On the transfinite symmetric strong diameter two property

TL;DR

This paper explores transfinite extensions of the symmetric strong diameter two property in Banach spaces, analyzing their stability under various operations and characterizing spaces with these properties using cardinal functions.

Contribution

It introduces and studies transfinite analogues of the symmetric strong diameter two property, including stability results and characterizations for spaces of the form C_0(X).

Findings

Stability of transfinite properties under c_0, ℓ_∞ sums, and tensor products.

Characterization of C_0(X) spaces with these properties via cardinal functions.

Examples of Banach spaces that do or do not have these properties.

Abstract

We study transfinite analogues of the symmetric strong diameter two property. We investigate the stability of these properties under $c_0$, $\ell_\infty$ sums and under projective tensor products. Moreover, we characterize Banach spaces of the form $C_0(X)$, where $X$ is a T$_4$ locally compact space, which posses these transfinite properties via cardinal functions over $X$. As an application, we are able to produce a variety of examples of Banach spaces which enjoy or fail these properties.

Equations

$${\ell }_{\infty }(\mathbb{N},X_n):=\left\{x\in \prod _{n=1}^{\infty }{X}_n:(\forall n\in \mathbb{N})x(n)\in X_n\text{ and }\underset{n}{\sup }\Vert x(n)\Vert <\infty \right\}$$

$$c_0(\mathbb{N},X_n):=\left\{x\in {\ell }_{\infty }(\mathbb{N},X_n):\underset{n}{\lim }\Vert x(n)\Vert =0\right\}.$$

$$\sum _{1\le n\le n_{\alpha }}\Vert x_{\alpha }^{\ast }(n)\Vert \ge (1-\epsilon {)}^{\frac{1}{2}}.$$

$$x_{\alpha }:=\sum _{1\le n\le n_{\alpha }}{x}_{\alpha }^n{e}_n\in B_{c_0(\mathbb{N},X_n)}$$

$$x_{\alpha }^{\ast }(x_{\alpha })=\sum _{1\le n\le n_{\alpha }}{x}_{\alpha }^{\ast }(n)(x_{\alpha }^n)\ge (1-\epsilon {)}^{\frac{1}{2}}\sum _{1\le n\le n_{\alpha }}\Vert x_{\alpha }^{\ast }(n)\Vert \ge 1-\epsilon ,$$

$$\Vert x_{\alpha }\pm y\Vert =\max \left\{\Vert x_{\alpha }^m\pm y^m\Vert ,\underset{1\le n\ne m\le n_{\alpha }}{\sup }\Vert x_{\alpha }^n\Vert \right\}\le 1$$

$$A:=\{x^{\ast }{e}_n:n\in \mathbb{N}\text{ and }{x}^{\ast }\in A_n\}\subset S_{{\ell }_1(\mathbb{N},X_n^{\ast })}$$

$$\Vert x(n)\pm y(n)\Vert \le \Vert x\pm y\Vert \le 1,$$

$$x_{\alpha }:=\sum _{n=1}^{\infty }{x}_{\alpha }^n{e}_n\in B_{{\ell }_{\infty }(\mathbb{N},X_n)}.$$

$$\Vert x_{\alpha }\pm y\Vert ={\left(\sum _{\lambda <\kappa }|x_{\alpha }(\lambda ){|}^m+1\right)}^{\frac{1}{m}}=2^{\frac{1}{m}}\le 1+\epsilon$$

$$B^{\prime }:=\frac{B(\cdot \otimes y_B)}{\Vert B(\cdot \otimes y_B)\Vert }\in S_{X^{\ast }}.$$

$$B^{\prime \prime }:=\frac{B(x_B^{\prime }\otimes \cdot )}{\Vert B(x_B^{\prime }\otimes \cdot )\Vert }\in S_{Y^{\ast }}.$$

$$B(u_B)$$

$$\ge (1-\epsilon {)}^{\frac{2}{3}}\Vert B(\cdot \otimes y_B)\Vert \ge (1-\epsilon {)}^{\frac{2}{3}}B(x_B\otimes y_B)\ge 1-\epsilon ,$$

$$x_n^{\ast }(m)\ne 0\text{ and }\Vert x_n^{\ast }(n)\Vert \ge \Vert x_n^{\ast }(m)\Vert \text{ for all }n,m\in \mathbb{N}.$$

$$U_{n,m}:=\{x\in B_{c_0(\mathbb{N},X_n)}:x_n^{\ast }(x)>1-m^{-1}\Vert x_n^{\ast }(m)\Vert \}$$

$$1-m^{-1}\Vert x_p^{\ast }(m)\Vert$$

$$=1-\Vert x_p^{\ast }(p)\Vert +x_p^{\ast }(p)(x_{p,m}(p)),$$

$$x_p^{\ast }(p)(x_{p,m}(p))\ge \Vert x_p^{\ast }(p)\Vert -m^{-1}\Vert x_p^{\ast }(m)\Vert ,$$

$$\Vert x_{p,m}(p)\Vert \ge 1-m^{-1}\frac{\Vert x_p^{\ast }(m)\Vert }{\Vert x_p^{\ast }(p)\Vert }\ge 1-m^{-1}.$$

$$1\ge \Vert x_{p,m}\pm y\Vert \ge \Vert x_{p,m}(p)\pm y(p)\Vert .$$

$${\delta }_{X_p}(2-2\epsilon )$$

$$\le \underset{m}{\inf }\left(1-\frac{\Vert (x_{p,m}(p)+y(p))+(x_{p,m}(p)-y(p))\Vert }{2}\right)$$

$$=\underset{m}{\inf }(1-\Vert x_{p,m}(p)\Vert )\le \underset{m}{\inf }{m}^{-1}$$

$$=0,$$

$$\left\{\sum _{i=1}^{n_U}{r}_i{x}_i:x_{i,U}^{\ast }(x_i)>1-{\epsilon }_U\text{ for all }1\le i\le n_U\right\}\subset U.$$

$$\{x\in B_X:|x_{i,V}^{\ast }(x)|\le {\epsilon }_V\text{ for all }1\le i\le n_V\}\subset V.$$

$$x_U:=\sum _{i=1}^{n_U}{r}_i{x}_{i,U}\in B_X$$

$$\Vert x_U\pm y\Vert =\Vert \sum _{i=1}^{n_U}{r}_i(x_{i,U}\pm y)\Vert \le \sum _{i=1}^{n_U}{r}_i\Vert x_{i,U}\pm y\Vert =1.$$

$$1=\Vert x_{i,V}\pm y\Vert \ge x_{i,V}^{\ast }(x_{i,V}\pm y)\ge 1-{\epsilon }_V\pm x_{i,V}^{\ast }(y),$$

$$d(X):=\min \{|\mathcal{D}|:\mathcal{D}\subset X\text{ is dense}\}+{\aleph }_0.$$

$$c(X):=\sup \{|\mathcal{C}|:\mathcal{C}\text{ is a cellular family in }X\}+{\aleph }_0.$$

$$\Vert g\Vert \ge 2/3,f_x(x)\ge 2/3\text{ and }\Vert f_x\pm g\Vert \le 1.$$

Full text

This work was supported by the Estonian Research Council grants (PSG487) and (PRG1501).

1. Introduction

Given an infinite-dimensional real Banach space $X$, its topological dual, its unit ball and its unit sphere are denoted by $X^{*}$, $B_{X}$ and $S_{X}$, respectively.

Definition 1.1**.**

A Banach space $X$ has the symmetric strong diameter two property ($\operatorname{SSD2P}$) if, and only if, for every $x^{*}_{1},\ldots,x^{*}_{n}\in S_{X^{*}}$ and $\varepsilon>0$, there are $x_{1},\ldots,x_{n},y\in B_{X}$ such that $\|y\|\geq 1-\varepsilon$, $x_{i}\pm y\in B_{X}$ and $x^{*}_{i}(x_{i})\geq 1-\varepsilon$ for all $1\leq i\leq n$.

The $\operatorname{SSD2P}$ was introduced in [2], but the original definition contains the additional requirement that $x^{*}_{i}(x_{i}\pm y)\geq 1-\varepsilon$, which is redundant. Indeed, if we require that $x^{*}_{i}(x_{i})\geq 1-\varepsilon/2$, since $x_{i}\pm y\in B_{X}$, then $|x^{*}_{i}(y)|\leq\varepsilon/2$ and, therefore, Definition 1.1 is equivalent to the original one.

Examples of Banach spaces enjoying the $\operatorname{SSD2P}$ include Lindenstrauss spaces, uniform algebras, almost square Banach spaces, Banach spaces with an infinite dimensional centralizer, somewhat regular subspaces of $C_{0}(X)$ spaces, where $X$ is an infinite locally compact Hausdorff space, and Müntz spaces (see [8]).

In [3] transfinite analogues of the $\operatorname{SSD2P}$ were defined, but, before recalling these definitions, let us introduce some notation. Given $r\in\mathbb{(}0,1)$, $B\subset B_{X}$ and $A\subset S_{X^{*}}$, we say that $B$ $r$-norms $A$ if, for every $x^{*}\in A$, there is $x\in B$ such that $x^{*}(x)\geq r$. In addition, we say that $B$ norms $A$ if it $r$-norms it for all $r\in(0,1)$.

Definition 1.2**.**

[3, Definition 5.3] Let $X$ be a Banach space and $\kappa$ an infinite cardinal.

• (i)

$X$ has the $\operatorname{SSD2P}_{\kappa}$ if, for every set $A\subset S_{X^{*}}$ of cardinality $<\kappa$ and $\varepsilon>0$, there are $B\subset B_{X}$, which $(1-\varepsilon)$-norms $A$, and $y\in B_{X}$ satisfying $B\pm y\subset B_{X}$ with $\|y\|\geq 1-\varepsilon$.

• (ii)

$X$ has the $\operatorname{1-ASSD2P}_{\kappa}$ if, for every set $A\subset S_{X^{*}}$ of cardinality $<\kappa$, there are $B\subset S_{X}$, which norms $A$, and $y\in S_{X}$ satisfying $B\pm y\subset S_{X}$.

In the following, we aim to investigate these transfinite extensions of the $\operatorname{SSD2P}$ and, in particular, to show differences in their behaviour when compared to the regular $\operatorname{SSD2P}$.

Now, let us also recall the transfinite extensions of almost squareness and the strong diameter two property.

Definition 1.3**.**

[3, Definition 2.1] Let $X$ be a Banach space and $\kappa$ a cardinal.

• (i)

$X$ is $\operatorname{ASQ}_{\kappa}$ if, for every set $A\subset S_{X}$ of cardinality $<\kappa$ and $\varepsilon>0$, there exists $y\in S_{X}$ such that $\|x\pm y\|\leq 1+\varepsilon$ holds for all $x\in A$.

• (ii)

$X$ is $\operatorname{SQ}_{\kappa}$ if, for every set $A\subset S_{X}$ of cardinality $<\kappa$, there exists $y\in S_{X}$ such that $\|x\pm y\|\leq 1$ holds for all $x\in A$.

Definition 1.4**.**

[5, Definitions 2.11 and 2.12] Let $X$ be a Banach space and $\kappa$ an infinite cardinal.

• (i)

$X$ has the $\operatorname{SD2P}_{\kappa}$ if, for every set $A\subset S_{X^{*}}$ of cardinality $<\kappa$ and $\varepsilon>0$, there are $B\subset B_{X}$, which $(1-\varepsilon)$-norms $A$, and $x^{*}\in B_{X^{*}}$ satisfying $x^{*}(x)\geq 1-\varepsilon$ for all $x\in B$.

• (ii)

$X$ has the $\operatorname{1-ASD2P}_{\kappa}$ if, for every set $A\subset S_{X^{*}}$ of cardinality $<\kappa$, there are $B\subset S_{X}$, which norms $A$, and $x^{*}\in S_{X^{*}}$ satisfying $x^{*}(x)=1$ for all $x\in B$.

It is clear that every $\operatorname{ASQ}_{\kappa}$ ($\operatorname{SQ}_{\kappa}$, respectively) Banach space enjoys the $\operatorname{SSD2P}_{\kappa}$ ($\operatorname{1-ASSD2P}_{\kappa}$, respectively). Moreover, it was shown in [3, Proposition 5.4] that the $\operatorname{SSD2P}_{\kappa}$ ($\operatorname{1-ASSD2P}_{\kappa}$, respectively) implies the $\operatorname{SD2P}_{\kappa}$ ($\operatorname{1-ASD2P}_{\kappa}$, respectively). To sum up, the following implications hold true.

1.1. Content of the paper

In Section 2, we study the stability of the transfinite $\operatorname{SSD2P}$ with respect to operations between Banach spaces.

We provide a complete description concerning $c_{0}$ and $\ell_{\infty}$ sums (see Theorems 2.1 and 2.2), which informally state that these sums of Banach spaces enjoy the $\operatorname{SSD2P}_{\kappa}$ if, and only if, we can always find one component which satisfies a property which arbitrarily well approximates the $\operatorname{SSD2P}_{\kappa}$. Thanks to these characterizations, we show that, for example, the Banach spaces $c_{0}(\mathbb{N},\ell_{n}(\kappa))$ and $\ell_{\infty}(\mathbb{N},\ell_{n}(\kappa))$ enjoy the $\operatorname{SSD2P}_{\kappa}$ (see Example 2.4).

We also investigate the behaviour of the $\operatorname{SSD2P}_{\kappa}$ under projective tensor products. Namely, we prove that the Banach space $X\hat{\otimes}_{\pi}Y$ has the $\operatorname{SSD2P}_{\kappa}$, whenever $X$ and $Y$ enjoy the property.

We conclude this section by studying the difference in behaviour of the transfinite $\operatorname{SSD2P}$ compared to the finite $\operatorname{SSD2P}$. In particular, we prove that, for the transfinite case, it is not possible to replace the functionals with relatively weakly open sets in Definition 1.2, even though it is possible for the traditional $\operatorname{SSD2P}$ (see Fact 2.6 (ii)). Moreover, we prove that an equivalent internal description of the $\operatorname{SSD2P}$ (see Fact 2.6 (iii)) also fails in the transfinite case.

Section 3 is dedicated to extend the class of known examples which possess the transfinite $\operatorname{SSD2P}$. To this aim, we search for a description in the class of $C_{0}(X)$ spaces, whenever $X$ is a T4 locally compact space. The main result of this section states that the Banach space $C_{0}(X)$ fails the $\operatorname{SSD2P}_{\kappa}$, where $\kappa$ is the successor cardinal of the density character of $X$, but it enjoys the $\operatorname{1-ASSD2P}_{\mu}$, where $\mu$ is the cellularity of $X$ (see Theorem 3.1).

Thanks to this result, new examples are provided, e.g. $C[0,1]$ and $\ell_{\infty}$ fail the $\operatorname{SSD2P}_{\aleph_{1}}$, $C(\beta\mathbb{N}\setminus\mathbb{N})$ enjoys the $\operatorname{1-ASSD2P}_{2^{\aleph_{0}}}$ and $\ell_{\infty}(\kappa)$ has the $\operatorname{1-ASSD2P}_{\kappa}$, whenever $\kappa>\aleph_{0}$.

1.2. Notation

Given a sequence of Banach spaces $(X_{n})$ we define

[TABLE]

endowed with the usual supremum norm. Moreover, we set

[TABLE]

Eventually, given a cardinal $\kappa$, we call $\operatorname{cf}(\kappa)$ its cofinality and $\kappa^{+}$ its successor cardinal.

2. Stability results

In this section we investigate the behaviour of the transfinite $\operatorname{SSD2P}$ with respect to operations between Banach spaces.

2.1. Direct sums

Given a sequence of Banach spaces $(X_{n})$, it is known that the $c_{0}$ sum $c_{0}(\mathbb{N},X_{n})$ is always $\operatorname{ASQ}$ [1, Example 3.1] and, therefore, has the $\operatorname{SSD2P}$. Moreover, it was proved in [3, Proposition 4.3] that the $c_{0}$ sum $c_{0}(\mathscr{A},X_{\alpha})$ of a family of Banach spaces $\{X_{\alpha}:\alpha\in\mathscr{A}\}$ is $\operatorname{SQ}_{|\mathscr{A}|}$ and thus has the $\operatorname{1-ASSD2P}_{|\mathscr{A}|}$, whenever $|\mathscr{A}|>\aleph_{0}$. For these reasons, in the following we will focus only on countable $c_{0}$ sums with respect to the $\operatorname{SSD2P}_{\kappa}$, for $\kappa>\aleph_{0}$.

Theorem 2.1**.**

Let $(X_{n})$ be a sequence of Banach spaces and $\kappa>\aleph_{0}$. If, for every $r\in[0,1)$, there is $n\in\mathbb{N}$ such that, for every set $A\subset S_{X^{*}_{n}}$ of cardinality $<\kappa$, there exist $B\subset B_{X_{n}}$ and $y\in B_{X_{n}}$ such that $\|y\|\geq r$, $B$ $r$-norms $A$ and $B\pm y\subset B_{X_{n}}$, then $c_{0}(\mathbb{N},X_{n})$ enjoys the $\operatorname{SSD2P}_{\kappa}$. If in addition $\operatorname{cf}(\kappa)>\aleph_{0}$, then the vice-versa also holds.

Proof.

Fix a set $A\subset S_{\ell_{1}(\mathbb{N},X^{*}_{n})}$ of cardinality $<\kappa$ and $\varepsilon>0$. Let $\{x^{*}_{\alpha}:\alpha\in\mathscr{A}\}$ be an enumeration of $A$ and find $m\in\mathbb{N}$ as in the statement for $r=1-\varepsilon$.

By assumption there are $y^{m}\in B_{X_{m}}$ and $x^{m}_{\alpha}\in B_{X_{m}}$ such that $\|y^{m}\|\geq 1-\varepsilon$, $x^{m}_{\alpha}\pm y^{m}\in B_{X_{m}}$ and $x^{*}_{\alpha}(m)(x^{m}_{\alpha})\geq(1-\varepsilon)^{\frac{1}{2}}\|x^{*}_{\alpha}(m)\|$ hold for every $\alpha\in\mathscr{A}$.

For each $m\not=n\in\mathbb{N}$ and $\alpha\in\mathscr{A}$, find $x^{n}_{\alpha}\in B_{X_{n}}$ satisfying $x^{*}_{\alpha}(n)(x^{n}_{\alpha})\geq(1-\varepsilon)^{\frac{1}{2}}\|x^{*}_{\alpha}(n)\|$. Moreover, since $x^{*}_{\alpha}\in\ell_{1}(\mathbb{N},X_{n}^{*})$, there exists $n_{\alpha}\geq m$ such that

[TABLE]

Now call

[TABLE]

and $y:=y^{m}e_{m}\in B_{c_{0}(\mathbb{N},X_{n})}$.

Notice that

[TABLE]

which means that the set $\{x_{\alpha}:\alpha\in\mathscr{A}\}$ $(1-\varepsilon)$-norms $A$.

On the other hand, $\|y\|=\|y^{m}\|\geq 1-\varepsilon$ and

[TABLE]

holds for all $\alpha\in\mathscr{A}$. Therefore, $c_{0}(\mathbb{N},X_{n})$ enjoys the $\operatorname{SSD2P}_{\kappa}$.

For the vice-versa, fix $\varepsilon>0$ and, for every $n\in\mathbb{N}$, $A_{n}\subset S_{X_{n}^{*}}$ of cardinality $<\kappa$. Call

[TABLE]

and notice that $|A|\leq\aleph_{0}\cdot\sup|A_{n}|<\kappa$, because $\operatorname{cf}(\kappa)>\aleph_{0}$. Therefore, there exist a set $B\subset B_{c_{0}(\mathbb{N},X_{n})}$, which $(1-\varepsilon)$-norms $A$, and $y\in B_{c_{0}(\mathbb{N},X_{n})}$ such that $\|y\|\geq 1-\varepsilon$ and $B\pm y\subset B_{c_{0}(\mathbb{N},X_{n})}$.

Since $\|y\|\geq 1-\varepsilon$, there exists $n\in\mathbb{N}$ satisfying $\|y(n)\|\geq 1-\varepsilon$. Moreover, from the fact that, in particular, $B$ $(1-\varepsilon)$-norms the set $\{x^{*}\delta_{n}:x^{*}\in A_{n}\}$, we deduce that the set $B_{n}:=\{x(n):x\in B\}\subset B_{X_{n}}$ $(1-\varepsilon)$-norms $A_{n}$.

Eventually, notice that, given $x(n)\in B_{n}$,

[TABLE]

which concludes the proof. ∎

Notice that the same proof can be adjusted to $\ell_{\infty}$ sums too. As a matter of fact, it is not needed to find $n_{\alpha}$ as in the proof of Theorem 2.1 and one can define

[TABLE]

By doing so, the following theorem is easily proved, up to a few minor changes.

Theorem 2.2**.**

Let $\{X_{\alpha}:\alpha\in\mathscr{A}\}$ be a family of Banach spaces and $\kappa>\aleph_{0}$. If, for every $r\in[0,1)$, there is $\alpha\in\mathscr{A}$ such that, for every set $A\subset S_{X^{*}_{\alpha}}$ of cardinality $<\kappa$, there exist $B\subset B_{X_{\alpha}}$ and $y\in B_{X_{\alpha}}$ such that $\|y\|\geq r$, $B$ $r$-norms $A$ and $B\pm y\subset B_{X_{\alpha}}$, then $\ell_{\infty}(\mathscr{A},X_{\alpha})$ enjoys the $\operatorname{SSD2P}_{\kappa}$. If in addition $\operatorname{cf}(\kappa)>|\mathscr{A}|$, then the vice-versa also holds.

Corollary 2.3**.**

Let $X$ and $Y$ be Banach spaces and $\kappa>\aleph_{0}$. Either $X$ or $Y$ enjoy the $\operatorname{SSD2P}_{\kappa}$ if, and only if, $X\oplus_{\infty}Y$ enjoys the $\operatorname{SSD2P}_{\kappa}$.

Proof.

Apply Theorem 2.2 with $|\mathscr{A}|=2$. ∎

Example 2.4**.**

Let $\kappa>\aleph_{0}$. We claim that $c_{0}(\mathbb{N},\ell_{n}(\kappa))$ enjoys the $\operatorname{SSD2P}_{\kappa}$, despite the fact that it is a sum of reflexive spaces. To this aim, fix $\varepsilon>0$ and choose any $m\in\mathbb{N}$ satisfying $2^{\frac{1}{m}}\leq 1+\varepsilon$.

Now fix a set $A\subset S_{\ell_{m}(\kappa)^{*}}$ of cardinality $<\kappa$ and let $\{x^{*}_{\alpha}:\alpha\in\mathscr{A}\}$ be an enumeration for $A$. Moreover, find, for each $\alpha\in\mathscr{A}$, $x_{\alpha}\in S_{\ell_{m}(\kappa)}$ satisfying $x^{*}_{\alpha}(x_{\alpha})\geq 1-\varepsilon$.

Since the support of the $x_{\alpha}$’s is at most countable and $\kappa>\aleph_{0}$, there exists an ordinal $\mu<\kappa$ such that $x_{\alpha}(\mu)=0$ holds for all $\alpha\in\mathscr{A}$. Call $y:=\delta_{\lambda}\in B_{\ell_{m}(\kappa)}$ and notice that

[TABLE]

holds for every $\alpha\in\mathscr{A}$. Notice that, up to a small perturbation argument, we showed that the Banach spaces $\ell_{n}(\kappa)$’s satisfy the hypothesis of Theorem 2.1, thus the claim is proved.

It is then clear that also the Banach space $\ell_{\infty}(\mathbb{N},\ell_{n}(\kappa))$ enjoys the $\operatorname{SSD2P}_{\kappa}$ thanks to Theorem 2.2.

Remark 2.5*.*

One might wonder whether Theorem 2.1 can be pushed further and used to obtain $c_{0}$ sums which possess the $\operatorname{1-ASSD2P}_{\kappa}$. Unfortunately this doesn’t happen, as a matter of fact, the space $c_{0}(\mathbb{N},\ell_{n}(\kappa))$ fails the $\operatorname{1-ASSD2P}_{\kappa}$ because, if by absurd it had the property, then Theorem 2.11 would apply and this would lead to a contradiction when combined with Theorem 2.9.

2.2. Tensor product

It is known that the $\operatorname{SSD2P}$ is preserved by taking projective tensor products [10, Theorem 2.2]. In the cited paper, the authors’ proof relies on the following characterization of the $\operatorname{SSD2P}$.

Fact 2.6**.**

[8, Theorem 2.1]** Let $X$ be a Banach space. The following assertions are equivalent:

• (i)

$X$* has the $\operatorname{SSD2P}$.*

• (ii)

Given non-empty relatively weakly open sets $U_{1},\ldots,U_{n}$ in $B_{X}$ and $\varepsilon>0$, there exist $x_{1},\ldots,x_{n},y\in B_{X}$ such that $\|y\|\geq 1-\varepsilon$, $x_{i}\pm y\in B_{X}$ and $x_{i}\in U_{i}$ for all $1\leq i\leq n$.

• (iii)

Given $x_{1},\ldots x_{n}\in S_{X}$, there exist nets $(y^{i}_{\alpha})$ and $(z_{\alpha})$ in $S_{X}$ such that $\lim\|y^{i}_{\alpha}\pm z_{\alpha}\|=1$ and, with respect to the weak topology on $X$, $\lim z_{\alpha}=0$ and $\lim y^{i}_{\alpha}=x_{i}$ hold for all $1\leq i\leq n$.

As we will later demonstrate, Fact 2.6 doesn’t hold true for the $\operatorname{SSD2P}_{\kappa}$ whenever $\kappa>\aleph_{0}$. Therefore, a different proof is required to extend [10, Theorem 2.2] to the transfinite setting.

Theorem 2.7**.**

Let $X$ and $Y$ be Banach spaces and $\kappa>\aleph_{0}$. If $X$ and $Y$ have the $\operatorname{SSD2P}_{\kappa}$, then the projective tensor product $X\hat{\otimes}_{\pi}Y$ enjoys the $\operatorname{SSD2P}_{\kappa}$.

Proof.

Fix a set $\mathscr{B}\subset S_{(X\hat{\otimes}_{\pi}Y)^{*}}$ of cardinality $<\kappa$ and $\varepsilon>0$. Recall that the Banach space $(X\hat{\otimes}_{\pi}Y)^{*}$ is isometrically isomorphic to the space of bounded bilinear forms acting on $X\times Y$ [12, Theorem 2.9], hence, for every $B\in\mathscr{B}$, there exists $x_{B}\otimes y_{B}\in S_{X}\otimes S_{Y}$ satisfying $B(x_{B}\otimes y_{B})\geq(1-\varepsilon)^{\frac{1}{3}}$.

Given $B\in\mathscr{B}$, define

[TABLE]

Since $X$ has the $\operatorname{SSD2P}_{\kappa}$, there are $x$ and $x^{\prime}_{B}$’s in $B_{X}$ such that $\|x\|\geq(1-\varepsilon)^{\frac{1}{2}}$ and, for all $B\in\mathscr{B}$, $x^{\prime}_{B}\pm x\in B_{X}$ and $B^{\prime}(x^{\prime}_{B})\geq(1-\varepsilon)^{\frac{1}{3}}$.

Now, given $B\in\mathscr{B}$, call

[TABLE]

Since $Y$ has the $\operatorname{SSD2P}_{\kappa}$, there are $y$ and $y^{\prime\prime}_{B}$’s in $B_{Y}$ such that $\|y\|\geq(1-\varepsilon)^{\frac{1}{2}}$ and, for all $B\in\mathscr{B}$, $y^{\prime\prime}_{B}\pm y\in B_{Y}$ and $B^{\prime\prime}(y^{\prime\prime}_{B})\geq(1-\varepsilon)^{\frac{1}{3}}$.

Define $u_{B}:=x^{\prime}_{B}\otimes y^{\prime\prime}_{B}\in B_{X\hat{\otimes}_{\pi}Y}$ and $v:=x\otimes y\in B_{X\hat{\otimes}_{\pi}Y}$. Notice that $\|v\|=\|x\|\|y\|\geq 1-\varepsilon$, moreover, the fact that $u_{B}\pm v\in B_{X\hat{\otimes}_{\pi}Y}$ is due to [10, Lemma 2.1]. Eventually, let us prove that the set $\{u_{B}:B\in\mathscr{B}\}$ $(1-\varepsilon)$-norms $\mathscr{B}$.

[TABLE]

which proves the claim and thus concludes the proof. ∎

Remark 2.8*.*

It is known that requiring only one component to have the $\operatorname{SSD2P}$ is not enough in order to ensure the projective tensor product to enjoy the $\operatorname{SSD2P}$ [11, Corollary 3.9]. Up to a few changes, the same ideas can be used to show that requiring in the statement of Theorem 2.7 only one component to enjoy the $\operatorname{SSD2P}_{\kappa}$ is not enough. Let us sketch the argument required to prove this statement.

We will later show that $\ell_{\infty}(\kappa)$ has the $\operatorname{1-ASSD2P}_{\kappa}$ (see Example 3.3), nevertheless, we claim that the Banach space $X:=\ell_{\infty}(\kappa)\hat{\otimes}_{\pi}\ell_{3}^{3}$ doesn’t enjoy the $\operatorname{SSD2P}_{\kappa}$.

Since $\ell_{3}^{3}$ is not finitely representable in $\ell_{1}$, so it is not finitely representable in $\ell_{1}(\kappa)$ either (notice that each finite dimensional subspace of $\ell_{1}(\kappa)$ is isometrically isomorphic to some finite dimensional subspace of $\ell_{1}$, and vice-versa). Thanks to a simple transfinite analogue of [11, Lemma 3.7] (replacing finite dimensional spaces with spaces of density $<\kappa$) we conclude that $\ell_{1}(\kappa)\hat{\otimes}_{\varepsilon}\ell_{3^{*}}^{3}$ is not $\kappa$-octahedral (see [4, Definition 5.3]), moreover, we can infer that $X=(\ell_{1}(\kappa)\hat{\otimes}_{\varepsilon}\ell_{3^{*}}^{3})^{*}$ [12, Theorem 5.3]. Therefore, by applying [5, Theorem 3.2], we conclude that $X$ fails the $\operatorname{SSD2P}_{\kappa}$.

2.3. Some more remarks

Previously we claimed that the transfinite analogue of Fact 2.6 doesn’t hold true. Let us now prove this statement for the implication (i)$\iff$(ii) by continuing the investigation began in Example 2.4. As a matter of fact, the Banach space $c_{0}(\mathbb{N},\ell_{n}(\kappa))$ enjoys the $\operatorname{SSD2P}_{\kappa}$, nevertheless we claim that it fails condition (ii) from Fact 2.6 with respect to $\aleph_{1}$. This claim follows from the following theorem.

Theorem 2.9**.**

Let $(X_{n})$ be a sequence of Banach spaces. If, given any sequence of relatively weakly open sets $(U_{n})$ in $B_{c_{0}(\mathbb{N},X_{n})}$ and $\varepsilon>0$, there exist $(x_{n})$ and $y$ in $B_{c_{0}(\mathbb{N},X_{n})}$ such that $\|y\|\geq 1-\varepsilon$, $x_{n}\pm y\in B_{c_{0}(\mathbb{N},X_{n})}$ and $x_{n}\in U_{n}$ for all $n\in\mathbb{N}$, then there exists $m\in\mathbb{N}$ such that $X_{m}$ is not uniformly convex.

Proof.

Let $A:=\{x^{*}_{n}:n\in\mathbb{N}\}\subset S_{\ell_{1}(\mathbb{N},X_{n}^{*})}$, where the $x^{*}_{n}$’s are any chosen elements satisfying the following conditions:

[TABLE]

Now consider the relatively weakly open sets

[TABLE]

Fix $\varepsilon>0$ and find $x_{n,m}\in U_{n,m}$ and $y\in B_{c_{0}(\mathbb{N},X_{n})}$ such that $\|y\|\geq 1-\varepsilon$ and $y\pm x_{n,m}\in B_{c_{0}(\mathbb{N},X_{n})}$ hold for all $n,m\in\mathbb{N}$.

Since $\|y\|\geq 1-\varepsilon$, we can find $p\in\mathbb{N}$ such that $\|y(p)\|\geq 1-\varepsilon$. On the other hand, since $x_{p,m}\in U_{p,m}$, we have that

[TABLE]

hence

[TABLE]

therefore

[TABLE]

Now, the fact that $x_{p,m}\pm y\in B_{c_{0}(\mathbb{N},X_{n})}$ implies

[TABLE]

Finally, let us compute the modulus of convexity of $X_{p}$.

[TABLE]

which implies that $X_{p}$ is not uniformly convex. ∎

Let us now turn our attention to the implication (i)$\iff$(iii) from Fact 2.6. We claim that also this fails in the transfinite context.

Example 2.10**.**

We will prove that $\ell_{\infty}(\kappa)$ fails the $\operatorname{SSD2P}_{\kappa^{+}}$ (see Example 3.3). Nevertheless, condition (iii) from Fact 2.6 is satisfied in a very strong way. In fact, fix $x\in\ S_{\ell_{\infty}(\kappa)}$, an ordinal $\mu<\kappa$ and call $y^{x}_{\mu}:=x-x(\mu)e_{\mu}\in B_{\ell_{\infty}(\kappa)}$ and $z_{\mu}=\delta_{\mu}\in S_{\ell_{\infty}(\kappa)}$. It is then clear that $y^{x}_{\mu}\pm z_{\mu}\in S_{X}$ and that, with respect to the weak topology, $\lim z_{\mu}=0$ and $\lim y^{x}_{\mu}=x$ holds for every $x\in S_{\ell_{\infty}(\kappa)}$. In other words, since $|\ell_{\infty}(\kappa)|=2^{\kappa}$, we showed that $\ell_{\infty}(\kappa)$ satisfies condition (iii) from Fact 2.6 with respect to $2^{\kappa}$.

Despite Theorem 2.9 and Example 2.10, it is possible to recover some transfinite analogue of Fact 2.6, but only for the $\operatorname{1-ASSD2P}_{\kappa}$.

Proposition 2.11**.**

Let $X$ be a Banach space and $\kappa>\aleph_{0}$. Consider the following statements:

• (i)

$X$* has the $\operatorname{1-ASSD2P}_{\kappa}$.*

• (ii)

Given a family $\mathscr{U}$ consisting of $<\kappa$ many relatively weakly open sets in $B_{X}$, a relatively weakly open neighborhood $V$ of [math] in $B_{X}$ and $\varepsilon>0$, there are $\{x_{U}:U\in\mathscr{U}\}$ and $y\in V\cap S_{X}$ satisfying $x_{U}\in U$ and $x_{u}\pm y\in B_{X}$ for all $U\in\mathscr{U}$.

• (iii)

Given $A\subset S_{X}$ of cardinality $<\kappa$, there are nets $\{(y^{x}_{\alpha}):x\in A\}$ and $(z_{\alpha})$ in $S_{X}$ satisfying $\lim\|z_{\alpha}\pm y^{x}_{\alpha}\|=1$ and, with respect to the weak topology, $\lim z_{\alpha}=0$ and $\lim y^{x}_{\alpha}=x$ for all $x\in A$.

Then (i)$\implies$(ii)$\implies$(iii).

Proof.

(i)$\implies$(ii). Fix a family $\mathscr{U}$ consisting of $<\kappa$ many relatively weakly open sets in $B_{X}$, a relatively weakly open neighborhood $V$ of [math] in $B_{X}$ and $\varepsilon>0$. For every $U\in\mathscr{U}$, thanks to Bourgain’s lemma [7, Lemma II.1], we can find functionals $x^{*}_{1,U},\ldots,x^{*}_{n_{U},U}\in S_{X^{*}}$, $\varepsilon_{U}>0$ and convex coefficients $r_{1,U},\ldots,r_{n_{U},U}$ such that

[TABLE]

Moreover, we can find $x^{*}_{1,V},\ldots,x^{*}_{n_{V},V}\in S_{X^{*}}$ and $\varepsilon_{V}>0$ satisfying

[TABLE]

Since $X$ has the $\operatorname{1-ASSD2P}_{\kappa}$ and $|\{x^{*}_{i,U}:1\leq i\leq n_{U}\text{ and }U\in\mathscr{U}\cup\{V\}\}|\leq\aleph_{0}\cdot|\mathscr{U}|<\kappa$, there exist $\{x_{i,U}:1\leq i\leq n_{U}\text{ and }U\in\mathscr{U}\cup\{V\}\}$ and $y$ in $S_{X}$ satisfying $x^{*}_{i,U}(x_{i,U})\geq 1-\varepsilon_{U}$ and $x_{i,U}\pm y\in S_{X}$ for all $1\leq i\leq n_{U}$ and $U\in\mathscr{U}\cup\{V\}$. Now, given $U\in\mathscr{U}$, define

[TABLE]

and notice that $x_{U}\in U$. Moreover,

[TABLE]

In order to conclude, it only remains to prove that $y\in V$. But this is clear because, for every $1\leq i\leq n_{V}$ we have that

[TABLE]

which means that $|x^{*}_{i,V}(y)|\leq\varepsilon_{V}$, hence $y\in V$.

(ii)$\implies$(iii). Fix a set $A\subset S_{X}$ of cardinality $<\kappa$ and temporarily fix a weak neighborhood $U$ of [math]. Define $\mathscr{U}:=\{(x+U)\cap B_{X}:x\in A\}$ and find $\{y_{U}^{x}:x\in A\}\subset B_{X}$ and $z_{U}\in U\cap S_{X}$ satisfying $y_{U}^{x}\in x+U$ and $y_{U}^{x}\pm z_{U}\in B_{X}$ for all $x\in A$.

Now semi-order the family of weakly open neighborhoods of [math] with respect to the inclusion and consider the nets $(y_{U}^{x})$ and $(z_{U})$. It is clear that $\lim y^{x}_{U}=x$, $\lim z_{U}=0$ and $\lim\|z_{U}\pm y^{x}_{U}\|=1$ holds for all $x\in A$. Moreover, up to a perturbation argument, we can assume that all $y_{U}^{x}$’s belong to $S_{X}$. Thus the claim is proved. ∎

Let us show that the implication (iii)$\implies$(ii) from Proposition 2.11 fails. As already witnessed by Example 2.10, $\ell_{\infty}(\kappa)$ satisfies condition (iii) in a very strong way, nevertheless it fails the $\operatorname{SSD2P}_{\kappa^{+}}$. Therefore, we only need to notice that condition (ii) with respect to $\kappa^{+}$ clearly implies possessing the $\operatorname{SSD2P}_{\kappa^{+}}$, thus the claim is proved.

It remains unclear whether the implication (ii)$\implies$(i) hold.

3. $C_{0}(X)$ spaces

In [2] it was proved that $C_{0}(X)$, for $X$ infinite Hausdorff locally compact, always has the $\operatorname{SSD2P}$. In this section we aim to extend the class of examples which enjoy the transfinite $\operatorname{SSD2P}$ by trying to characterize under which conditions $C_{0}(X)$ spaces have this property. Before doing so, let us introduce a bit of notation about some cardinal functions.

Let $X$ be a topological space. Define the density character of $X$ as

[TABLE]

A cellular family in $X$ is a family of mutually disjoint open sets in $X$. Define the cellularity of $X$ as

[TABLE]

It is well known that $\operatorname{c}(X)\leq\operatorname{d}(X)$. We refer the reader to [9] for a detailed treatment about these cardinal functions and more.

Before stating the main result of this section, let us recall that, thanks to the Riesz–Markov representation theorem, every continuous linear functional on $C_{0}(X)$ admits a unique representation as a regular countably additive Borel measure on $X$.

Theorem 3.1**.**

Let $X$ be a $T_{4}$ locally compact space.

• (i)

$C_{0}(X)$* fails the $\operatorname{SSD2P}_{\operatorname{d}(X)^{+}}$.*

• (ii)

If $\operatorname{c}(X)>\aleph_{0}$, then $C_{0}(X)$ has the $\operatorname{1-ASSD2P}_{\operatorname{c}(X)}$.

Proof.

$(i)$. Let $\mathscr{D}$ be dense in $X$. Consider the set $\{\delta_{x}:x\in\mathscr{D}\}\subset S_{C_{0}(X)^{*}}$ and suppose by contradiction that $C_{0}(X)$ has the $\operatorname{SSD2P}_{\operatorname{d}(X)^{+}}$. Then we can find functions $\{f_{x}:x\in\mathscr{D}\}\subset B_{C_{0}(X)}$ and $g\in B_{C_{0}(X)}$ satisfying

[TABLE]

Since $\mathscr{D}$ is dense, then we can find $x\in\mathscr{D}$ such that $|g(x)|>1/3$, which contradicts the fact that $|f_{x}(x)\pm g(x)|\leq 1$.

$(ii)$. Fix $\lambda<\operatorname{c}(X)$ and a set $\mathscr{M}\subset S_{C_{0}(X)^{*}}$ of cardinality $\lambda$. Find a cellular family $\mathscr{C}$ in $X$ of size $\lambda<|\mathscr{C}|\leq\operatorname{c}(X)$ and, given any $m\in\mathbb{N}$ and $\mu\in\mathscr{M}$, define

[TABLE]

Notice that

[TABLE]

Therefore, there is $C\in\mathscr{C}$ satisfying $|\mu|(C)=0$ for every $\mu\in\mathscr{M}$. Notice that, without loss of generality, we can assume that $|\mu|(\overline{C})=0$. In fact, if that’s not the case, then we can replace $C$ with some non-empty open set $C^{\prime}$ satisfying $\overline{C^{\prime}}\subset C$.

Find functions $\{f_{m,\mu}:m\in\mathbb{N}\text{ and }\mu\in\mathscr{M}\}\subset S_{C_{0}(X)}$ such that $\mu(f_{m,\mu})\geq 1-(3m)^{-1}$ and, since $\mu$’s are regular, compact sets $\{K_{m,\mu}:m\in\mathbb{N}\text{ and }\mu\in\mathscr{M}\}\subset X\setminus\overline{C}$ satisfying $|\mu|(K_{m,\mu})\geq 1-(3m)^{-1}$. Now construct Urysohn’s functions $\{g_{m,\mu}:m\in\mathbb{N}\text{ and }\mu\in\mathscr{M}\}$ and $h$ in $S_{C_{0}(X)}$ satisfying

[TABLE]

Define

[TABLE]

and notice that $i_{m,\mu}\pm h\in S_{C_{0}(X)}$. Moreover, given any $m\in\mathbb{N}$ and $\mu\in\mathscr{M}$,

[TABLE]

∎

It remains unclear whether the statement of Theorem 3.1 can be written using only one cardinal function. Namely, we don’t know the answer to the following two questions.

Question 3.2**.**

Let $X$ be a $T_{4}$ locally compact space. Is it true that $C_{0}(X)$ fails the $\operatorname{SSD2P}_{\operatorname{c}(X)^{+}}$? Is it true that $C_{0}(X)$ enjoys the $\operatorname{1-ASSD2P}_{\operatorname{d}(X)}$, whenever $\operatorname{d}(X)>\aleph_{0}$?

Example 3.3**.**

Let us now employ Theorem 3.1 to produce some new examples of spaces enjoying or failing the transfinite $\operatorname{SSD2P}$.

• (i)

Let $X$ be a separable locally compact Hausdorff space. It is clear that $\operatorname{c}(X)\leq\operatorname{d}(X)=\aleph_{0}$, hence $C_{0}(X)$ fails the $\operatorname{SSD2P}_{\aleph_{1}}$.

• (ii)

It is known that $\operatorname{c}(\beta\mathbb{N}\setminus\mathbb{N})=2^{\aleph_{0}}$ [9, 7.22], therefore $C(\beta\mathbb{N}\setminus\mathbb{N})$ enjoys the $\operatorname{1-ASSD2P}_{2^{\aleph_{0}}}$.

• (iii)

Let $B$ be a Boolean algebra and let $S(B)$ be the Stone space associated to $B$. It is clear that the set $\{\{b\}:b\in B\}\subset S(B)$ defines a cellular family in $S(B)$.

Now let us consider a regular positive Borel measure $\mu$ over some T4 locally compact space $X$. Call $\mathfrak{B}_{\mu}$ the set of measurable sets modulo the negligible sets in $X$. It is known that $L_{\infty}(\mu)$ is isometrically isomorphic to $C(S(\mathfrak{B}_{\mu}))$ (see e.g. pages 27–29 in [6]), therefore we conclude that $L_{\infty}(\mu)$ enjoys the $\operatorname{1-ASSD2P}_{|\mathfrak{B}_{\mu}|}$, whenever $|\mathfrak{B}_{\mu}|>\aleph_{0}$.

In particular, whenever $\kappa>\aleph_{0}$ and $\mu$ is the counting measure over $\kappa$, $|\mathfrak{B}_{\mu}|=\kappa$, thus it follows that $\ell_{\infty}(\kappa)$ enjoys the $\operatorname{1-ASSD2P}_{\kappa}$, but it fails the $\operatorname{SSD2P}_{\kappa^{+}}$, because $\operatorname{d}(\ell_{\infty}(\kappa))=\kappa$.

To conclude this section, let us provide a criterion to identify cellular families in particular classes of topological spaces, including Alexandrov-discrete spaces.

Proposition 3.4**.**

Let $X$ be a $T_{2\frac{1}{2}}$ space and $\kappa$ an infinite cardinal. If there are $\kappa$ many points in $X$ such that every non-empty intersection of at most $\kappa$ many neighborhoods is still a neighborhood, then $\operatorname{c}(X)\geq\kappa$.

Proof.

Let $A\subset X$ be a set of cardinality $\kappa$ such that every non-empty intersection of at most $\kappa$ many neighborhoods of $x$ is still a neighborhood for every $x\in A$. For every distinct $x,y\in A$ find a closed neighborhood $U_{x,y}$ of $x$ which doesn’t contain $y$. By assumption

[TABLE]

is an open neighborhood of $x$. Notice that, given distinct $x,y\in A$ we have that

[TABLE]

In other words, $\{U_{x}:x\in A\}$ defines a cellular family of size $\kappa$. ∎

Notice that the assumption in Proposition 3.4 is far from being necessary. As a matter of fact, it is consistent with ZFC that $\beta\mathbb{N}\setminus\mathbb{N}$ contains no P-points, that is, points for which every $G_{\delta}$ containing them is a neighborhood, nevertheless, as already recalled in Example 3.3, $\beta\mathbb{N}\setminus\mathbb{N}$ has a cellular family of cardinality $2^{\aleph_{0}}$.