On the Transfinite Symmetric Strong Diameter Two Property

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 Hausdorff locally compact space, which possesses 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.

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 (${{\,\textrm{SSD2P}\,}}$) 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 $\Vert y\Vert \ge 1-\varepsilon$, $x_i\pm y\in B_X$ and $x^*_i(x_i)\ge 1-\varepsilon$ for all $1\le i\le n$.

The ${{\,\textrm{SSD2P}\,}}$ was introduced in [2], but the original definition contains the additional requirement that $x^*_i(x_i\pm y)\ge 1-\varepsilon$, which is redundant. Indeed, if we require that $x^*_i(x_i)\ge 1-\varepsilon /2$, since $x_i\pm y\in B_X$, then $|x^*_i(y)|\le \varepsilon /2$ and, therefore, Definition 1.1 is equivalent to the original one.

Examples of Banach spaces enjoying the ${{\,\textrm{SSD2P}\,}}$ 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 analogs of the ${{\,\textrm{SSD2P}\,}}$ 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 $x^*(x)\ge r$. 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 ${{\,\textrm{SSD2P}\,}}_{\kappa }$ 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 $\Vert y\Vert \ge 1-\varepsilon$.

• (ii) X has the ${{\,\mathrm{1-ASSD2P}\,}}_{\kappa }$ 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 $B\pm y\subset S_X$.

Here 1-A stands for 1-norming and attaining, respectively. In the following, we aim to investigate these transfinite extensions of the ${{\,\textrm{SSD2P}\,}}$ and, in particular, to show differences in their behavior when compared to the regular ${{\,\textrm{SSD2P}\,}}$.

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 ${{\,\textrm{ASQ}\,}}_{\kappa }$ if, for every set $A\subset S_X$ of cardinality $<\kappa$ and $\varepsilon >0$, there exists $y\in S_X$ such that $\Vert x\pm y\Vert \le 1+\varepsilon$ holds for all $x\in A$.

• (ii) X is ${{\,\textrm{SQ}\,}}_{\kappa }$ if, for every set $A\subset S_X$ of cardinality $<\kappa$, there exists $y\in S_X$ such that $\Vert x\pm y\Vert \le 1$ holds for all $x\in A$.

It should be noted that a small change of notation is here applied. We denote (A)SQ$_\kappa$ what was written (A)SQ$_{<\kappa }$ in [3, Definition 2.1].

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 ${{\,\textrm{SD2P}\,}}_{\kappa }$ 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 $x^*(x)\ge 1-\varepsilon$ for all $x\in B$.

• (ii) X has the ${{\,\mathrm{1-ASD2P}\,}}_{\kappa }$ 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 ${{\,\textrm{ASQ}\,}}_{\kappa }$ (${{\,\textrm{SQ}\,}}_{\kappa }$, respectively) Banach space enjoys the ${{\,\textrm{SSD2P}\,}}_{\kappa }$ (${{\,\mathrm{1-ASSD2P}\,}}_{\kappa }$, respectively). Moreover, it was shown in [3, Proposition 5.4] that the ${{\,\textrm{SSD2P}\,}}_{\kappa }$ (${{\,\mathrm{1-ASSD2P}\,}}_{\kappa }$, respectively) implies the ${{\,\textrm{SD2P}\,}}_{\kappa }$ (${{\,\mathrm{1-ASD2P}\,}}_{\kappa }$, respectively). To sum up, the following implications hold true.

1.1 Content of the Paper

In Sect. 2, we study the stability of the transfinite ${{\,\textrm{SSD2P}\,}}$ 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 ${{\,\textrm{SSD2P}\,}}_{\kappa }$ if, and only if, we can always find one component which satisfies a property which arbitrarily well approximates the ${{\,\textrm{SSD2P}\,}}_{\kappa }$. Thanks to these characterizations, we show that, for example, the Banach spaces $c_0({\mathbb {N}}_{\ge 2},\ell _n(\kappa ))$ and $\ell _\infty ({\mathbb {N}}_{\ge 2},\ell _n(\kappa ))$ enjoy the ${{\,\textrm{SSD2P}\,}}_{\kappa }$ (see Example 2.4).

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

We conclude Sect. 2 by studying the difference in the behavior of the transfinite ${{\,\textrm{SSD2P}\,}}$ compared to the finite ${{\,\textrm{SSD2P}\,}}$. 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 ${{\,\textrm{SSD2P}\,}}$ (see Fact 2.5 (ii)). Moreover, we prove that an equivalent internal description of the ${{\,\textrm{SSD2P}\,}}$ (see Fact 2.5 (iii)) also fails in the transfinite case.

Section 3 is dedicated to extending the class of known examples which possess the transfinite ${{\,\textrm{SSD2P}\,}}$. To this aim, we search for a description in the class of $C_0(X)$ spaces, whenever X is a Hausdorff locally compact space. The main result of this section states that the Banach space $C_0(X)$ fails the ${{\,\textrm{SSD2P}\,}}_{\kappa }$, where $\kappa$ is the successor cardinal of the density character of X, but it enjoys the ${{\,\mathrm{1-ASSD2P}\,}}_{\mu }$, 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 ${{\,\textrm{SSD2P}\,}}_{\aleph _1}$, $C(\beta {\mathbb {N}}{\setminus }{\mathbb {N}})$ enjoys the ${{\,\mathrm{1-ASSD2P}\,}}_{2^{\aleph _0}}$ and $\ell _\infty (\kappa )$ has the ${{\,\mathrm{1-ASSD2P}\,}}_{\kappa }$, whenever $\kappa >\aleph _0$.

1.2 Notation

Given a sequence of Banach spaces $(X_n)$ we define

$$\begin{aligned} \ell _\infty ({\mathbb {N}},X_n):=\left\{ x\in \prod _{n=1}^\infty X_n:(\forall n\in {\mathbb {N}})\hspace{0.1cm}x(n)\in X_n\text { and }\sup _{n}\Vert x(n)\Vert <\infty \right\} \end{aligned}$$

endowed with the usual supremum norm. Moreover, we set

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

Finally, given a cardinal $\kappa$, we define ${{\,\textrm{cf}\,}}(\kappa )$ its cofinality and $\kappa ^+$ its successor cardinal.

2 Stability Results

In this section, we investigate the behavior of the transfinite ${{\,\textrm{SSD2P}\,}}$ 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 ${{\,\textrm{ASQ}\,}}$ [1, Example 3.1] and, therefore, has the ${{\,\textrm{SSD2P}\,}}$. 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 ${{\,\textrm{SQ}\,}}_{|{\mathscr {A}}|}$ and thus has the ${{\,\mathrm{1-ASSD2P}\,}}_{|{\mathscr {A}}|}$, whenever $|{\mathscr {A}}|>\aleph _0$. For these reasons, in the following, we will focus only on countable $c_0$ sums with respect to the ${{\,\textrm{SSD2P}\,}}_{\kappa }$, 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 $\Vert y\Vert \ge r$, B r-norms A and $B\pm y\subset B_{X_n}$, then $c_0({\mathbb {N}},X_n)$ enjoys the ${{\,\textrm{SSD2P}\,}}_{\kappa }$. If in addition ${{\,\textrm{cf}\,}}(\kappa )>\aleph _0$, then the converse 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 $r=(1-\varepsilon )^\frac{1}{2}$.

By assumption, there are $y^m\in B_{X_m}$ and $x^m_\alpha \in B_{X_m}$ such that $\Vert y^m\Vert \ge 1-\varepsilon$, $x^m_\alpha \pm y^m\in B_{X_m}$ and $x^*_\alpha (m)(x^m_\alpha )\ge (1-\varepsilon )^\frac{1}{2}\Vert x^*_\alpha (m)\Vert$ 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 $x^*_\alpha (n)(x^n_\alpha )\ge (1-\varepsilon )^\frac{1}{2}\Vert x^*_\alpha (n)\Vert$. Moreover, since $x^*_\alpha \in \ell _1({\mathbb {N}},X_n^*)$, there exists $n_\alpha \ge m$ such that

$$\begin{aligned} \sum _{1\le n\le n_\alpha }\Vert x^*_\alpha (n)\Vert \ge (1-\varepsilon )^\frac{1}{2}. \end{aligned}$$

Now define

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

and $y:=y^me_m\in B_{c_0({\mathbb {N}},X_n)}$.

Notice that

$$\begin{aligned} x^*_\alpha (x_\alpha )=\sum _{1\le n\le n_\alpha }x^*_\alpha (n)(x^n_\alpha )\ge (1-\varepsilon )^\frac{1}{2}\sum _{1\le n\le n_\alpha }\Vert x^*_\alpha (n)\Vert \ge 1-\varepsilon , \end{aligned}$$

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

On the other hand, $\Vert y\Vert =\Vert y^m\Vert \ge 1-\varepsilon$ and

$$\begin{aligned} \Vert x_\alpha \pm y\Vert =\max \left\{ \Vert x_\alpha ^m\pm y^m\Vert ,\sup _{1\le n\not =m\le n_\alpha }\Vert x_\alpha ^n\Vert \right\} \le 1 \end{aligned}$$

holds for all $\alpha \in {\mathscr {A}}$. Therefore, $c_0({\mathbb {N}},X_n)$ enjoys the ${{\,\textrm{SSD2P}\,}}_{\kappa }$.

For the converse, fix $\varepsilon >0$ and, for every $n\in {\mathbb {N}}$, $A_n\subset S_{X_n^*}$ of cardinality $<\kappa$. Define

$$\begin{aligned} A:=\{x^*e_n:n\in {\mathbb {N}}\text { and }x^*\in A_n\}\subset S_{\ell _1({\mathbb {N}},X_n^*)} \end{aligned}$$

and notice that $|A|\le \aleph _0\cdot \sup |A_n|<\kappa$, because ${{\,\textrm{cf}\,}}(\kappa )>\aleph _0$. 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 $\Vert y\Vert \ge 1-\varepsilon$ and $B\pm y\subset B_{c_0({\mathbb {N}},X_n)}$.

Since $\Vert y\Vert \ge 1-\varepsilon$, there exists $n\in {\mathbb {N}}$ satisfying $\Vert y(n)\Vert \ge 1-\varepsilon$. Moreover, from the fact that, in particular, B $(1-\varepsilon )$-norms the set $\{x^*e_n:x^*\in A_n\}$, we deduce that the set $B_n:=\{x(n):x\in B\}\subset B_{X_n}$ $(1-\varepsilon )$-norms $A_n$.

Finally, notice that, given $x(n)\in B_n$,

$$\begin{aligned} \Vert x(n)\pm y(n)\Vert \le \Vert x\pm y\Vert \le 1, \end{aligned}$$

which concludes the proof. $\square$

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

$$\begin{aligned} x_\alpha :=\sum _{n=1}^\infty x_\alpha ^ne_n\in B_{\ell _\infty ({\mathbb {N}},X_n)}. \end{aligned}$$

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 $\Vert y\Vert \ge r$, B r-norms A and $B\pm y\subset B_{X_\alpha }$, then $\ell _\infty ({\mathscr {A}},X_\alpha )$ enjoys the ${{\,\textrm{SSD2P}\,}}_{\kappa }$. If in addition ${{\,\textrm{cf}\,}}(\kappa )>|{\mathscr {A}}|$, then the converse also holds.

Corollary 2.3

Let X and Y be Banach spaces and $\kappa >\aleph _0$. Either X or Y enjoy the ${{\,\textrm{SSD2P}\,}}_{\kappa }$ if, and only if, $X\oplus _\infty Y$ enjoys the ${{\,\textrm{SSD2P}\,}}_{\kappa }$.

Proof

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

Example 2.4

Let $\kappa >\aleph _0$. We claim that $c_0({\mathbb {N}}_{\ge 2},\ell _n(\kappa ))$ enjoys the ${{\,\textrm{SSD2P}\,}}_{\kappa }$, despite the fact that it is a sum of reflexive spaces. To this aim, observe that the claim is immediately achieved thanks to [3, Example 3.1], but, for the sake of providing an application of Theorem 2.1, let us prove it here again. Notice that, it suffices to show that the Banach spaces $\ell _n(\kappa )$’s satisfy the hypothesis of Theorem 2.1. To this purpose, fix $\varepsilon >0$ and choose any $m\in {\mathbb {N}}$ satisfying $2^\frac{1}{m}\le 1+\varepsilon$. 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 $x^*_\alpha (x_\alpha )\ge 1-\varepsilon$.

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 $\alpha \in {\mathscr {A}}$. Let $y\in B_{\ell _m(\kappa )}$ be defined by $y(\mu ):=\delta _\lambda ^\mu$ and notice that

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

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 $\ell _\infty ({\mathbb {N}}_{\ge 2},\ell _n(\kappa ))$ enjoys the ${{\,\textrm{SSD2P}\,}}_{\kappa }$, thanks to Theorem 2.2.

2.2 Tensor Product

It is known that the ${{\,\textrm{SSD2P}\,}}$ 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 ${{\,\textrm{SSD2P}\,}}$.

Fact 2.5

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

• (i) X has the ${{\,\textrm{SSD2P}\,}}$.

• (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 $\Vert y\Vert \ge 1-\varepsilon$, $x_i\pm y\in B_X$ and $x_i\in U_i$ for all $1\le i\le n$.

• (iii) Given $x_1,\ldots x_n\in S_X$, there exist nets $(y^i_\alpha )$ and $(z_\alpha )$ in $S_X$ such that $\lim \Vert y^i_\alpha \pm z_\alpha \Vert =1$ and, with respect to the weak topology on X, $\lim z_\alpha =0$ and $\lim y^i_\alpha =x_i$ hold for all $1\le i\le n$.

As we will later demonstrate, Fact 2.5 doesn’t hold true for the ${{\,\textrm{SSD2P}\,}}_{\kappa }$ whenever $\kappa >\aleph _0$. Therefore, a different proof is required to extend [10, Theorem 2.2] to the transfinite setting.

Theorem 2.6

Let X and Y be Banach spaces and $\kappa >\aleph _0$. If X and Y have the ${{\,\textrm{SSD2P}\,}}_{\kappa }$, then the projective tensor product $X{\hat{\otimes }}_{\pi }Y$ enjoys the ${{\,\textrm{SSD2P}\,}}_{\kappa }$.

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 $X\times Y$ [13, Theorem 2.9], hence, for every $B\in {\mathscr {B}}$, there exists $x_B\otimes y_B\in S_X\otimes S_Y$ satisfying $B(x_B\otimes y_B)\ge (1-\varepsilon )^\frac{1}{3}$.

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

$$\begin{aligned} B':=\frac{B(\cdot \otimes y_B)}{\Vert B(\cdot \otimes y_B)\Vert }\in S_{X^*}. \end{aligned}$$

Since X has the ${{\,\textrm{SSD2P}\,}}_{\kappa }$, there are x and $x'_B$’s in $B_X$ such that $\Vert x\Vert \ge (1-\varepsilon )^\frac{1}{2}$ and, for all $B\in {\mathscr {B}}$, $x'_B\pm x\in B_X$ and $B'(x'_B)\ge (1-\varepsilon )^\frac{1}{3}$.

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

$$\begin{aligned} B'':=\frac{B(x'_B\otimes \cdot )}{\Vert B(x'_B\otimes \cdot )\Vert }\in S_{Y^*}. \end{aligned}$$

Since Y has the ${{\,\textrm{SSD2P}\,}}_{\kappa }$, there are y and $y''_B$’s in $B_Y$ such that $\Vert y\Vert \ge (1-\varepsilon )^\frac{1}{2}$ and, for all $B\in {\mathscr {B}}$, $y''_B\pm y\in B_Y$ and $B''(y''_B)\ge (1-\varepsilon )^\frac{1}{3}$.

Define $u_B:=x'_B\otimes y''_B\in B_{X{\hat{\otimes }}_{\pi }Y}$ and $v:=x\otimes y\in B_{X{\hat{\otimes }}_{\pi }Y}$. Notice that $\Vert v\Vert =\Vert x\Vert \Vert y\Vert \ge 1-\varepsilon$. Moreover, the fact that $u_B\pm v\in B_{X{\hat{\otimes }}_{\pi }Y}$ is due to [12, Lemma 2.2]. Finally, let us prove that the set $\{u_B:B\in {\mathscr {B}}\}$ $(1-\varepsilon )$-norms ${\mathscr {B}}$.

$$\begin{aligned} B(u_B)&=B(x'_B\otimes y''_B)\ge (1-\varepsilon )^\frac{1}{3}\Vert B(x'_B\otimes \cdot )\Vert \ge (1-\varepsilon )^\frac{1}{3}B(x'_B\otimes y_B)\\&\ge (1-\varepsilon )^\frac{2}{3}\Vert B(\cdot \otimes y_B)\Vert \ge (1-\varepsilon )^\frac{2}{3}B(x_B\otimes y_B)\ge 1-\varepsilon , \end{aligned}$$

which proves the claim and thus concludes the proof. $\square$

Remark 2.7

It is known that requiring only one component to have the ${{\,\textrm{SSD2P}\,}}$ is not enough in order to ensure the projective tensor product enjoys the ${{\,\textrm{SSD2P}\,}}$ [11, Corollary 3.9]. Up to a few changes, the same ideas can be used to show that requiring in the statement of Theorem 2.6 only one component to enjoy the ${{\,\textrm{SSD2P}\,}}_{\kappa }$ is not enough. Let us sketch the argument required to prove this statement.

We will later show that $\ell _\infty (\kappa )$ has the ${{\,\mathrm{1-ASSD2P}\,}}_{\kappa }$ (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 ${{\,\textrm{SSD2P}\,}}_{\kappa }$.

Since $\ell _3^3$ is not finitely representable in $\ell _1$, 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 converse). 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 $X=(\ell _1(\kappa ){\hat{\otimes _\varepsilon }}(\ell _{3}^3)^*)^*$ [13, Theorem 5.3]. Therefore, by applying [5, Theorem 3.2], we conclude that X fails the ${{\,\textrm{SSD2P}\,}}_{\kappa }$.

2.3 Some More Remarks

Previously we claimed that the transfinite analog of Fact 2.5 doesn’t hold true. Let us now prove this statement for the implication (i)$\iff$(ii) by continuing the investigation that we began in Example 2.4. As a matter of fact, the Banach space $c_0({\mathbb {N}}_{\ge 2},\ell _n(\kappa ))$ enjoys the ${{\,\textrm{SSD2P}\,}}_{\kappa }$. Nevertheless, we claim that it fails condition (ii) from Fact 2.5 with respect to $\aleph _1$. This claim follows from the following theorem:

Theorem 2.8

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 $\Vert y\Vert \ge 1-\varepsilon$, $x_n\pm y\in B_{c_0({\mathbb {N}}, X_n)}$ 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:

$$\begin{aligned} x^*_n(m)\not =0\text { and }\Vert x^*_n(n)\Vert \ge \Vert x^*_n(m)\Vert \text { for all }n,m\in {\mathbb {N}}. \end{aligned}$$

Now consider the relatively weakly open sets

$$\begin{aligned} U_{n,m}:=\{x\in B_{c_0({\mathbb {N}}, X_n)}:x^*_n(x)>1-m^{-1}\Vert x^*_n(m)\Vert \} \end{aligned}$$

Fix $\varepsilon >0$ and find $x_{n,m}\in U_{n,m}$ and $y\in B_{c_0({\mathbb {N}}, X_n)}$ such that $\Vert y\Vert \ge 1-\varepsilon$ and $y\pm x_{n,m}\in B_{c_0({\mathbb {N}}, X_n)}$ hold for all $n,m\in {\mathbb {N}}$.

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

$$\begin{aligned} 1-m^{-1}\Vert x^*_p(m)\Vert&\le x^*_p(x_{p,m})\le \sum _{n\not =p}\Vert x^*_p(n)\Vert +x^*_p(p)(x_{p,m}(p))\\&=1-\Vert x^*_p(p)\Vert +x^*_p(p)(x_{p,m}(p)), \end{aligned}$$

hence

$$\begin{aligned} x^*_p(p)(x_{p,m}(p))\ge \Vert x^*_p(p)\Vert -m^{-1}\Vert x^*_p(m)\Vert , \end{aligned}$$

therefore

$$\begin{aligned} \Vert x_{p,m}(p)\Vert \ge 1-m^{-1}\frac{\Vert x^*_p(m)\Vert }{\Vert x^*_p(p)\Vert }\ge 1-m^{-1}. \end{aligned}$$

Now, the fact that $x_{p,m}\pm y\in B_{c_0({\mathbb {N}}, X_n)}$ implies

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

Finally, let us compute the modulus of convexity of $X_p$.

$$\begin{aligned} \delta _{X_p}(2-2\varepsilon )&:=\inf \left\{ 1-\left\| \frac{u+v}{2}\right\| :u,v\in B_{X_p}\text { and }\Vert u-v\Vert \ge 2-2\varepsilon \right\} \\&\le \inf _m\left( 1-\frac{\Vert (x_{p,m}(p)+y(p))+(x_{p,m}(p)-y(p))\Vert }{2}\right) \\&=\inf _m(1-\Vert x_{p,m}(p)\Vert )\le \inf _m m^{-1}\\&=0, \end{aligned}$$

which implies that $X_p$ is not uniformly convex. $\square$

Let us now turn our attention to the implication (i)$\iff$(iii) from Fact 2.5. We claim that also this fails in the transfinite context.

Example 2.9

We will prove that $\ell _\infty (\kappa )$ fails the ${{\,\textrm{SSD2P}\,}}_{\kappa ^+}$ (see Example 3.3). Nevertheless, condition (iii) from Fact 2.5 is satisfied in a very strong way. In fact, fix $x\in \ S_{\ell _\infty (\kappa )}$, an ordinal $\mu <\kappa$ and define $y^x_\mu :=x-x(\mu )e_\mu \in B_{\ell _\infty (\kappa )}$ and $z_\mu :=e_{\mu }\in S_{\ell _\infty (\kappa )}$. 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 2.5, where, instead of fixing a finite set in the unit sphere, we can fix any subset of the unit sphere of cardinality at most $2^\kappa$.

Despite Theorem 2.8 and Example 2.9, it is possible to recover some transfinite analog of Fact 2.5, but only for the ${{\,\mathrm{1-ASSD2P}\,}}_{\kappa }$.

Proposition 2.10

Let X be a Banach space and $\kappa >\aleph _0$. Consider the following statements:

• (i) X has the ${{\,\mathrm{1-ASSD2P}\,}}_{\kappa }$.

• (ii) Given a family ${\mathscr {U}}$ consisting of $<\kappa$ many relatively weakly open sets in $B_X$, a relatively weakly open neighborhood V of 0 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 $\lim \Vert z_\alpha \pm y^x_\alpha \Vert =1$ 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 0 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

$$\begin{aligned} \left\{ \sum _{i=1}^{n_U}r_ix_i:x^*_{i,U}(x_i)>1-\varepsilon _U\text { for all }1\le i\le n_U\right\} \subset U. \end{aligned}$$

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

$$\begin{aligned} \{x\in B_X:|x^*_{i,V}(x)|\le \varepsilon _V\text { for all }1\le i\le n_V\}\subset V. \end{aligned}$$

Since X has the ${{\,\mathrm{1-ASSD2P}\,}}_{\kappa }$ and $|\{x^*_{i,U}:1\le i\le n_U\text { and }U\in {\mathscr {U}}\cup \{V\}\}|\le \aleph _0\cdot |{\mathscr {U}}|<\kappa$, there exist $\{x_{i,U}:1\le i\le n_U\text { and }U\in {\mathscr {U}}\cup \{V\}\}$ and y in $S_X$ satisfying $x^*_{i,U}(x_{i,U})\ge 1-\varepsilon _U$ and $x_{i,U}\pm y\in S_X$ for all $1\le i\le n_U$ and $U\in {\mathscr {U}}\cup \{V\}$. Now, given $U\in {\mathscr {U}}$, define

$$\begin{aligned} x_U:=\sum _{i=1}^{n_U}r_ix_{i,U}\in B_X \end{aligned}$$

and notice that $x_U\in U$. Moreover,

$$\begin{aligned} \Vert x_U\pm y\Vert =\left\| \sum _{i=1}^{n_U}r_i(x_{i,U}\pm y)\right\| \le \sum _{i=1}^{n_U}r_i\Vert x_{i,U}\pm y\Vert =1. \end{aligned}$$

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

$$\begin{aligned} 1=\Vert x_{i,V}\pm y\Vert \ge x^*_{i,V}(x_{i,V}\pm y)\ge 1-\varepsilon _V\pm x^*_{i,V}(y), \end{aligned}$$

which means that $|x^*_{i,V}(y)|\le \varepsilon _V$, 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 0. 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 0 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 $\lim \Vert z_U\pm y^x_U\Vert =1$ 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. $\square$

Let us show that the implication (iii)$\implies$(ii) from Proposition 2.10 fails. As already witnessed by Example 2.9, $\ell _\infty (\kappa )$ satisfies condition (iii) in a very strong way, nevertheless it fails the ${{\,\textrm{SSD2P}\,}}_{\kappa ^+}$. Therefore, we only need to notice that condition (ii) with respect to $\kappa ^+$ clearly implies possessing the ${{\,\textrm{SSD2P}\,}}_{\kappa ^+}$, thus the claim is proved.

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

Remark 2.11

One might have wondered whether Theorem 2.1 can be pushed further and used to obtain $c_0$ sums which possess the ${{\,\mathrm{1-ASSD2P}\,}}_{\kappa }$. Unfortunately, this doesn’t happen, as a matter of fact, the space $c_0({\mathbb {N}}_{\ge 2},\ell _n(\kappa ))$ fails the ${{\,\mathrm{1-ASSD2P}\,}}_{\kappa }$ because, if it had the property, then Theorem 2.10 would apply and this would lead to a contradiction when combined with Theorem 2.8.

3 $C_0(X)$ Spaces

In [2], it was proved that $C_0(X)$, for X infinite Hausdorff locally compact, always has the ${{\,\textrm{SSD2P}\,}}$. In this section, we aim to extend the class of examples that enjoy the transfinite ${{\,\textrm{SSD2P}\,}}$ 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

$$\begin{aligned} {{\,\textrm{d}\,}}(X):=\min \{|{\mathscr {D}}|:{\mathscr {D}}\subset X\text { is dense}\}+\aleph _0. \end{aligned}$$

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

$$\begin{aligned} {{\,\textrm{c}\,}}(X):=\sup \{|{\mathscr {C}}|:{\mathscr {C}}\text { is a cellular family in }X\}+\aleph _0. \end{aligned}$$

It is well known that ${{\,\textrm{c}\,}}(X)\le {{\,\textrm{d}\,}}(X)$. We refer the reader to [9] for a detailed treatment of 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 Hausdorff locally compact space.

• (i) $C_0(X)$ fails the ${{\,\textrm{SSD2P}\,}}_{{{\,\textrm{d}\,}}(X)^+}$.

• (ii) If ${{\,\textrm{c}\,}}(X)>\aleph _0$, then $C_0(X)$ has the ${{\,\mathrm{1-ASSD2P}\,}}_{{{\,\textrm{c}\,}}(X)}$.

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 for contradiction that $C_0(X)$ has the ${{\,\textrm{SSD2P}\,}}_{{{\,\textrm{d}\,}}(X)^+}$. Then we can find functions $\{f_x:x\in {\mathscr {D}}\}\subset B_{C_0(X)}$ and $g\in B_{C_0(X)}$ satisfying

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

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

(ii). Fix $\lambda <{{\,\textrm{c}\,}}(X)$ and a set ${\mathscr {M}}\subset S_{C_0(X)^*}$ of cardinality $\lambda$. Find a cellular family ${\mathscr {C}}$ in X of size $\lambda <|{\mathscr {C}}|\le {{\,\textrm{c}\,}}(X)$ and, given any $m\in {\mathbb {N}}$ and $\mu \in {\mathscr {M}}$, define

$$\begin{aligned} {\mathscr {C}}_{m,\mu }:=\{C\in {\mathscr {C}}:|\mu |(C)>m^{-1}\}. \end{aligned}$$

Notice that

$$\begin{aligned} |\{{\mathscr {C}}_{m,\mu }:m\in {\mathbb {N}}\text { and }\mu \in {\mathscr {M}}\}|\le \aleph _0\cdot \lambda <|{\mathscr {C}}|. \end{aligned}$$

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 $C'$ satisfying $\overline{C'}\subset C$.

Find functions $\{f_{m,\mu }:m\in {\mathbb {N}}\text { and }\mu \in {\mathscr {M}}\}\subset S_{C_0(X)}$ such that $\mu (f_{m,\mu })\ge 1-(3m)^{-1}$ 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 $|\mu |(K_{m,\mu })\ge 1-(3m)^{-1}$. 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

$$\begin{aligned} g_{m,\mu }|_{K_{m,\mu }}=1,\ g_{m,\mu }|_{{\overline{C}}}=0\text { and }h|_{X\setminus C}=0. \end{aligned}$$

Define

$$\begin{aligned} i_{m,\mu }:=\frac{f_{m,\mu }\cdot g_{m,\mu }}{\Vert f_{m,\mu }\cdot g_{m,\mu }\Vert }\in S_{C_0(X)} \end{aligned}$$

and notice that $i_{m,\mu }\pm h\in S_{C_0(X)}$. Moreover, given any $m\in {\mathbb {N}}$ and $\mu \in {\mathscr {M}}$,

$$\begin{aligned} \mu (i_{m,\mu })&\ge \int _X f_{m,\mu }\cdot g_{m,\mu }d\mu \ge \int _{K_{m,\mu }}f_{m,\mu }d\mu -(3m)^{-1}\\&\ge \int _X f_{m,\mu }d\mu -2\cdot (3m)^{-1}\ge 1-m^{-1}. \end{aligned}$$

$\square$

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 Hausdorff locally compact space. Is it true that $C_0(X)$ fails the ${{\,\textrm{SSD2P}\,}}_{{{\,\textrm{c}\,}}(X)^+}$? Is it true that $C_0(X)$ enjoys the ${{\,\mathrm{1-ASSD2P}\,}}_{{{\,\textrm{d}\,}}(X)}$, whenever ${{\,\textrm{d}\,}}(X)>\aleph _0$?

Example 3.3

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

• (i) Let X be a separable locally compact Hausdorff space. It is clear that ${{\,\textrm{c}\,}}(X)\le {{\,\textrm{d}\,}}(X)=\aleph _0$, hence $C_0(X)$ fails the ${{\,\textrm{SSD2P}\,}}_{\aleph _1}$.

• (ii) It is known that ${{\,\textrm{c}\,}}(\beta {\mathbb {N}}\setminus {\mathbb {N}})=2^{\aleph _0}$ [9, 7.22], therefore $C(\beta {\mathbb {N}}\setminus {\mathbb {N}})$ enjoys the ${{\,\mathrm{1-ASSD2P}\,}}_{2^{\aleph _0}}$.

• (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 Hausdorff locally compact space X. define ${\mathfrak {B}}_\mu$ 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 ${{\,\mathrm{1-ASSD2P}\,}}_{|{\mathfrak {B}}_\mu |}$, 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 ${{\,\mathrm{1-ASSD2P}\,}}_{\kappa }$, but it fails the ${{\,\textrm{SSD2P}\,}}_{\kappa ^+}$, because ${{\,\textrm{d}\,}}(\ell _\infty (\kappa ))=\kappa$.

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 ${{\,\textrm{c}\,}}(X)\ge \kappa$.

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 $x\in A$. Since X is $T_{2\frac{1}{2}}$, for every distinct $x,y\in A$ we can find a closed neighborhood $U_{x,y}$ of x which doesn’t contain y. By assumption

$$\begin{aligned} U_x:=\left( \bigcap _{y\in A\setminus \{x\}}U_{x,y}^\circ \right) \cap \left( \bigcap _{y\in A\setminus \{x\}}X\setminus U_{y,x}\right) \end{aligned}$$

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

$$\begin{aligned} U_x\cap U_y\subset U_{x,y}^\circ \cap (X\setminus U_{y,x})\cap U_{y,x}^\circ \cap (X\setminus U_{x,y})=\emptyset \end{aligned}$$

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

Notice that the assumption in Proposition 3.4 is far from being necessary. 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 $2^{\aleph _0}$.