{m(c)}\subset\sigma^{-1}(\{c\})$.

We define

[TABLE]

and

[TABLE]

It is clear that $\tilde{\sigma}$ satisfies (6). Since for every $c\in\mathcal{C}$,

[TABLE]

$\tilde{\sigma}$ does not satisfy (4). In order to verify that (5) is not satisfied, assume $L\subset\Gamma$ is a line, and let $\{c_{1},c_{2},\ldots,c_{p}\}$ be the image of $L$ under $\sigma$. By construction,

[TABLE]

which is also finite. ∎

Proof of Theorem 3.2.

We assume that $\sigma:\Gamma=\Gamma_{1}\times\Gamma_{2}\times\ldots\times\Gamma_{n}\to\mathcal{C}$ is a map which fails both (4) and (5). By Lemma 3.3 we may also assume that $\sigma$ satisfies (6). For each $a\in\Gamma$ we fix an $i(a)\in\{1,2,\ldots,n\}$ so that \sigma^{-1}\big{(}\{\sigma(a)\}\big{)}\subset H(a,i(a)). It is important to note that, since (5) is not satisfied, it follows that each line $L$, whose direction is some $j\in\{1,2,\ldots,n\}$, can only have finitely many elements $b$ for which $i(b)=j$. Indeed, if $i(b)=j$ then $b$ is uniquely determined by the value $\sigma(b)$. To continue with the the proof the following Reduction Lemma will be essential.

Lemma 3.4**.**

Let $\beta$ be an uncountable regular cardinal. Assume that $\tilde{\Gamma}_{1}\subset\Gamma_{1}$, $\tilde{\Gamma}_{2}\subset\Gamma_{2},\ldots,\tilde{\Gamma}_{n}\subset\Gamma_{n}$ are such that $|\tilde{\Gamma}_{i}|\geq\exp_{+}(\beta,n)$, for all $i\in\{1,\dots,n\}$.

Then, for any $i\in\{1,2\ldots,n\}$ there are a number $K_{i}\in\mathbb{N}$, and subsets $\Gamma_{1}^{\prime}\subset\tilde{\Gamma}_{1},\Gamma_{2}^{\prime}\subset\tilde{\Gamma}_{2},\ldots,\Gamma_{n}^{\prime}\subset\tilde{\Gamma}_{n},$ with $|\Gamma^{\prime}_{j}|\geq\beta$, so that

[TABLE]

Proof.

We assume without loss of generality that $i=n$. Abbreviate $\beta_{j}=\exp_{+}(\beta,j)$, for $j=1,2\ldots n$. We choose subsets $\tilde{\Gamma}_{j}^{(0)}\subset\tilde{\Gamma}_{j}$, for which \big{|}\tilde{\Gamma}^{(0)}_{j}\big{|}=\beta_{n+1-j}.

Since the $\beta_{j}$’s are regular, it follows for each $j=1,2\ldots,n-1$ that

[TABLE]

Abbreviate for $i=1,\ldots,n$.

[TABLE]

and consider the function

[TABLE]

For fixed $a_{1}\in\Gamma_{1}^{(0)}$, $\phi_{1}(a_{1},\cdot)\in\mathcal{F}_{2}$, and the cardinality of $\mathcal{F}_{2}$ is by the above estimates smaller than the cardinality of $\tilde{\Gamma}_{1}^{(0)}$, which is regular. Therefore we can find a function $\Phi_{2}\in\mathcal{F}_{2}$ and a subset $\Gamma^{\prime}_{1}\subset\tilde{\Gamma}_{i}^{(0)}$ of cardinality $\beta_{n}$ so that $\phi_{1}(a_{1},\dot{)}=\phi_{2}$ for all $a_{1}\in\Gamma^{\prime}_{1}$. We continue the process and find $\Gamma^{\prime}_{j}\subset\tilde{\Gamma}_{j}^{(0)}$, for $j=1,2\ldots,n-2$ of cardinality $\beta_{n+1-j}$ and functions $\phi_{j}\in F_{j}$, for $j=1,2,\ldots,n-1$, so that for all $(a_{1},a_{2},\ldots,a_{n-2})\in\prod_{j=1}^{n-2}\Gamma^{\prime}_{j}$ and $a_{n-1}\in\tilde{\Gamma}_{n-1}^{(0)}$, we have

[TABLE]

Then, since $\phi_{n-1}$ is $\mathbb{N}$ valued, we can finally choose an $K_{n}\in\mathbb{N}$ and a subset $\Gamma^{\prime}_{n-1}$, of cardinality at least $\beta$, so that $\phi_{n-1}(a_{1})\leq K_{n}$, which finishes our argument. ∎

Continuation of the proof of Theorem 3.2. We apply Lemma 3.4 successively to all $i\in\{1,2\ldots n\}$, and the cardinals $\beta^{(i)}=\exp_{+}(\omega_{1},n(n-i))$. We obtain numbers $K_{1},K_{2},\ldots K_{n}$ in $\mathbb{N}$ and infinite sets $\Lambda_{j}\subset\Gamma_{i}$, for $j\leq n$, so that for all $i=1,2\ldots n$ and all $a\in\prod_{j=1}^{n}\Lambda_{j}$

[TABLE]

In order to deduce a contradiction choose for each $j=1,\ldots n$ a subset $A_{j}$ of $\Lambda_{j}$ of cardinality $l_{j}=(n+1)K_{j}$. Then it follows that

[TABLE]

which is a contradiction and finishes the proof of the Theorem. ∎

We can now state the ZFC version of Theorem 4.1. in [6], in which it was shown that for weakly compact cardinalities $\kappa_{0}$ the space $C(K_{\kappa_{0}})$, where $K_{\kappa_{0}}$ was defined at the end of Section 2, cannot be uniformly (or coarsely) embedded into any $c_{0}(\Gamma)$, where $\Gamma$ has any cardinality. Since the only property of $\kappa_{0}$, which is needed in [6], is the fact that $(P(\kappa_{0}))$ holds, we deduce

Corollary 3.5**.**

$C(K_{\lambda})$* does not coarsely (or uniformly) embed into $c_{0}(\Gamma)$, for any cardinality $\Gamma$.*

4 Proof of Theorem A

In this section we use our combinatorial Theorem B from Section 3 to show Theorem A.

Recall that a long Schauder basis of a Banach space $X$ is a transfinite sequence $\{e_{\gamma}\}_{\gamma=0}^{\Gamma}$ such that for every $x\in X$ there exists a unique transfinite sequence of scalars $\{a_{\gamma}\}_{\gamma=0}^{\Gamma}$ such that $x=\sum_{\gamma=0}^{\Gamma}a_{\gamma}e_{\gamma}$. Similarly, a long Schauder basic sequence in a Banach space $X$ is a transfinite sequence $\{e_{\gamma}\}_{\gamma=0}^{\Gamma}$ which is a long Schauder basis of its closed linear span. Recall that the $w^{*}-\text{dens}(X^{*})$ is the smallest cardinal such that there exists a $w^{*}$-dense subset of $X^{*}$. Analogously to the classical Mazur construction of a Schauder basic sequence in a separable Banach space we have the following result, proved e.g. in [4, p.135] (the fact that the basis is normalized, i.e. $\|e_{\gamma}\|=1$, is not a part of the statement in [4], but it is easy to get it by normalizing the existing basis).

Theorem 4.1**.**

Let $X$ be a Banach space with $\Gamma=w^{*}-\text{dens}X^{*}>\omega_{0}$. Then $X$ contains a monotone normalized long Schauder basic sequence of length $\Gamma$.

Proof of Theorem A.

Using the Hahn-Banach theorem it is easy to see that $w^{*}-\text{dens}X^{*}\leq\text{dens}X$. On the other hand, since every $x\in X$ is uniquely determined by its values on a $w^{*}$-dense subset of $X^{*}$, it is clear that

[TABLE]

It follows that for $\lambda$ defined in (1) we get that $\lambda=w^{*}-\text{dens}X^{*}$ holds if and only if $\lambda=\text{dens}X$. In order to prove Theorem A we may assume without loss of generality that $X$ has a long normalized and monotone Schauder basis $(e_{\mu})_{\mu<\lambda}$, of length $\lambda$, i.e. $\Gamma=\lambda$.

Set

[TABLE]

Suppose that $F=\{\gamma_{1},\dots,\gamma_{n}\}$ where $\gamma_{1}<\dots<\gamma_{n}$ are elements of $[0,\lambda)$ arranged in an increasing order. Consider the corresponding finite set $M_{F}=\{\sum_{i=1}^{n}\varepsilon_{i}e_{\gamma_{i}}:\varepsilon_{i}\in\{-1,1\}\}$, containing $2^{n}$ distinct vectors of $X$, and put a linear order $\prec$ on this set according to the arrangement of the signs $\varepsilon_{i}$, setting

[TABLE]

if and only if for the minimal $i$, such that $\varepsilon_{i}\neq\tilde{\varepsilon}_{i}$, it holds $\varepsilon_{i}<\tilde{\varepsilon}_{i}$. In order to prove Theorem A it suffices to show that if $M=\cup_{F\in D_{n},n\in\mathbb{N}}M_{F}\subset X$ has the coarse Stone property then $X$ fails to have nontrivial cotype. To this end, starting with $\mathcal{U}=\{B_{2}(x):x\in M\}$ we find a uniform bounded cover $\mathcal{V}$, which is point finite and so that $\mathcal{U}$ refines $\mathcal{V}$, i.e., for all $x\in M$ there is a $V_{x}\in\mathcal{V}$ with $B_{2}(x)\subset V_{x}$. Let $r>0$ be such that each $V\in\mathcal{V}$ is a subset of a ball of radius $r$.

Let $\mathcal{C}$ be the set consisting of all finite tuples $(V^{1},\dots,V^{m})$, where $V^{j}\in\mathcal{V}$. We now define the function $\sigma:M\to\mathcal{C}$ as follows. If $F\in D_{n},F=\{\gamma_{1},\dots,\gamma_{n}\}$ where $\gamma_{1}<\dots<\gamma_{n}$, we let

[TABLE]

where $y_{1}\prec\dots\prec y_{2^{n}}$ are the elements of $M_{F}$ arranged in the increasing order. Applying Theorem B to the function $\sigma$, for a fixed $n\in\mathbb{N}$, yields one of two possibilities. Either there is an $F=\{\gamma_{1},\dots,\gamma_{n-1}\}$, where $\gamma_{1}<\dots<\gamma_{n-1}$, so that \sigma\big{(}\big{\{}F\cup\{\tau\}:\tau\in\lambda\setminus F\big{\}}\big{)} is infinite. In this case, pick an infinite sequence of distinct $\{\tau_{j}\}_{j=1}^{\infty}$ witnessing the desired property. By passing to a subsequence, we may assume without loss of generality that either there exists $k,1\leq k\leq n-1$, so that for all $j\in\mathbb{N}$, $\gamma_{k}<\tau_{j}<\gamma_{k+1}$, or $\tau_{j}<\gamma_{1}$ for all $j\in\mathbb{N}$, or $\gamma_{n-1}<\tau_{j}$ for all $j\in\mathbb{N}$. For simplicity of notation, assume the last case, i.e. $\gamma_{1}<\dots<\gamma_{n-1}<\tau_{j}$ holds for all $j\in\mathbb{N}$. Denoting $F^{j}=\{\gamma_{1},\dots,\gamma_{n-1},\tau_{j}\}$, we conclude that there exists a fixed selection of signs $\varepsilon_{1},\dots,\varepsilon_{n}$ such that the set

[TABLE]

is infinite. Indeed, otherwise the set of values $\{\sigma(\{\gamma_{1},\dots,\gamma_{n-1},\tau_{j}\}),j\in\mathbb{N}\}$, which are determined by the definition (9), would have only a finite set of options for each coordinate, and would therefore have to be finite. This is a contradiction with the point finiteness of the system $\mathcal{V}$, because

[TABLE]

It remains to consider the other case when there is a sequence $(F_{j})$ of pairwise disjoint elements of $[\lambda]^{n}$, with $\sigma(F_{i})\!=\!\sigma(F_{j})$, for any $i,j\in\mathbb{N}$. In fact, it suffices to choose just a pair of such disjoint elements (written in an increasing order of ordinals) $F=\{\gamma_{1},\dots,\gamma_{n}\}$, $G=\{\beta_{1},\dots,\beta_{n}\}$, such that $\sigma(F)=\sigma(G)$. This means, in particular, that for every fixed selection of signs $\varepsilon_{1},\dots,\varepsilon_{n}$,

[TABLE]

By our assumption, the elements of $\mathcal{V}$ are contained in a ball of radius $r$, hence

[TABLE]

holds for any selection of signs $\varepsilon_{1},\dots,\varepsilon_{n}$. Let $u_{j}=e_{\gamma_{j}}-e_{\beta_{j}}$, $j\in\{1,\dots,n\}$. Because $\{e_{\gamma}\}$ is a monotonne normalized long Schauder basis, we have the trivial estimate $1\leq\|u_{j}\|\leq 2$. The equation (10) means that

[TABLE]

holds for any selection of signs $\varepsilon_{1},\dots,\varepsilon_{n}$. Since norm functions are convex, this means that for the unite vector ball $B_{E}$ of of $E=\text{span}(u_{i}:i\leq)$ it follows that

[TABLE]

which means that $(u_{j})_{j=1}^{n}$ is $4r$-equivalent to the unit vector basis of $\ell_{\infty}^{n}$. ∎

In fact, our proof gives a much stronger condition than just failing cotype, because our copies of $\ell_{\infty}^{k}$ are formed by vectors of the type $e_{\alpha}-e_{\beta}$. This fact can be used to obtain much stronger structural results for spaces with special bases. Recall that a long Schauder basis $\{e_{\gamma}\}_{\gamma=1}^{\Lambda}$ is said to be symmetric if

[TABLE]

for any selection of $a_{i}\in\mathbb{R}$, and any pair of sets $\{\gamma_{i}\}_{i=1}^{n}\subset[1,\Lambda)$, $\{\beta_{i}\}_{i=1}^{n}\subset[1,\Lambda)$. It is well-known (c.f. [9, Prop. II.22.2]), that each symmetric basis is automatically unconditional, i.e. there exists $K>0$ such that

[TABLE]

In particular,

[TABLE]

whenever $A\subset B$.

Theorem 4.2**.**

Let $X$ be a Banach space of density $\Lambda\geq\lambda$, with a symmetric basis $\{e_{\gamma}\}_{\gamma=1}^{\Lambda}$, which coarsely (or uniformly) embeds into some $c_{0}(\Gamma)$.

Then $X$ is linearly isomorphic with $c_{0}(\Lambda)$.

Proof.

By the proof of the above results, if $X$ embeds into $c_{0}(\Gamma)$, there exists an $C>0$, such that for each $k\in\mathbb{N}$ there are some vectors $\{v_{i}\}_{i=1}^{k}$ of the form $v_{i}=e_{\gamma_{i}}-e_{\beta_{i}}$ satisfying the conditions

[TABLE]

Using the fact that the basis $\{e_{\gamma}\}$ is unconditional, (and symmetric) we obtain by an easy manipulation that there exist some constants $A,B>0$ such that

[TABLE]

Combining (12) and (13) we finally obtain that for some $D\geq 1$, and any $k\in\mathbb{N}$,

[TABLE]

for all $\{\gamma_{1},\dots,\gamma_{k}\}\subset[1,\Lambda)$, which proves our claim. ∎

5 Final comments and open problems

Let us mention in this final section some problems of interest.

First of all, we do not know whether or not Theorem A is true if we replace $\lambda$ by smaller cardinal numbers.

Problem 1**.**

Assume that $X$ is a Banach space with $\text{\rm dens}(X)\geq\omega_{1}$, and assume that $X$ coarsely embeds into $c_{0}(\Gamma)$ for some cardinal number $\Gamma$. Does $X$ have trivial co-type? If moreover $X$ has a symmetric basis, must it be isomorphic to $c_{0}(\omega_{1})$?

Of course Problem 1 would have a positive answer if the following is true.

Problem 2**.**

Is Theorem B true for $\omega_{1}$?

Connected to Problems 1 and 2 is the following

Problem 3**.**

Does $\ell_{\infty}$ coarsely embed into $c_{0}(\kappa)$ for some uncountable cardinal number $\kappa$.

Another line of interesting problems asks which isomorphic properties do non separable Banach spaces have which coarsely embed into $c_{0}(\Gamma)$

Problem 4**.**

Does a non separable Banach space which coarsely embeds into some $c_{0}(\Gamma)$, $\Gamma$ being uncountable, contain copies $c_{0}$, or even $c_{0}(\omega_{1})$.