3.1 Johnson-Lindenstrauss spaces, Boolean algebras, and Alexandrov-Urysohn compacta

We denote by $\omega = \{0, 1, 2 \dots , \}$ the set of non-negative integers and let $\operatorname {\mathrm {fin}}( \omega )$ be the family of all finite subsets of $\omega $ . Given $\mathcal {F} \subseteq \mathcal {P}(\omega )$ , we let $[\mathcal {F}]$ denote the smallest Boolean subalgebra of $\mathcal {P}(\omega )$ containing $\mathcal {F}$ .

A family $\mathcal {B}$ of infinite subsets of $\omega $ (or of any other countably infinite set) is called almost disjoint if for every distinct pair $A, B \in \mathcal {B}$ the intersection $A \cap B$ is finite. Given an almost disjoint family $\mathcal {B}$ , the Johnson-Lindenstrauss space $\operatorname {\mathrm {JL}} (\mathcal {B})$ is the closed linear span inside of $\ell _\infty $ of the set of characteristic functions

where stands for . Alternatively, let $\mathfrak {B} = [\mathcal {B \cup \operatorname {\mathrm {fin}}( \omega )}]$ , then $\operatorname {\mathrm {JL}}(\mathcal {B})$ is the closed linear span of simple $\mathfrak {B}$ -measurable functions, that is, functions of the form , where $n \in \omega $ , $r_i \in \mathbb {R}$ and $B_i \in \mathfrak {B}$ . It is known that $\operatorname {\mathrm {JL}}(\mathcal {B})$ is isometrically isomorphic to a $C(K)$ -space. The underlying compact Hausdorff space $\operatorname *{\mathrm {ult}}(\mathfrak {B})$ is the Stone space of the Boolean algebra $\mathfrak {B}$ . Equivalently, this space can be explicitly realized as follows. Define $K_{\mathcal {B}} = \omega \cup \{p_B: B \in \mathcal {B\}} \cup \{\infty \}$ , and specify a topology on $K_{\mathcal {B}}$ by

• • points in $\omega $ are isolated,

• • for every $B \in \mathcal {B}$ , a neighborhoods basis of $p_B$ is given by the sets $p_B \cup (B \setminus F)$ where $F \in \operatorname {\mathrm {fin}}( \omega )$ ,

• • $K_{\mathcal {B}}$ is the one-point compactification of the locally compact Hausdorff space $\omega \cup \{p_B: B \in \mathcal {B\}}$ , endowed with the above topology.

It is easy to see that this space is a separable, scattered, compact Hausdorff space with empty third Cantor-Bendixson derivative. Observe that the dual of $\operatorname {\mathrm {JL}}(\mathcal {B})$ is isometrically isomorphic to the space $M(\mathfrak {B})$ of real-valued finitely additive measures on $\mathfrak {B}$ of finite variation.

3.2 The space $M(\mathfrak {B})$

The measures in $M(\mathfrak {B})$ have a nice description. Since $\operatorname {\mathrm {JL}}(\mathcal {B})$ is isometrically isomorphic to $C(K_{\mathcal {B}})$ , where $K_{\mathcal {B}}$ is scattered, the Riesz Representation Theorem and Rudin’s theorem [Reference Rudin7] imply that

$$ \begin{align*} M(\mathfrak{B}) = \operatorname{\mathrm{JL}}(\mathcal{B})^* = C(K_{\mathcal{B}})^* = \ell_1(K_{\mathcal{B}}); \end{align*} $$

that is, every measure $\nu \in M(\mathfrak {B})$ can be expressed as

$$ \begin{align*} \nu = \sum_{s \in K_{\mathcal{B}}} c_s \delta_s \end{align*} $$

for some scalars $c_s \in \mathbb {R}$ , so that

$$ \begin{align*} \left\lVert\nu\right\rVert_1 = \sum_{s \in K_{\mathcal{B}}} |c_s|. \end{align*} $$

In particular, we can decompose $\nu $ as

$$ \begin{align*} \nu = \mu + \overline{\nu}, \hspace{5pt } \text{ where } \hspace{5pt }\mu = \sum_{s \in \omega} c_s \delta_s \hspace{5pt} \text{ and } \hspace{5pt} \overline{\nu} = \sum_{s \in K_{\mathcal{B}} \setminus \omega} c_s \delta_s. \end{align*} $$

We will call $\mu $ the $\ell _1(\omega )$ -part of $\nu $ .

For our purposes, we will usually have an enumeration of the almost disjoint family $\mathcal {B} = \{B_\xi : \xi < \mathfrak {c\}}$ , in that case, we can further identify

$$ \begin{align*} M(\mathfrak{B}) = \ell_1(\omega) \oplus \ell_1(\mathfrak{c}) \end{align*} $$

in a canonical way. It is worthwhile to pause briefly to expand on this representation of $M(\mathfrak {B})$ , as this will play an important role later.

Observe that given an element $x \in \ell _1(\omega )$ , we can naturally consider it as a measure $\mu \in M(\mathcal {P}(\omega ))$ by

$$ \begin{align*} \mu(B) = \sum_{s \in B} x(s), \hspace{5pt} B \subseteq \omega. \end{align*} $$

Furthermore, considering the restriction of $\mu $ to the Boolean subalgebra $\mathfrak {B}$ of $\mathcal {P}(\omega )$ , we can consider $\mu \in M(\mathfrak {B})$ .

Now, given a measure $\nu \in M(\mathfrak {B})$ , we can decompose $\nu = \mu + \overline {\nu }$ , where $\mu \in M(\mathfrak {B})$ arises from an element x of $\ell _1(\omega )$ and $\overline {\nu }$ vanishes on finite sets. Moreover, since $\overline {\nu }$ vanishes on finite sets, $|\overline {\nu }|(B) \neq 0$ for at most countably many $B \in \mathcal {B}$ . Indeed, if $|\overline {\nu }|(B) \neq 0$ for uncountably many $B \in \mathcal {B}$ , we could find $\varepsilon> 0$ and distinct sets $(B_n)_{n \in \omega } \subseteq \mathcal {B}$ such that $|\overline {\nu }|(B_n)> \varepsilon $ for all $n \in \omega $ . Since the family $\mathcal {B}$ is almost disjoint, we can find finite subsets $U_n \subseteq \omega $ such that the family $(B_n \setminus U_n)_{n \in \omega }$ is disjoint (e.g., take $U_n = B_n \cap \left ( \bigcup _{k < n} B_k \right )$ ) and thus

$$ \begin{align*} |\overline{\nu}| \geq \sum_{n=1}^\infty |\overline{\nu}|(B_n \setminus U_n) = \sum_{n=1}^\infty |\overline{\nu}|(B_n)> \sum_{n=1}^\infty \varepsilon, \end{align*} $$

where we used that $|\overline {\nu }|(B_n \setminus U_n) = |\overline {\nu }|(B_n)$ since $\overline {\nu }$ vanishes on finite sets. This yields a contradiction. A similar argument shows, in fact, that $(\overline {\nu }(B_\alpha ))_{\alpha < \mathfrak {c}}$ is absolutely summable. Thus we may identify $\overline {\nu }$ with an element $y \in \ell _1(\mathfrak {c})$ by setting $y(\alpha ) = \overline {\nu }(B_\alpha )$ for $\alpha < \mathfrak {c}$ .

Conversely, given $y \in \ell _1(\mathfrak {c})$ , we may associate to it a measure $\overline {\nu } \in M(\mathfrak {B})$ in this way. Thus, we may represent any $\nu \in M(\mathfrak {B})$ by a pair $(x,y) \in \ell _1(\omega ) \oplus \ell _1(\mathfrak {c})$ , where x corresponds to $\mu $ and y to $\overline {\nu }$ . Conversely, every $(x,y) \in \ell _1(\omega ) \oplus \ell _1(\mathfrak {c})$ determines a measure $\nu = \mu + \overline {\nu } \in M(\mathfrak {B})$ .

We now record a crucial observation. For any ordinal $\xi < \mathfrak {c}$ consider the subalgebra $\mathfrak {B}(\xi )$ of $\mathfrak {B}$ given by

$$ \begin{align*} \mathfrak{B}(\xi) = [\{B_\alpha: \alpha < \xi \} \cup \operatorname{\mathrm{fin}}( \omega )] \subseteq \mathfrak{B}, \end{align*} $$

and let $\nu \in M(\mathfrak {B}(\xi ))$ . Then $\nu $ naturally extends to an element of $M(\mathfrak {B})$ . As above, we decompose $\nu = \mu + \overline {\nu }$ , where $\mu $ is the $\ell _1(\omega )$ -part of $\nu $ . Observe that $\mu $ naturally defines a measure in $M(\mathfrak {B})$ , while $\overline {\nu }$ can be extended to $M(\mathfrak {B})$ by declaring $\overline {\nu }(B_\alpha ) = 0$ for every $\xi \leq \alpha < \mathfrak {c} $ .

Conversely, suppose $\nu \in M(\mathfrak {B})$ and decompose $\nu = \mu + \overline {\nu }$ , where $\mu $ is the $\ell _1(\omega )$ -part of $\nu $ . Since $|\overline {\nu }|(B) \neq 0$ for only countably many $B \in \mathfrak {B}$ , there exists $\xi < \mathfrak {c}$ such that $|\overline {\nu }|(B_\alpha ) = 0$ for all $\alpha \geq \xi $ . In particular, we can identify $\nu $ with a measure in $M(\mathfrak {B}(\xi ))$ , corresponding to its restriction to $\mathfrak {B}(\xi )$ . Observe that the natural extension of the latter recovers $\nu $ .

3.3 Complemented subspaces

As in the original work of Plebanek and Salguero-Alarcón, it will be convenient to work with $\omega \times 2$ instead of $\omega $ . Every subset C of the form $C = C_0 \times 2$ for some $C_0 \subseteq \omega $ is called a cylinder. For $n \in \omega $ we denote by $c_n$ the cylinder $\{(n, 0), (n,1)\}$ . We say that a set $B \subseteq C$ splits the cylinder C if $|B \cap c_n| = 1$ whenever $c_n \subseteq C$ . Observe that in this case $C \setminus B$ also splits C, and we say that B and $C \setminus B$ is a splitting of C.

Suppose that $\mathcal {A} = \{A_\xi : \xi < \mathfrak {c\}}$ is an almost disjoint family of cylinders in $\omega \times 2$ , together with a splitting $B^i_\xi $ , $i = 0,1$ of each cylinder $A_\xi $ . Observe that then $\mathcal {B} = \{B^0_\xi , B^1_\xi : \xi < \mathfrak {c}\}$ is also an almost disjoint family. In this situation, we will say that the almost disjoint family $\mathcal {B}$ splits $\mathcal {A}$ .

In this setting, we can find a $1$ -complemented isometric copy of $\operatorname {\mathrm {JL}}(\mathcal {A})$ inside $\operatorname {\mathrm {JL}}(\mathcal {B})$ , see [Reference Plebanek and Salguero-Alarcón5, Section 3]. With a slight abuse of notation, we will continue to denote this copy by $\operatorname {\mathrm {JL}}(\mathcal {A})$ . More precisely, this copy is the closed linear subspace of $\ell _\infty (\omega \times 2)$ spanned by

Equivalently, in the definition of $\operatorname {\mathrm {JL}}(\mathcal {A})$ , we restrict to finite cylinders rather than all finite subsets of $\omega \times 2$ .

It is worthwhile to briefly recall how $\operatorname {\mathrm {JL}}(\mathcal {A})$ sits inside $\operatorname {\mathrm {JL}}(\mathcal {B})$ . Concretely, $\operatorname {\mathrm {JL}}(\mathcal {A})$ consists of those functions in $\operatorname {\mathrm {JL}}(\mathcal {B})$ which are constant on cylinders. The map $P: \operatorname {\mathrm {JL}}(\mathcal {B}) \to \operatorname {\mathrm {JL}}(\mathcal {B})$ defined by

$$ \begin{align*} Pf(n,0) = Pf(n,1) = \tfrac{1}{2}\big(f(n,0) + f(n,1)\big) \end{align*} $$

is a norm-one projection. We denote $Y = \ker P$ , which can be explicitly described as

(2) $$ \begin{align} Y = \{f \in \operatorname{\mathrm{JL}}(\mathcal{B}) : f(n,0) = -f(n,1) \text{ for all } n \in \omega\}. \end{align} $$

Thus, we obtain the decomposition

(3) $$ \begin{align} C(K_{\mathcal{B}}) = \operatorname{\mathrm{JL}}(\mathcal{B}) = \operatorname{\mathrm{JL}}(\mathcal{A}) \oplus Y, \end{align} $$

which is precisely the content of [Reference Plebanek and Salguero-Alarcón5, Proposition 3.1]. Observe that by the discussion in Subsections 3.1 and 3.2, we can naturally identify

$$ \begin{align*} \operatorname{\mathrm{JL}}(\mathcal{B})^* = M(\mathfrak{B}) = C(K_{\mathcal{B}})^* = \ell_1(\omega \times 2) \oplus \ell_1(\mathcal{B}) = \ell_1(\omega \times 2) \oplus \ell_1(\mathfrak{c} \times 2), \end{align*} $$

where $\mathfrak {B} = [\mathcal {B} \cup \operatorname {\mathrm {fin}}( \omega \times 2 )]$ .

3.4 Two spaces

As we mentioned before, we will need to carry out two simultaneous constructions and thus, to avoid confusion, it is convenient to use the distinctive notation $\tilde {\omega } \times 2$ for the second copy of $\omega \times 2$ in which the second construction will be carried out.

We emphasize that the absence of a tilde indicates we are working in $\omega \times 2$ , whereas a tilde denotes objects in $\tilde {\omega } \times 2$ . For instance, a cylinder C is understood to satisfy $C \subseteq \omega \times 2$ , while a cylinder $\tilde {C}$ satisfies $\tilde {C} \subseteq \tilde {\omega } \times 2$ . It will also be convenient to sometimes use this when we are working with index sets. For example, we may write $(\tilde {c}_{\tilde {n}})_{\tilde {n} \in \tilde {\omega }}$ . When there is no ambiguity, we may simply omit the tilde in the index. In all cases, the intended meaning should be clear from the context.

Therefore, in our construction, we will build two almost disjoint families $\mathcal {B} = \{B^0_\xi , B^1_\xi : \xi < \mathfrak {c} \} \subseteq \mathcal {P}(\omega \times 2)$ and $\tilde {\mathcal {B}} = \{\tilde {B}^0_{\xi }, \tilde {B}^1_{\xi }: \xi < \mathfrak {c} \} \subseteq \mathcal {P}(\tilde {\omega } \times 2)$ splitting almost disjoint families of cylinders $\mathcal {A} = \{A_\xi : \xi < \mathfrak {c}\}$ and $\tilde {\mathcal {A}} = \{\tilde {A}_{\xi }: \xi < \mathfrak {c}\}$ . From the discussion above (3), it follows that

$$ \begin{align*} \operatorname{\mathrm{JL}}(\mathcal{B}) = \operatorname{\mathrm{JL}}(\mathcal{A}) \oplus Y \hspace{5pt} \text{ and } \hspace{5pt} \operatorname{\mathrm{JL}}(\tilde{\mathcal{B}}) = \operatorname{\mathrm{JL}}(\tilde{\mathcal{A}}) \oplus \tilde{Y}, \end{align*} $$

so that

$$ \begin{align*} \operatorname{\mathrm{JL}}(\mathcal{B}) \oplus \operatorname{\mathrm{JL}}(\tilde{\mathcal{B}}) = (\operatorname{\mathrm{JL}}(\mathcal{A}) \oplus Y) \oplus (\operatorname{\mathrm{JL}}(\tilde{\mathcal{A}}) \oplus \tilde{Y}) = X \oplus \tilde{X}, \end{align*} $$

where $X := \operatorname {\mathrm {JL}}(\tilde {A}) \oplus Y$ and $\tilde {X} := \operatorname {\mathrm {JL}}(A) \oplus \tilde {Y}$ . Observe that this is precisely the setting in (1), bearing in mind that $\operatorname {\mathrm {JL}}(\cdot )$ are isometric to $C(K)$ -spaces.

For brevity, we write

$$ \begin{align*} \operatorname{\mathrm{JL}} (\mathcal{B}, \tilde{\mathcal{B}}) = \operatorname{\mathrm{JL}} (\mathcal{B}) \oplus \operatorname{\mathrm{JL}} (\tilde{\mathcal{B}}) \hspace{5pt} \text{ and } \hspace{5pt} M(\mathfrak{B}, \tilde{\mathfrak{B}}) = \operatorname{\mathrm{JL}} (\mathcal{B}, \tilde{\mathcal{B}})^* = M(\mathfrak{B}) \oplus M(\tilde{\mathfrak{B}}). \end{align*} $$

We denote by Q and $\tilde {Q}$ the norm-one projections of $\operatorname {\mathrm {JL}} (\mathcal {B}, \tilde {\mathcal {B}})$ onto X and $\tilde {X}$ , respectively.

Furthermore, we denote by $M_1(\mathfrak {B}, \tilde {\mathfrak {B}})$ the unit ball of $\operatorname {\mathrm {JL}} (\mathcal {B}, \tilde {\mathcal {B}})^*$ , equipped with the weak $^*$ topology. We can naturally view $M(\mathfrak {B})$ and $M(\tilde {\mathfrak {B}})$ as subsets of $M(\mathfrak {B}, \tilde {\mathfrak {B}})$ . In this spirit, any measure $\nu \in M(\mathfrak {B})$ can be regarded as an element of $M(\mathfrak {B}, \tilde {\mathfrak {B}})$ , and similarly for $\tilde {\nu } \in M(\tilde {\mathfrak {B}})$ . Naturally, any measure $\lambda \in M(\mathfrak {B}, \tilde {\mathfrak {B}})$ can be decomposed as $\lambda = \nu +\tilde {\nu }$ where $\nu \in M(\mathfrak {B})$ and $\tilde {\nu } \in M(\tilde {\mathfrak {B}})$ .

3.5 Norming sets and Banach lattices

Our overall aim is to prove that, for a suitable choice of the families $\mathcal {A}, \tilde {\mathcal {A}}, \mathcal {B}$ and $\tilde {\mathcal {B}}$ , the resulting spaces X and $\tilde {X}$ are not isomorphic to Banach lattices. Therefore, we recall the conditions from [Reference de Hevia, Martínez-Cervantes, Salguero-Alarcón and Tradacete1] that ensure that this occurs. We denote by $B_X$ the unit ball of a Banach space X.

Definition 3.2. Let X be a Banach space and $0 < c \leq 1$ . We say that a sequence $(e_n^*)_{n\in \omega } \subseteq B_{X^*}$ is c-norming if

$$ \begin{align*} \left\lVert x\right\rVert \geq c \sup_{n \in \omega} |\langle e^*_n, x\rangle| \end{align*} $$

for every $x \in X$ . If the constant c is not relevant, we simply say that the sequence $(e_n^*)_{n\in \omega } \subseteq B_{X^*}$ is norming.

We have the following definition [Reference de Hevia, Martínez-Cervantes, Salguero-Alarcón and Tradacete1, Definition 4.2], which encapsulates the fact that the absolute value of f does not exist.

Definition 3.3. We say that a Banach space X has the Desired Property of De Hevia, Martínez-Cervantes, Salguero-Alarcón, and Tradacete, abbreviated (DP), if for every norming sequence $(e_n^*)_{n\in \omega }$ in $B_{X^*}$ there exists an element $f \in X$ such that no element $g \in X$ satisfies

(DP) $$ \begin{align} \langle e_n^*, g \rangle = |\langle e_n^*, f \rangle| \quad \text{for every } n \in \omega. \end{align} $$

The motivation behind the previous definition is that, under certain conditions, satisfying (DP) implies that the space is not isomorphic to a Banach lattice. In particular, we have the following, see [Reference de Hevia, Martínez-Cervantes, Salguero-Alarcón and Tradacete1, Corollary 4.3].

We note that, for any almost disjoint families $\mathcal {A}, \tilde {\mathcal {A}}, \mathcal {B}$ and $\tilde {\mathcal {B}}$ , the spaces X and $\tilde {X}$ , defined in Subsection 3.4, satisfy the conditions of Proposition 3.4 (see Section 6 for details). Therefore, to prove they are not isomorphic to Banach lattices, it is enough to show that they satisfy (DP).

3.6 The spaces $M(\mathfrak {B}, \tilde {\mathfrak {B}})$ , $X^*$ and $\tilde {X}^*$

Let $\mathcal {A}, \tilde {\mathcal {A}}, \mathcal {B}$ and $\tilde {\mathcal {B}}$ be almost disjoint families and $\operatorname {\mathrm {JL}}(\mathcal {B}, \mathcal {B}) = X \oplus \tilde {X}$ as in the setting of Subsection 3.4. To ensure that X and $\tilde {X}$ are not isomorphic to Banach lattices, it will be necessary to have control over the elements in the duals $X^*$ and $\tilde {X}^*$ .

We start with the following observation, which is the analogue of [Reference Plebanek and Salguero-Alarcón5, Remark 3.2]. Recall that Q and $\tilde {Q}$ denote the projections onto X and $\tilde {X}$ , respectively.

Remark 3.5. Let $x^* \in X^*$ . Then $x^* Q \in \operatorname {\mathrm {JL}} (\mathcal {B}, \tilde {\mathcal {B}})^* = M(\mathfrak {B}, \tilde {\mathfrak {B}})$ , so that every functional on X can be seen as an element on $M(\mathfrak {B}, \tilde {\mathfrak {B}})$ vanishing on all cylinders from $\mathfrak {B}$ . In other words, we have an embedding

$$ \begin{align*} X^* \to M(\mathfrak{B}, \tilde{\mathfrak{B}}), \quad x^* \mapsto x^* Q. \end{align*} $$

Analogous considerations apply to $\tilde {x}^* \in \tilde {X}^*$ .

Thus, from now on, when we consider an element in the dual of X or $\tilde {X}$ , we will always think of it as the measure in $M(\mathfrak {B}, \tilde {\mathfrak {B}})$ representing it.

Remark 3.6. We will reserve the symbols $\lambda $ and $\gamma $ for elements in $M_1(\mathfrak {B}, \tilde {\mathfrak {B}})$ . Observe that we have a decomposition $\lambda = \nu + \tilde {\nu }$ where $\nu \in M_1(\mathfrak {B})$ and $\tilde {\nu } \in M_1(\tilde {\mathfrak {B}})$ (with their natural interpretations). From the discussion in Subsection 3.2, we can further express $\nu = \mu + \overline {\nu }$ where $\mu \in \ell _1(\omega \times 2)$ , $\overline {\nu }$ vanishes on finite sets and $|\overline {\nu }|(B) \not = 0$ for at most countably many $B \in \mathcal {B}$ . Identical considerations apply for $\tilde {\nu }$ . Using these decompositions we can write $\lambda = \nu + \tilde {\nu } = (\mu + \overline {\nu }) + (\tilde {\mu } + \tilde {\overline {\nu }})$ . Therefore, a measure $\lambda \in M_1(\mathfrak {B}, \tilde {\mathfrak {B}})$ can be seem as a quadruple $(\mu , \overline {\nu }; \tilde {\mu }, \tilde {\overline {\nu }})$ . We call $\mu $ and $\tilde {\mu }$ the $\ell _1(\omega \times 2)$ -part and $\ell _1(\tilde {\omega } \times 2)$ -part of $\lambda $ respectively.

Furthermore, from the discussion in Subsection 3.2, we can identify $\mu $ and $\tilde {\mu }$ with elements $x \in \ell _1(\omega \times 2)$ and $\tilde {x} \in \ell _1(\tilde {\omega } \times 2)$ , and $\overline {\nu }$ and $\tilde {\overline {\nu }}$ with elements $y \in \ell _1(\mathfrak {c} \times 2)$ and $\tilde {y} \in \ell _1(\tilde {\mathfrak {c}} \times 2)$ . In this way, the measure $\lambda $ can be coded by the quadruple

$$ \begin{align*} (x,y;\,\tilde{x},\tilde{y}) \in \ell_1(\omega \times 2) \oplus \ell_1(\mathfrak{c} \times 2) \oplus \ell_1(\tilde{\omega} \times 2) \oplus \ell_1(\tilde{\mathfrak{c}} \times 2). \end{align*} $$

4.1 Separating sets of measures

Since the nature of the underlying countable set is not important, in this section, we will discuss measures on $\omega $ and $\tilde {\omega }$ . Throughout this section, we fix Boolean subalgebras $\mathfrak {B} \subseteq \mathcal {P}(\omega )$ and $\tilde {\mathfrak {B}} \subseteq \mathcal {P}(\tilde {\omega })$ containing all finite subsets of $\omega $ and $\tilde {\omega }$ , respectively. We state our results in general form, for arbitrary finitely additive measures $\lambda $ on $\omega \cup \tilde {\omega }$ , which we implicitly regard as the sum of a measure on $\omega $ and a measure on $\tilde {\omega }$ . Accordingly, we adopt the slight abuse of notation $M_1(\mathcal {P}(\omega ), \mathcal {P}(\tilde {\omega }))$ for the collection of those measures with total variation at most one.

We shall employ the following analogue of separation, as given in [Reference Plebanek and Salguero-Alarcón5, Definition 4.1].

Definition 4.1. Let $M, M' \subseteq M_1(\mathcal {P}(\omega ), \mathcal {P}(\tilde {\omega }))$ and $\varepsilon> 0$ . We say that $(M, M')$ is $(\mathfrak {B}, \tilde {\mathfrak {B}})$ -separated with constant $\varepsilon $ if there exist $n, m \in \omega $ and $B_1, \dots , B_n \in \mathfrak {B}$ , $\tilde {B}_1, \dots , \tilde {B}_m \in \tilde {\mathfrak {B}}$ such that

$$ \begin{align*} \max( \max_{i \leq n} |\lambda(B_i) - \lambda'(B_i)|, \max_{j \leq m} |\lambda(\tilde{B}_j) - \lambda'(\tilde{B}_j)|) \geq \varepsilon \end{align*} $$

for any $\lambda \in M$ and $\lambda ' \in M'$ . If $M, M'$ are separated for some $\varepsilon $ , we say that they are $(\mathfrak {B}, \tilde {\mathfrak {B}})$ -separated.

With this definition, we can get the following analogue of [Reference Plebanek and Salguero-Alarcón5, Lemma 4.2]. The proof is identical and thus omitted.

Lemma 4.2. Let $M, M' \subseteq M_1(\mathcal {P}(\omega ), \mathcal {P}(\tilde {\omega }))$ and suppose there exist $\varepsilon> 0$ and simple $\mathfrak {B}$ -measurable and $\tilde {\mathfrak {B}}$ -measurable functions $g: \omega \to \mathbb {R}$ and $\tilde {g}: \tilde {\omega } \to \mathbb {R}$ such that

$$ \begin{align*} |\langle \lambda, g + \tilde{g}\rangle - \langle \lambda', g + \tilde{g}\rangle| \geq \varepsilon, \end{align*} $$

for any $\lambda \in M$ and $\lambda ' \in M'$ . Then $(M,M')$ is $(\mathfrak {B}, \tilde {\mathfrak {B}})$ -separated.

Likewise, we have the following counterpart of [Reference Plebanek and Salguero-Alarcón5, Lemma 4.3]. The proof is essentially the same, using now the compactness of $[-1,1]^n \times [-1,1]^m$ . We omit the details.

For any set $Z \subseteq \omega $ , we write $\mathfrak {B}[Z]$ for the algebra generated by $\mathfrak {B} \cup \{Z\}$ . Observe that every element in $\mathfrak {B}[Z]$ is of the form $(A \cap Z) \cup (B \cap Z^c)$ for some $A, B \in \mathfrak {B}$ . Similarly for $\tilde {Z} \subseteq \tilde {\omega }$ and $\tilde {\mathfrak {B}}$ .

Finally, we will need the following counterpart of [Reference Plebanek and Salguero-Alarcón5, Proposition 4.4], whose underlying ideas date back to a counting argument of Haydon [Reference Haydon3].

We give a brief outline of the proof, which is essentially the same as the proof of [Reference Plebanek and Salguero-Alarcón5, Proposition 4.4]. We do this for the almost disjoint family $\mathcal {Z}$ , the other case being identical. We start with an analogue to [Reference Plebanek and Salguero-Alarcón5, Lemma 4.5].

Lemma 4.5. Suppose that for a given $M, M' \subseteq M_1(\mathcal {P}(\omega ), \mathcal {P}(\tilde {\omega }))$ , there are:

1. 1. $n \in \omega $ , $m \in \omega $ , $\varepsilon> 0$ and $k> \frac {12n}{\varepsilon }$

2. 2. almost disjoint sets $Z_1, \dots , Z_k \subseteq \omega $ ,

3. 3. $A_1, B_1, \dots , A_n, B_n \in \mathfrak {B}$ and $\tilde {A}_1, \dots , \tilde {A}_m \in \tilde {\mathfrak {B}}$

such that, for every $1 \leq j \leq k$ , the pair $(M, M')$ is $(\mathfrak {B}[Z_j], \tilde {\mathfrak {B}})$ -separated, with constant $\varepsilon $ , by the sets

$$ \begin{align*} Y_{j,i} = (A_i \cap Z_j) \cup (B_i \cap Z_j^c), \hspace{5pt} 1 \leq i \leq n \hspace{10 pt} \text{ and } \tilde{A}_l, \hspace{10pt} 1 \leq l \leq m. \end{align*} $$

Then $(M, M')$ is $(\mathfrak {B}, \tilde {\mathfrak {B}})$ -separated.

Once we have this analogue of [Reference Plebanek and Salguero-Alarcón5, Lemma 4.5], we can prove the extension to $(\mathfrak {B}_j[Z], \tilde {\mathfrak {B}}_j)$ in Proposition 4.4, by Haydon’s counting argument. Suppose that for every $Z \in \mathcal {Z}$ the conclusion of Proposition 4.4 fails. This is witnessed by some $j \in I$ , $n \in \omega $ , $m \in \omega $ , some rational $\varepsilon> 0$ , and some collections $A_1, B_1, \dots , A_n, B_n \in \mathfrak {B}_j$ and $\tilde {A}_1, \dots , \tilde {A}_m \in \tilde {\mathfrak {B}}_j$ . Since the number of such witnesses has size less than $\mathfrak {c}$ and $|\mathcal {Z}| = \mathfrak {c}$ , there are infinitely many $Z \in \mathcal {Z}$ with the same witness. Lemma 4.5 gives that the pair $(M_j, M^{\prime }_j)$ is $(\mathfrak {B}_j, \tilde {\mathfrak {B}}_j)$ -separated, contrary to our assumptions.

4.2 Separation versus norming sets

By definition, the space X contains every function of the form for $n \in \omega $ , while the space $\tilde {X}$ contains every function of the form for $\tilde {n} \in \tilde {\omega }$ . Regarding elements in $B_{X^*}$ and $B_{\tilde {X}^*}$ as elements in $M_1(\mathfrak {B}, \tilde {\mathfrak {B}})$ , as in Remark 3.5, then the elements in $B_{X^*}$ vanish on all the cylinders from $\mathfrak {B}$ while the elements in $B_{\tilde {X}^*}$ vanish on all cylinders from $\tilde {\mathfrak {B}}$ .

From the above discussion, every norming sequence for X, regarded as a sequence in $M_1(\mathfrak {B}, \tilde {\mathfrak {B}})$ , must contain a subsequence $(\lambda _n)_{n \in \omega }$ such that $\inf _{n \in \omega }|\lambda _n(\{(n,0)\})| = \frac {1}{2}\inf _{n \in \omega }|\langle \lambda _n, f_n \rangle |> 0$ . Observe that if we decompose $\lambda _n = (\mu _n + \overline {\nu }_n) + \tilde {\nu }_n$ as in Remark 3.6, then $\lambda _n(\{(n,0)\}) = \mu _n(\{(n, 0)\})$ . Similar considerations apply for $\tilde {X}$ ; this motivates the following, see [Reference Plebanek and Salguero-Alarcón5, Definition 5.1].

For a sequence $(\mu _n)_{n \in \omega }$ and any set $J \subseteq \omega $ we write $\mu [J] = \{ \mu _n: n \in J\}$ . For convenience, we introduce the following notation: for $s, t \in \mathbb {R}$ and $\varepsilon> 0$ we say $s \approx _\varepsilon t$ if $|s - t| < \varepsilon $ . We have the following version of [Reference Plebanek and Salguero-Alarcón5, Lemma 5.3].

Lemma 4.7. Let $C = C_0 \times 2 \subseteq \omega \times 2$ be an infinite cylinder, $(\mu _n)_{n \in \omega }$ be an admissible sequence and $b = \inf _{n \in \omega } |\mu _n(\{(n,0)\})|$ . Then there exist:

• – three infinite sets $J_2 \subseteq J_1 \subseteq J_0 \subseteq C_0$ with $J_0 \setminus J_1$ , $J_1 \setminus J_2$ infinite,

• – a set Z splitting the cylinder $J_1 \times 2$ ,

• – two positive constants $a \geq b$ and $0 < \delta < b/11$ ,

such that the following holds:

(A1) $$ \begin{align} \mu_n(Z) - \mu_n((J_1 \times 2 )\setminus Z) \approx_{3\delta} \begin{cases} 2a & \text{if } n \in J_2, \\ -2a & \text{if } n \in J_1 \setminus J_2, \\ 0 & \text{if } n \in J_0 \setminus J_1. \end{cases} \end{align} $$

Similar considerations apply for a cylinder $\tilde {C} = \tilde {C}_0 \times 2 \subseteq \tilde {\omega } \times 2$ and a $\sim $ -admissible sequence $(\tilde {\mu }_{\tilde {n}})_{\tilde {n} \in \tilde {\omega }}$ .

Proof. The proof is identical to that of [Reference Plebanek and Salguero-Alarcón5, Lemma 5.3]. Since it is short and crucial for our purposes, we include it for the reader’s convenience.

Choose $0 < \delta < b/11$ . By Rosenthal’s lemma (see [Reference Rosenthal6, p. 18] and the version [Reference Plebanek and Salguero-Alarcón5, Theorem 3.5]), we can find an infinite subset $J_0 \subseteq C_0$ such that

$$ \begin{align*} |\mu_n|((J_0 \times 2) \setminus c_n) < \delta \hspace{5 pt} \text{ for every } n \in J_0. \end{align*} $$

We can now find an infinite subset $J_1 \subseteq J_0$ with $J_0 \setminus J_1$ infinite, and singletons $p_n \subseteq c_n$ so that the sequence $(\mu _n(p_n))_{n \in J_1}$ converges to some $a \geq b$ . Observe that these are precisely the singletons such that $\mu _n(p_n)> 0$ .

By removing a finite number of elements in $J_1$ , we may assume that

$$ \begin{align*} |\mu_n(p_n) - a| < \delta \text{ for every } n \in J_1. \end{align*} $$

Lastly, choose an infinite set $J_2 \subseteq J_1$ with $J_1 \setminus J_2$ infinite and set

$$ \begin{align*} Z = \left(\bigcup_{n \in J_2} p_n \right) \cup \left( \bigcup_{n \in J_1 \setminus J_2} (c_n \setminus p_n)\right). \end{align*} $$

By construction, Z splits the cylinder $J_1 \times 2$ , so that we only need to verify that (A1) is satisfied. For every $n \in J_0 \backslash J_1$ we have $J_1 \times 2 \subseteq (J_0 \times 2) \setminus c_n$ , which gives

$$ \begin{align*} |\mu_n(Z) - \mu_n((J_1 \times 2) \setminus Z)| \leq |\mu_n|((J_0 \times 2) \setminus c_n) < \delta, \end{align*} $$

equivalently $\mu _n(Z) - \mu _n((J_1 \times 2) \setminus Z) \approx _\delta 0$ .

Observe that for any $n \in J_1$ , we have $\mu _n(p_n) \approx _\delta a$ and $\mu _n(c_n \setminus p_n) \approx _\delta -a$ . Now, for every $n \in J_1 \backslash J_2$ , we have $(c_n \backslash p_n) \in Z$ and $p_n \in (J_1 \times 2) \setminus Z$ so that

$$ \begin{align*} \mu_n(Z) - \mu_n((J_1 \times 2) \setminus Z) \approx_\delta \mu_n(c_n \backslash p_n) - \mu_n(p_n) \approx_{2\delta} -2a. \end{align*} $$

Similarly, for $n \in J_2$ we have $p_n \in Z$ and $(c_n \setminus p_n) \in (J_1 \times 2) \setminus Z$ and thus

$$ \begin{align*} \mu_n(Z) - \mu_n((J_1 \times 2) \setminus Z) \approx_\delta \mu_n( p_n) - \mu_n( c_n \setminus p_n) \approx_{2\delta} 2a, \end{align*} $$

which finishes the proof.

We also recall the following, which is [Reference Plebanek and Salguero-Alarcón5, Remark 5.4], and follows by examining the proof of the previous lemma.

5.1 Notation and Setup

We now present the construction of the four almost disjoint families $\mathcal {A}$ , $\tilde {\mathcal {A}}$ , $\mathcal {B}$ and $\tilde {\mathcal {B}}$ , having the properties specified in Subsection 3.4. We closely follow the construction in [Reference Plebanek and Salguero-Alarcón5, Section 6], with modifications to adapt it to our setting as appropriate.

We will construct almost disjoint families $\mathcal {A} = \{A_\alpha : \alpha < \mathfrak {c} \}$ and $\tilde {\mathcal {A}} = \{\tilde {A}_\alpha : \alpha < \mathfrak {c} \}$ of cylinders of $\omega \times 2$ and $\tilde {\omega } \times 2$ , and splittings of every $A_\alpha $ and $\tilde {A}_\alpha $ into two sets $B_\alpha ^{i}$ and $\tilde {B}_\alpha ^{i}$ , $i=0,1$ . The construction will be carried out so that $X = \operatorname {\mathrm {JL}}(\tilde {\mathcal {A}}) \oplus Y$ and $\tilde {X} = \operatorname {\mathrm {JL}}(\mathcal {A}) \oplus \tilde {Y}$ fail to be isomorphic to Banach lattices.

We start by coding all the sequences of measures that will (eventually) appear in norming sequences of $X^*$ and $\tilde {X}^*$ . For $(x,y; \tilde {x}, \tilde {y}) \in \ell _1(\omega \times 2) \times \ell _1(\mathfrak {c} \times 2) \times \ell _1(\tilde {\omega } \times 2) \times \ell _1(\tilde {\mathfrak {c}} \times 2)$ we say that $(x,y; \tilde {x}, \tilde {y})$ is of

1. Type-1 if $x(n,0) + x(n,1) = 0$ for every $n \in \omega $ and $y(\alpha , 0) + y(\alpha ,1) = 0$ for every $\alpha < \mathfrak {c}$ ,

2. Type-2 if $\tilde {x}(\tilde {n},0) + \tilde {x}(\tilde {n},1) = 0$ for every $\tilde {n} \in \tilde {\omega }$ and $\tilde {y}(\tilde {\alpha }, 0) + \tilde {y}(\tilde {\alpha },1) = 0$ for every $\tilde {\alpha } < \tilde {\mathfrak {c}}$ .

We let $\Psi $ be the set of all $(x,y; \tilde {x}, \tilde {y})$ of Type-1 such that additionally $\left \lVert x\right \rVert _1 + \left \lVert y\right \rVert _1 + \left \lVert \tilde {x}\right \rVert _1 + \left \lVert \tilde {y}\right \rVert _1 \leq 1$ . Similarly $\tilde {\Psi }$ is the set of all $(x,y; \tilde {x}, \tilde {y})$ of Type-2 satisfying $\left \lVert x\right \rVert _1 + \left \lVert y\right \rVert _1 + \left \lVert \tilde {x}\right \rVert _1 + \left \lVert \tilde {y}\right \rVert _1 \leq 1$ . We will only care about those sequences of measures that (will eventually) appear as norming sequences of $X^*$ and $\tilde {X}^*$ , and thus we will make a further reduction.

Hence we let $\Phi $ be the set of all sequences $(x_n,y_n; \tilde {x}_n, \tilde {y}_n)_{n \in \omega }$ in $\Psi $ such that

$$ \begin{align*} \inf_{n \in \omega} |x_n(n,0)|> 0, \end{align*} $$

and similarly we let $\tilde {\Phi }$ be the set of all sequences $(x_{\tilde {n}},y_{\tilde {n}}; \tilde {x}_{\tilde {n}}, \tilde {y}_{\tilde {n}})_{{\tilde {n}} \in \tilde {\omega }}$ in $\tilde {\Psi }$ such that

$$ \begin{align*} \ \inf_{\tilde{n} \in \tilde{\omega}} |\tilde{x}_{\tilde{n}}(\tilde{n},0)|> 0. \end{align*} $$

We can enumerate

$$ \begin{align*} \Phi = \{\varphi_\alpha : \alpha < \mathfrak{c}\} \quad \end{align*} $$

in such a way that, if $\varphi _\alpha = (x^\alpha _n, y^\alpha _n; \tilde {x}^\alpha _n, \tilde {y}^\alpha _n)_{n \in \omega } \in \Phi $ , then

(4) $$ \begin{align} y^\alpha_n(\xi, i) = \tilde{y}^\alpha_n(\tilde{\xi}, i) = 0 \quad \text{for every } n \in \omega, i = 0,1 \text{ and every } \xi \geq \alpha \text{ and } \ \tilde{\xi} \geq \tilde{\alpha}. \end{align} $$

To achieve this, we follow the argument in [Reference Plebanek and Salguero-Alarcón5, Section 6]. Start with an enumeration $\{\theta _\alpha : \alpha < \mathfrak {c} \}$ of $\Phi $ where every element is repeated cofinally often. Then for $\theta _\alpha = (x^\alpha _n, z^\alpha _n; \tilde {x}^\alpha _n, \tilde {z}^\alpha _n)_{n \in \omega }$ put $\varphi _\alpha = (x^\alpha _n,y^\alpha _n; \tilde {x}^\alpha _n, \tilde {y}^\alpha _n)_{n \in \omega }$ where for every $n \in \omega $ and $i= 0,1$ , $y_n^\alpha (\beta , i) = z^\alpha _n(\beta ,i)$ and $\tilde {y}^\alpha _n(\tilde {\beta },i) = \tilde {z}^\alpha _n(\tilde {\beta },i)$ whenever $\beta < \alpha $ and $\tilde {\beta } < \tilde {\alpha }$ and $y^\alpha _n(\beta ,i) = \tilde {y}^\alpha _n(\tilde {\beta },i) = 0$ whenever $\beta \geq \alpha $ and $\tilde {\beta } \geq \tilde {\alpha }$ .

Similarly, we can enumerate

$$ \begin{align*} \tilde{\Phi} = \{\tilde{\varphi}_{\alpha} : \alpha < \mathfrak{c}\} \end{align*} $$

such that if $\tilde {\varphi }_\alpha = (x^\alpha _{\tilde {n}}, y^\alpha _{\tilde {n}}; \tilde {x}^\alpha _{\tilde {n}}, \tilde {y}^\alpha _{\tilde {n}})_{{\tilde {n}} \in \tilde {\omega }} \in \Phi $ then (4) holds for every $\tilde {n} \in \tilde {\omega }$ .

We can now exploit the following devilishly clever observation of Plebanek and Salguero-Alarcón: using these enumerations, the codings $\varphi _\alpha $ and $\varphi ^{\prime }_\alpha $ represent sequences of measures in $M_1(\mathfrak {B}, \tilde {\mathfrak {B}})$ even before the construction of $\mathcal {B}$ and $\mathcal {B}'$ has been completed. This is a delicate point which warrants discussion.

At induction step $\alpha $ , the sets $\{B_\beta ^0, B_\beta ^1: \beta < \alpha \}$ and $\{\tilde {B}_\beta ^0, \tilde {B}_\beta ^1: \beta < \alpha \}$ will have been constructed, and thus we can consider the Boolean algebras that they generate

$$ \begin{align*} \mathfrak{B}(\alpha) = [\{B_{\beta}^0, B_{\beta}^1: \beta < \alpha \} \cup \operatorname{\mathrm{fin}}( \omega \times 2 )] \hspace{5pt} \text{ and } \tilde{\mathfrak{B}}(\alpha) = [\{\tilde{B}_{\beta}^0, \tilde{B}_{\beta}^1: \beta < \alpha \} \cup \operatorname{\mathrm{fin}}( \omega \times 2 )]. \end{align*} $$

Observe now that the element $\varphi _\alpha = (x^\alpha _n, y^\alpha _n; \tilde {x}^\alpha _n, \tilde {y}^\alpha _n)_{n \in \omega }$ can, at this point, be considered to code a sequence $(\lambda _n^\alpha )_{n \in \omega }$ of measures in $M_1(\mathfrak {B}(\alpha ), \tilde {\mathfrak {B}}(\alpha ))$ since (4) holds. Finally, the coup de grâce: once the construction of $\mathcal {B}$ and $\tilde {\mathcal {B}}$ is complete, the algebras $\mathfrak {B}(\alpha )$ and $\tilde {\mathfrak {B}}(\alpha )$ will be Boolean subalgebras of $\mathfrak {B}$ and $\tilde {\mathfrak {B}}$ . Thus, arguing as in Subsection 3.2, the sequence of measures $(\lambda _n^\alpha )_{n \in \omega }$ in $M_1(\mathfrak {B}(\alpha ), \tilde {\mathfrak {B}}(\alpha ))$ extends uniquely to a sequence of measures in $M_1(\mathfrak {B}, \tilde {\mathfrak {B}})$ , which, in an abuse of notation, we still call $(\lambda _n^\alpha )_{n \in \omega }$ . This is precisely what we mean when we say that $\varphi _\alpha $ codes a sequence of measure $(\lambda _n^\alpha )_{n \in \omega }$ in $M_1(\mathfrak {B}, \tilde {\mathfrak {B}})$ .

Conversely, observe that any sequence in $M_1(\mathfrak {B}, \tilde {\mathfrak {B}})$ arises as the extension of a sequence in $M_1(\mathfrak {B}(\alpha ), \tilde {\mathfrak {B}}(\alpha ))$ for some $\alpha < \mathfrak {c}$ . For details of how this works, we refer back to Subsection 3.2, together with the observation that the cofinality of the continuum cannot be $\omega $ .

During the inductive construction, for a subset $V \subseteq \mathfrak {c}$ we will denote the Boolean subalgebras

$$ \begin{align*} \mathfrak{B}(V) = [\{B_{\beta}^0, B_{\beta}^1: \beta \in V \} \cup \operatorname{\mathrm{fin}}( \omega \times 2 )] \hspace{5pt} \text{ and } \tilde{\mathfrak{B}}(V) = [\{\tilde{B}_{\beta}^0, \tilde{B}_{\beta}^1: \beta \in V \} \cup \operatorname{\mathrm{fin}}( \omega \times 2 )], \end{align*} $$

provided all the elements $B_{\beta }^i, \tilde {B}_{\beta }^i$ , $i = 0,1$ , $\beta \in V$ , have been constructed at that point of the induction process.

5.2 The main construction

Before we start, we fix two almost disjoint families $\mathcal {R} = \{R_\alpha : \alpha < \mathfrak {c} \}$ and $\tilde {\mathcal {R}} = \{\tilde {R}_\alpha : \alpha < \mathfrak {c} \}$ of infinite subsets of $\omega $ and $\tilde {\omega }$ respectively.

From now on, we will write $(\lambda ^\alpha _{n})_{n \in \omega }$ and $(\gamma ^\alpha _{\tilde {n}})_{\tilde {n} \in \tilde {\omega }}$ to refer to the sequences in $M_1(\mathfrak {B}, \tilde {\mathfrak {B}})$ coded by $\varphi _\alpha $ and $\tilde {\varphi }_{\alpha }$ respectively. This should be understood in the sense explained above. For these sequences, we write

$$ \begin{align*} \lambda^\alpha_{n} = (\mu^\alpha_n + \overline{\nu}^\alpha_n) + \tilde{\nu}^\alpha_n \hspace{5pt} \text{ and } \hspace{5pt} \gamma^\alpha_{\tilde{n}}= \nu^\alpha_{\tilde{n}} + (\tilde{\mu}^\alpha_{\tilde{n}} + \tilde{\overline{\nu}}^\alpha_{\tilde{n}}), \end{align*} $$

as in Remark 3.6.

Using transfinite induction, we will construct for every $\xi < \mathfrak {c}$ :

1. (i) three infinite subsets $J_2^\xi \subseteq J_1^\xi \subseteq J_0^\xi \subseteq R_\xi $ ,

2. (ii) three infinite subsets $\tilde {J}_2^\xi \subseteq \tilde {J}_1^\xi \subseteq \tilde {J}_0^\xi \subseteq \tilde {R}_\xi $ ,

3. (iii) splittings $B_\xi ^0$ , $B_\xi ^1$ and $\tilde {B}_\xi ^0$ , $\tilde {B}_\xi ^1$ of the cylinders $A_\xi =J_1^\xi \times 2$ and $\tilde {A}_\xi = \tilde {J}_1^\xi \times 2$ ,

4. (iv) positive constants $a_\xi \geq b_\xi := \inf _{n \in \omega } |\mu ^\xi _n(\{(n,0)\})|$ , $\tilde {a}_\xi \geq \tilde {b}_\xi := \inf _{\tilde {n} \in \tilde {\omega }} |\tilde {\mu }^\xi _{\tilde {n} }(\{(\tilde {n} ,0)\})|$ , $0 < \delta _\xi < b_\xi /11$ and $0 < \tilde {\delta }_\xi < \tilde {b}_\xi /11$ ;

so that the following conditions hold:

1. (B1)

   $$ \begin{align*} \mu^\xi_n(B_\xi^0) - \mu^\xi_n(B_\xi^1) \approx_{3\delta_\xi} \begin{cases} 2a_\xi & \text{if } n \in J^\xi_2, \\ -2a_\xi & \text{if } n \in J^\xi_1 \setminus J^\xi_2, \\ 0 & \text{if } n \in J^\xi_0 \setminus J^\xi_1. \end{cases} \end{align*} $$

2. (B2)

   $$ \begin{align*} \tilde{\mu}^\xi_{\tilde{n}}(\tilde{B}_\xi^0) - \tilde{\mu}^\xi_{\tilde{n}}(\tilde{B}_\xi^1) \approx_{3\tilde{\delta}_\xi} \begin{cases} 2\tilde{a}_\xi & \text{if } \tilde{n} \in \tilde{J}^\xi_2, \\ -2\tilde{a}_\xi & \text{if } \tilde{n} \in \tilde{J}^\xi_1 \setminus \tilde{J}^\xi_2, \\ 0 & \text{if } \tilde{n} \in \tilde{J}^\xi_0 \setminus \tilde{J}^\xi_1. \end{cases} \end{align*} $$

3. (B3) For every $\alpha \leq \xi $ , the pairs

   $$ \begin{align*} (\lambda^\alpha[J_2^\alpha], \lambda^\alpha[J_1^\alpha \setminus J_2^\alpha]) \text{ and } (\lambda^\alpha[J_1^\alpha], \lambda^\alpha[J_0^\alpha \setminus J_1^\alpha]) \end{align*} $$
   are not $(\mathfrak {B}(\xi +1 \setminus \{\alpha \}), \tilde {\mathfrak {B}}(\xi +1))$ -separated; and the pairs
   $$ \begin{align*} (\gamma^\alpha[\tilde{J}_2^\alpha], \gamma^\alpha[\tilde{J}_1^\alpha \setminus \tilde{J}_2^\alpha]) \text{ and } (\gamma^\alpha[\tilde{J}_1^\alpha], \gamma^\alpha[\tilde{J}_0^\alpha \setminus \tilde{J}_1^\alpha]) \end{align*} $$
   are not $(\mathfrak {B}(\xi +1), \tilde {\mathfrak {B}}(\xi +1 \setminus \{\alpha \}))$ -separated, where we recall that $\mathfrak {B}(\xi +1 \setminus \{\alpha \})$ denotes the Boolean subalgebra of $\mathcal {P}(\omega \times 2)$ given by
   $$ \begin{align*} \mathfrak{B}(\xi+1 \setminus \{\alpha \}) = [\{B_\eta^0, B_\eta^1: \eta \leq \xi, \eta \not = \alpha\} \cup \operatorname{\mathrm{fin}}( \omega \times 2 )], \end{align*} $$
   and $\mathfrak {B}(\xi +1)$ denotes the Boolean subalgebra of $\mathcal {P}(\omega \times 2)$ given by
   $$ \begin{align*} \mathfrak{B}(\xi+1) = [\{B_\eta^0, B_\eta^1: \eta \leq \xi\} \cup \operatorname{\mathrm{fin}}( \omega \times 2 )]. \end{align*} $$

   Analogous definitions apply to $\tilde {\mathfrak {B}}(\xi +1)$ and $\tilde {\mathfrak {B}}(\xi +1 \setminus \{ \alpha \})$ .

We now describe how to carry out the construction for an ordinal $\xi <\mathfrak {c}$ assuming that we have already carried out the construction for all ordinals $\alpha < \xi $ . That is, for every $\alpha < \xi $ , we have chosen objects as specified in (i)-(iv) so that conditions (B1)-(B3) hold. Note that this allows us to deal with both the initial and the induction steps at the same time. Furthermore, at this point of the induction process, the Boolean algebras $\mathfrak {B}(\xi )$ and $\tilde {\mathfrak {B}}(\xi )$ are defined. We show how to construct the objects at stage $\xi $ .

Fix almost disjoint families $\mathcal {C}$ and $\tilde {\mathcal {C}}$ , each of size $\mathfrak {c}$ , consisting of infinite cylinders contained in $R_\xi \times 2$ and $\tilde {R}_\xi \times 2$ , respectively. Since $(\mu ^\xi _{n})_{n \in \omega }$ , the $\ell _1(\omega \times 2)$ -part of $(\lambda ^\xi _{n})_{n \in \omega }$ , is admissible, we can apply Lemma 4.7 continuum many times—each time for a given cylinder $C = C_0 \times 2 \in \mathcal {C}$ —to obtain sets $J_2(C) \subseteq J_1(C) \subseteq J_0(C) \subseteq C_0$ and $Z(C) \subseteq J_1(C) \times 2$ , together with positive scalars $a(C)$ and $\delta (C)$ satisfying the conditions stated in the lemma. Furthermore, by Lemma 4.3 and Remark 4.8 we may assume that the pairs

$$ \begin{align*} (\lambda^\xi[J_2(C)], \lambda^\xi[J_1(C) \setminus J_2(C)]) \text{ and } (\lambda^\xi[J_1(C)], \lambda^\xi[J_0(C) \setminus J_1(C)]) \end{align*} $$

are not $(\mathfrak {B}(\xi ), \tilde {\mathfrak {B}}(\xi ))$ -separated.

Observe that for any $\alpha < \xi $ we have that the pairs

$$ \begin{align*} (\lambda^\alpha[J_2^\alpha], \lambda^\alpha[J_1^\alpha \setminus J_2^\alpha]) \text{ and } (\lambda^\alpha[J_1^\alpha], \lambda^\alpha[J_0^\alpha \setminus J_1^\alpha]) \end{align*} $$

are not $(\mathfrak {B}(\xi \setminus \{\alpha \}), \tilde {\mathfrak {B}}(\xi ))$ -separated; and the pairs

$$ \begin{align*} (\gamma^\alpha[\tilde{J}_2^\alpha], \gamma^\alpha[\tilde{J}_1^\alpha \setminus \tilde{J}_2^\alpha]) \text{ and } (\gamma^\alpha[\tilde{J}_1^\alpha], \gamma^\alpha[\tilde{J}_0^\alpha \setminus \tilde{J}_1^\alpha]) \end{align*} $$

are not $(\mathfrak {B}(\xi ), \tilde {\mathfrak {B}}(\xi \setminus \{\alpha \}))$ -separated. This follows from the fact that, by the induction hypothesis, (B3) holds for any ordinal $\eta < \xi $ . Indeed, if $\xi $ is a successor ordinal, say $\xi = \eta + 1$ , then this is just (B3) for $\eta $ . Otherwise, if $\xi $ is a limit ordinal, this follows from the fact that an increasing family of nonseparating algebras is also not separating.

Therefore, we can apply Proposition 4.4 to the almost disjoint family $\mathcal {Z} = \{Z(C): C \in \mathcal {C}\}$ and

• – the family of algebra pairs $(\mathfrak {B}(\xi \setminus \{ \alpha \}), \tilde {\mathfrak {B}}(\xi ))$ and the lists

  $$ \begin{align*} (\lambda^\alpha[J_2^\alpha], \lambda^\alpha[J_1^\alpha \setminus J_2^\alpha]) \text{ and } (\lambda^\alpha[J_1^\alpha], \lambda^\alpha[J_0^\alpha \setminus J_1^\alpha]) \end{align*} $$
  for $\alpha < \xi $ .

• – The family of algebra pairs $(\mathfrak {B}(\xi ), \tilde {\mathfrak {B}}(\xi \setminus \{ \alpha \}))$ and the lists

  $$ \begin{align*} (\gamma^\alpha[\tilde{J}_2^\alpha], \gamma^\alpha[\tilde{J}_1^\alpha \setminus \tilde{J}_2^\alpha]) \text{ and } (\gamma^\alpha[\tilde{J}_1^\alpha], \gamma^\alpha[\tilde{J}_0^\alpha \setminus \tilde{J}_1^\alpha]) \end{align*} $$
  for $\alpha < \xi $ .

Thus, we can find $C \in \mathcal {C}$ such that:

• – for every $\alpha < \xi $ , the pairs

  $$ \begin{align*} (\lambda^\alpha[J_2^\alpha], \lambda^\alpha[J_1^\alpha \setminus J_2^\alpha]) \hspace{5pt}\text{ and } \hspace{5pt} (\lambda^\alpha[J_1^\alpha], \lambda^\alpha[J_0^\alpha \setminus J_1^\alpha]) \end{align*} $$
  are not separated even by the larger algebra pair $(\mathfrak {B}(\xi \setminus \{ \alpha \})[Z(C)], \tilde {\mathfrak {B}}(\xi ))$ .

• – For every $\alpha < \xi $ , the pairs

  $$ \begin{align*} (\gamma^\alpha[\tilde{J}_2^\alpha], \gamma^\alpha[\tilde{J}_1^\alpha \setminus \tilde{J}_2^\alpha]) \hspace{5pt}\text{ and } \hspace{5pt} (\gamma^\alpha[\tilde{J}_1^\alpha], \gamma^\alpha[\tilde{J}_0^\alpha \setminus \tilde{J}_1^\alpha]) \end{align*} $$
  are not separated even by the larger algebra pair $(\mathfrak {B}(\xi )[Z(C)], \tilde {\mathfrak {B}}(\xi \setminus \{ \alpha \}))$ .

Having chosen $C \in \mathcal {C}$ , we repeat the procedure in the second copy. Since $(\tilde {\mu }_{\tilde {n}})_{\tilde {n} \in \tilde {\omega }}$ , the $\ell _1(\tilde {\omega } \times 2)$ -part of $(\gamma ^\xi _{\tilde {n}})_{\tilde {n} \in \tilde {\omega }}$ , is $\sim $ -admissible, we can apply Lemma 4.7 continuum many times, each time for a cylinder $\tilde {C} = \tilde {C}_0 \times 2 \in \tilde {\mathcal {C}}$ , to obtain sets $\tilde {J}_2(\tilde {C}) \subseteq \tilde {J}_1(\tilde {C}) \subseteq \tilde {J}_0(\tilde {C}) \subseteq \tilde {C}_0$ and $\tilde {Z}(\tilde {C}) \subseteq \tilde {J}_1(\tilde {C}) \times 2$ , together with positive scalars $\tilde {a}(\tilde {C})$ and $\tilde {\delta }(\tilde {C})$ satisfying the conditions stated in the lemma. Again, by Lemma 4.3 and Remark 4.8 we may assume that the pairs

$$ \begin{align*} (\gamma^\xi[\tilde{J}_2(\tilde{C})], \gamma^\xi[\tilde{J}_1(\tilde{C}) \setminus \tilde{J}_2(\tilde{C})]) \text{ and } (\gamma^\xi[\tilde{J}_1(\tilde{C})], \gamma^\xi[\tilde{J}_0(\tilde{C}) \setminus \tilde{J}_1(\tilde{C})]) \end{align*} $$

are not $(\mathfrak {B}(\xi )[Z(C)], \tilde {\mathfrak {B}}(\xi ))$ -separated.

We can invoke Proposition 4.4, this time for the almost disjoint family $\tilde {\mathcal {Z}} = \{\tilde {Z}[\tilde {C}]: \tilde {C} \in \tilde {\mathcal {C}} \}$ and

• – the family of algebra pairs $(\mathfrak {B}(\xi \setminus \{ \alpha \})[Z(C)], \tilde {\mathfrak {B}}(\xi ))$ and the lists

  $$ \begin{align*} (\lambda^\alpha[J_2^\alpha], \lambda^\alpha[J_1^\alpha \setminus J_2^\alpha]) \text{ and } (\lambda^\alpha[J_1^\alpha], \lambda^\alpha[J_0^\alpha \setminus J_1^\alpha]) \end{align*} $$
  for $\alpha < \xi $ .

• – The algebra pair $(\mathfrak {B}(\xi ), \tilde {\mathfrak {B}}(\xi ))$ and the pair

  $$ \begin{align*} (\lambda^\xi[J_2(C)], \lambda^\xi[J_1(C) \setminus J_2(C)]) \text{ and } (\lambda^\xi[J_1(C)], \lambda^\xi[J_0(C) \setminus J_1(C)]). \end{align*} $$

• – The family of algebra pairs $(\mathfrak {B}(\xi )[Z(C)], \tilde {\mathfrak {B}}(\xi ))$ and the lists

  $$ \begin{align*} (\gamma^\alpha[\tilde{J}_2^\alpha], \gamma^\alpha[\tilde{J}_1^\alpha \setminus \tilde{J}_2^\alpha]) \hspace{5pt}\text{ and } \hspace{5pt} (\gamma^\alpha[\tilde{J}_1^\alpha], \gamma^\alpha[\tilde{J}_0^\alpha \setminus \tilde{J}_1^\alpha]) \end{align*} $$
  for $\alpha < \xi $ .

Hence, we can find $\tilde {C} \in \tilde {\mathcal {C}}$ such that:

• – for every $\alpha < \xi $ , the pairs

  $$ \begin{align*} (\lambda^\alpha[J_2^\alpha], \lambda^\alpha[J_1^\alpha \setminus J_2^\alpha]) \hspace{5pt}\text{ and } \hspace{5pt} (\lambda^\alpha[J_1^\alpha], \lambda^\alpha[J_0^\alpha \setminus J_1^\alpha]) \end{align*} $$
  are not separated even by the larger algebra pair $(\mathfrak {B}(\xi \setminus \{ \alpha \})[Z(C)], \tilde {\mathfrak {B}}(\xi )[\tilde {Z}(\tilde {C})])$ .

• – The pairs

  $$ \begin{align*} (\lambda^\xi[J_2(C)], \lambda^\xi[J_1(C) \setminus J_2(C)]) \text{ and } (\lambda^\xi[J_1(C)], \lambda^\xi[J_0(C) \setminus J_1(C)]). \end{align*} $$
  are not separated even by the larger algebra pair $(\mathfrak {B}(\xi ), \tilde {\mathfrak {B}}(\xi )[\tilde {Z}(\tilde {C})])$ .

• – For every $\alpha < \xi $ , the pairs

  $$ \begin{align*} (\gamma^\alpha[\tilde{J}_2^\alpha], \gamma^\alpha[\tilde{J}_1^\alpha \setminus \tilde{J}_2^\alpha]) \hspace{5pt}\text{ and } \hspace{5pt} (\gamma^\alpha[\tilde{J}_1^\alpha], \gamma^\alpha[\tilde{J}_0^\alpha \setminus \tilde{J}_1^\alpha]) \end{align*} $$
  are not separated even by the larger algebra pair $(\mathfrak {B}(\xi )[Z(C)], \tilde {\mathfrak {B}}(\xi \setminus \{ \alpha \})[\tilde {Z}(\tilde {C})])$ .

Finally, set

1. (i’) $J_2^\xi = J_2(C)$ , $J_1^\xi = J_1(C)$ , $J_0^\xi = J_0(C)$ ,

2. (ii’) $\tilde {J}_2^\xi = \tilde {J}_2(\tilde {C})$ , $\tilde {J}_1^\xi = \tilde {J}_1(\tilde {C})$ , $\tilde {J}_0^\xi = \tilde {J}_0(\tilde {C})$ ,

3. (iii’) $A_\xi = J_1(C) \times 2$ , $B_\xi ^0 = Z(C)$ , $B_\xi ^1 = A_\xi \setminus B_\xi ^0$ ; $\tilde {A}_\xi = \tilde {J}_1(\tilde {C}) \times 2$ , $\tilde {B}_\xi ^0 = \tilde {Z}(\tilde {C})$ , $\tilde {B}_\xi ^1 = \tilde {A}_\xi \setminus \tilde {B}_\xi ^0$ ,

4. (iv’) $a_\xi = a(C)$ , $\tilde {a}_\xi = \tilde {a}(\tilde {C})$ , $\delta _\xi = \delta (C)$ and $\tilde {\delta }_\xi = \tilde {\delta }(\tilde {C})$ .

By construction, conditions (B1)–(B3) are satisfied. Indeed, (B1)–(B2) are precisely (A1) in Lemma 4.7, which hold by our choices in the construction. Moreover, (B3) follows directly from the construction, since we ensured an appropriate choice guaranteeing separation of measures. This completes the induction.

With the construction completed according to the inductive procedure, we now define the almost disjoint families

$$ \begin{align*} \mathcal{A} = \{A_\xi : \xi < \mathfrak{c}\}, \hspace{3pt} \tilde{\mathcal{A}} = \{\tilde{A}_\xi : \xi < \mathfrak{c}\}, \hspace{3pt} \mathcal{B} = \{B_\xi^0, B_\xi^1 : \xi < \mathfrak{c}\}, \text{ and } \tilde{\mathcal{B}} = \{\tilde{B}_\xi^0, \tilde{B}_\xi^1 : \xi < \mathfrak{c}\}. \end{align*} $$

These families will ensure that the corresponding X and $\tilde {X}$ are not isomorphic to Banach lattices, as we will demonstrate in the next section.