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The complete separation of the two finer asymptotic $\boldsymbol {\ell _{p}}$ structures for $ \boldsymbol {1\le p}$<$ \boldsymbol {\infty }$

Published online by Cambridge University Press:  19 December 2022

Spiros A. Argyros
Affiliation:
Department of Mathematics, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece; E-mail: sargyros@math.ntua.gr
Alexandros Georgiou
Affiliation:
Department of Mathematics, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece; E-mail: ale.grgu@gmail.com
Antonis Manoussakis
Affiliation:
School of Chemical and Environmental Engineering, Technical University of Crete, 73100 Chania, Greece; E-mail: amanousakis@tuc.gr
Pavlos Motakis
Affiliation:
Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto ON M3J 1P3, Canada; E-mail: pmotakis@yorku.ca

Abstract

For $1\le p <\infty $ , we present a reflexive Banach space $\mathfrak {X}^{(p)}_{\text {awi}}$ , with an unconditional basis, that admits $\ell _p$ as a unique asymptotic model and does not contain any Asymptotic $\ell _p$ subspaces. Freeman et al., Trans. AMS. 370 (2018), 6933–6953 have shown that whenever a Banach space not containing $\ell _1$ , in particular a reflexive Banach space, admits $c_0$ as a unique asymptotic model, then it is Asymptotic $c_0$ . These results provide a complete answer to a problem posed by Halbeisen and Odell [Isr. J. Math. 139 (2004), 253–291] and also complete a line of inquiry of the relation between specific asymptotic structures in Banach spaces, initiated in a previous paper by the first and fourth authors. For the definition of $\mathfrak {X}^{(p)}_{\text {awi}}$ , we use saturation with asymptotically weakly incomparable constraints, a new method for defining a norm that remains small on a well-founded tree of vectors which penetrates any infinite dimensional closed subspace.

Type
Analysis
Creative Commons
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

1 Introduction

The purpose of this article is to provide an answer to the following problem of Halbeisen and Odell from [Reference Halbeisen and Odell20] and is, in particular, the last step towards the complete separation of a list of asymptotic structures from [Reference Argyros and Motakis9]. Given a Banach space X, let $\mathscr {F}_0(X)$ denote the family of normalised weakly null sequences in X and $\mathscr {F}_b(X)$ denote the family of normalised block sequences of a fixed basis, if X has one.

Problem 1. Let X be a Banach space that admits a unique asymptotic model with respect to $\mathscr {F}_0(X)$ , or with respect to $\mathscr {F}_b(X)$ if X has a basis. Does X contain an Asymptotic $\ell _p$ , $1\le p<\infty $ or an Asymptotic $c_0$ subspace?

The following definition from [Reference Argyros and Motakis9] provides a more general setting in which we will describe this problem, as well as other previous separation results. A property of a Banach space is called hereditary if it is inherited by all of its closed and infinite dimensional subspaces.

Definition 1.1. Let (P) and (Q) be two hereditary properties of Banach spaces, and assume that (P) implies (Q).

  1. (i) If (Q) $\not \Rightarrow $ (P), that is, there exists a Banach space satisfying (Q) and failing (P), then we say that (P) is separated from (Q).

  2. (ii) If there exists a Banach space satisfying (Q) and whose every infinite dimensional closed subspace fails (P), then we say that (P) is completely separated from (Q) and write (Q) (P).

We consider properties that are classified into the following three categories: the sequential asymptotic properties, the array asymptotic properties and the global asymptotic properties.

Sequential asymptotic properties are related to the notion of a spreading model from [Reference Brunel and Sucheston15], which describes the asymptotic behaviour of a sequence in a Banach space. We say that a Banach space admits a unique spreading model with respect to some family of normalised sequences $\mathscr {F}$ , if whenever two sequences from $\mathscr {F}$ generate spreading models, then those must be equivalent. If this equivalence happens with some uniform constant, then we say that the space admits a uniformly unique spreading model.

The category of array asymptotic structures concerns the asymptotic behaviour of arrays of sequences $(x^i_j)_j$ , $i\in \mathbb {N}$ , in a Banach space. Notions that describe this behaviour are those of asymptotic models from [Reference Halbeisen and Odell20] and joint spreading models from [Reference Argyros, Georgiou and Motakis8]. We define the uniqueness of asymptotic models and the uniform uniqueness of joint spreading models in a similar manner to the uniqueness and uniform uniqueness of spreading models, respectively. Although asymptotic models and joint spreading models are not identical notions, they are strongly related. As Sari pointed out, a Banach space X admits a uniformly unique joint spreading model with respect to $\mathscr {F}_b(X)$ or $\mathscr {F}_0(X)$ if and only if it admits a unique asymptotic model with respect to $\mathscr {F}_b(X)$ or $\mathscr {F}_0(X)$ , respectively (see, e.g. [Reference Argyros, Georgiou, Lagos and Motakis6, Remark 4.21] or [Reference Argyros and Motakis9, Proposition 3.12]). Notably, the property that a Banach space X with a basis admits some $\ell _p$ as a uniformly unique joint spreading model with respect to $\mathscr {F}_b(X)$ can be described by the following statement. The case where this happens with respect to $\mathscr {F}_0(X)$ is given by an easy modification.

Proposition 1.2 (Lemma 3.4).

Let $1\le p\le \infty $ . A Banach space X with a basis admits $\ell _p$ (or $c_0$ for $p=\infty $ ) as a uniformly unique joint spreading model with respect to $\mathscr {F}_b(X)$ if and only if there exist constants $c,C>0$ , such that for every $\ell \in \mathbb {N}$ , any choice of successive families $(F_j)_j$ of normalised blocks in X with $\#F_j=\ell $ , there is an infinite subset of the naturals $M=\{m_1<m_2<\ldots \}$ , such that for any choice of $x_j\in F_j$ , $j\in M$ , every $G\subset M$ with $m_k\le G$ and $\#G\le k$ , for $k\in \mathbb {N}$ , and any choice of scalars $a_j$ , $j\in G$ , we have

$$\begin{align*}c\|(a_j)_{j\in G}\|_p \le \| \sum_{j\in G}a_jx_j \| \le C\| (a_j)_{j\in G} \|_p. \end{align*}$$

Even though this property is very close to the weaker one that X admits $\ell _p$ or $c_0$ as a uniformly unique spreading model, it was shown in [Reference Argyros and Motakis9] that these two properties are in fact completely separated for all $1\le p\le \infty $ .

Finally, global asymptotic properties describe the behaviour of finite block sequences that are chosen sufficiently far apart in a space with a basis. We recall the following definition from [Reference Milman and Tomczak-Jaegermann25].

Definition 1.3. Let X be a Banach space with a basis $(e_i)_i$ and $1\le p\le \infty $ . We say that the basis $(e_i)_i$ of X is asymptotic $\ell _p$ (asymptotic $c_0$ when $p=\infty $ ) if there exist positive constants $D_1$ and $D_2$ , such that for all $n\in \mathbb {N}$ , there exists $N(n)\in \mathbb {N}$ with the property that whenever $N(n)\le x_1< \cdots < x_n$ are vectors in X, then

$$\begin{align*}\frac{1}{D_1}(\sum_{i=1}^n\|x_i\|^p)^{\frac{1}{p}}\le \|\sum_{i=1}^nx_i\|\le D_2 (\sum_{i=1}^n\|x_i\|^p )^{\frac{1}{p}}, \end{align*}$$

where for $p=\infty $ , the above inequality concerns the $\|\cdot \|_{\infty }$ . Specifically, we say that $(e_i)_i$ is D-asymptotic $\ell _p$ (D-asymptotic $c_0$ when $p=\infty $ ) for $D=D_1D_2$ .

This definition is given with respect to a fixed basis of the space. The coordinate-free notion of Asymptotic $\ell _p$ and $c_0$ spaces was introduced in [Reference Maurey, Milman and Tomczak-Jaegermann24], generalising the aforementioned one to spaces with or without a basis (note the difference between the terms asymptotic $\ell _p$ and Asymptotic $\ell _p$ ). Moreover, this property is hereditary and any Asymptotic $\ell _p$ (or $c_0$ ) space is asymptotic $\ell _p$ (respectively, $c_0$ ) saturated. Given a Banach space X with a basis, we focus on the following properties, where $1\le p\le \infty $ and whenever $p=\infty $ , then $\ell _p$ should be replaced with $c_0$ .

  • (a) $_p$ The space X is Asymptotic $\ell _p$ .

  • (b) $_p$ The space X admits $\ell _p$ as a uniformly unique joint spreading model (or equivalently, a unique asymptotic model, as mentioned above) with respect to $\mathscr {F}_b(X)$ .

  • (c) $_p$ The space X admits $\ell _p$ as a uniformly unique spreading model with respect to $\mathscr {F}_b(X)$ .

  • (d) $_p$ The space X admits $\ell _p$ as a unique spreading model with respect to $\mathscr {F}_b(X)$ .

Note that it is fairly straightforward to see that the following implications hold for all $1\le p\le \infty $ : (a) $_p\Rightarrow $ (b) $_p\Rightarrow $ (c) $_p\Rightarrow $ (d) $_p$ . It is also easy to see that (d) $_p\not \Rightarrow $ (c) $_p$ for all $1\le p<\infty $ . In [Reference Baudier, Lancien, Motakis and Schlumprecht14] it was shown that (c) $_p\not \Rightarrow $ (b) $_p$ for all $1\le p\le \infty $ and that (b) $_p\not \Rightarrow $ (a) $_p$ for all $1<p<\infty $ . The latter was also shown in [Reference Argyros, Georgiou and Motakis8], as well as that (b) $_1\not \Rightarrow $ (a) $_1$ along with an even stronger result, namely, the existence of a Banach space with a basis satisfying (b) $_1$ and, such that any infinite subsequence of its basis generates a non-Asymptotic $\ell _1$ subspace. However, it was proved in [Reference Androulakis, Odell, Schlumprecht and Tomczak-Jaegermann12] that (d) $_{\infty }\Leftrightarrow $ (c) $_{\infty }$ and a remarkable result from [Reference Freeman, Odell, Sari and Zheng18] states that (b) $_{\infty }\Leftrightarrow $ (a) $_{\infty }$ for Banach spaces not containing $\ell _1$ . Towards the complete separation of these properties, it was shown in [Reference Argyros and Motakis9] that (c) (b) $_p$ for all $1\le p\le \infty $ and that (d) (c) $_p$ for all $1\le p <\infty $ . Hence, the only remaining open question was whether (b) (a) $_p$ for $1\le p<\infty $ . We prove this in the affirmative and, in particular, we show the following.

Theorem 1.4. For $1\le p<\infty $ , there exists a reflexive Banach space $\mathfrak {X}_{\text {awi}}^{(p)}$ with an unconditional basis that admits $\ell _p$ as a uniformly unique joint spreading model with respect to $\mathscr {F}_b(\mathfrak {X}_{\text {awi}}^{(p)})$ and contains no Asymptotic $\ell _p$ subspaces.

To construct these spaces, we use a saturation method with asymptotically weakly incomparable constraints. This method, initialised in [Reference Argyros, Georgiou and Motakis8], employs a tree structure, penetrating every subspace of $\mathfrak {X}_{\text {awi}}^{(p)}$ , that admits segments with norm strictly less than the $\ell _p$ -norm. Thus, we are able to prove that no subspace of $\mathfrak {X}_{\text {awi}}^{(p)}$ is an Asymptotic $\ell _p$ space. This saturation method is different from the method of saturation with increasing weights from [Reference Argyros and Motakis9], used to define spaces with no subspaces admitting a unique asymptotic model. It does not seem possible to use the method of increasing weights to construct a space with a unique asymptotic model, that is, it is not appropriate for showing (b) (a) $_p$ . On the other hand, the method of asymptotically weakly incomparable constraints yields spaces with a unique asymptotic model, and thus it cannot be used to show (c) (b) $_p$ . This method will be discussed in detail in Part 1.

In the case of $1<p<\infty $ , it is possible to obtain a stronger result. Namely, for every countable ordinal $\xi $ , the space separating the two asymptotic properties additionally satisfies the property that every block subspace contains an $\ell _1$ -tree of order $\omega ^{\xi }$ . This is achieved using the attractors method, which was first introduced in [Reference Argyros, Arvanitakis, Tolias, Castillo and Johnson3] and later also used in [Reference Argyros, Manoussakis and Pelczar-Barwacz10]. The precise statement of this result is the following.

Theorem 1.5 ([Reference Argyros, Georgiou, Manoussakis and Motakis7]).

For every $1<p<\infty $ and every infinite countable ordinal $\xi $ , there exists a hereditarily indecomposable reflexive Banach space $\mathfrak {X}^{(p)}_{\xi }$ that admits $\ell _p$ as a uniformly unique joint spreading model with respect to the family of normalised block sequences and whose every subspace contains an $\ell _1$ -block tree of order $\omega ^{\xi }$ .

However, in the case of $\ell _1$ , we are not able to construct a space whose every subspace contains a well-founded tree which is either $\ell _p$ for some $1<p<\infty $ or $c_0$ . This case is more delicate, since as we mentioned, the two properties are in fact equivalent in its dual problem for spaces not containing $\ell _1$ .

The paper is organised as follows: In Section 2, we recall the notions of Schreier families and special convex combinations and prove some of their basic properties, while Section 3 contains the precise definitions of the aforementioned asymptotic structures. In Section 4, we recall certain combinatorial results concerning measures on countably branching well-founded trees from [Reference Argyros, Georgiou and Motakis8], which are a key ingredient in the proof that $\mathfrak {X}^{(p)}_{\text {awi}}$ admits $\ell _p$ as a unique asymptotic model for $1\le p <\infty $ . We then split the remainder of the paper into two main parts, each dedicated to the definition and properties of $\mathfrak {X}_{\text {awi}}^{(1)}$ and $\mathfrak {X}_{\text {awi}}^{(p)}$ for $p=2$ , respectively. The construction of $\mathfrak {X}_{\text {awi}}^{(p)}$ for $1<p<\infty $ and $p\neq 2$ follows as an easy modification of our construction and is omitted. Each of these parts contains an introduction in which we describe the main key points of each construction. Finally, we include two appendices containing variants of the basic inequality, which has been used repeatedly in the past in several related constructions (see, e.g. [Reference Argyros, Arvanitakis, Tolias, Castillo and Johnson3], [Reference Argyros and Motakis9], [Reference Argyros, Manoussakis and Pelczar-Barwacz10] and [Reference Deliyanni and Manoussakis16]).

2 Preliminaries

In this section, we recall some necessary definitions, namely, the Schreier families $(\mathcal {S}_n)_n$ [Reference Alspach and Argyros2] and the corresponding repeated averages $\{a(n,L):n\in \mathbb {N},\; L\in [\mathbb {N}]^{\infty }\}$ [Reference Argyros, Mercourakis and Tsarpalias11] which we call n-averages, as well as the notion of special convex combinations. For a more thorough discussion of the above, we refer the reader to [Reference Argyros and Tolias13]. We begin with some useful notation.

Notation. By $\mathbb {N}=\{1,2,\ldots \}$ , we denote the set of all positive integers. We will use capital letters, such as $L,M,N,\ldots $ (respectively, lower case letters, such as $s,t,u,\ldots $ ) to denote infinite subsets (respectively, finite subsets) of $\mathbb {N}$ . For every infinite subset L of $\mathbb {N}$ , the notation $[L]^{\infty }$ (respectively, $[L]^{<\infty }$ ) stands for the set of all infinite (respectively, finite) subsets of L. For every $s\in [\mathbb {N}]^{<\infty }$ , by $|s|$ , we denote the cardinality of s. For $L\in [\mathbb {N}]^{\infty }$ and $k\in \mathbb {N}$ , $[L]^k$ (respectively, $[L]^{\le k}$ ) is the set of all $s\in [L]^{<\infty }$ with $|s|=k$ (respectively, $|s|\le k$ ). For every $s,t\in [\mathbb {N}]^{<\infty }$ , we write $s<t$ if at least one of them is the empty set, or $\max s<\min t$ . Also for $\emptyset \neq s\in [\mathbb {N}]^{< \infty }$ and $n\in \mathbb {N}$ , we write $n<s$ if $n<\min s$ . We shall identify strictly increasing sequences in $\mathbb {N}$ with their corresponding range, that is, we view every strictly increasing sequence in $\mathbb {N}$ as a subset of $\mathbb {N}$ and, conversely, every subset of $\mathbb {N}$ as the sequence resulting from the increasing order of its elements. Thus, for an infinite subset $L=\{l_1<l_2<\ldots \}$ of $\mathbb {N}$ and $i\in \mathbb {N}$ , we set $L(i)=l_i$ and, similarly, for a finite subset $s=\{n_1<\ldots <n_k\}$ of $\mathbb {N}$ and for $1\le i\le k$ , we set $s(i)=n_i$ .

Finally, throughout the paper, we follow [Reference Lindenstrauss and Tzafriri23] (see also [Reference Albiac and Kalton1]) for standard notation and terminology concerning Banach space theory. For $x\in c_{00}(\mathbb {N})$ , we denote $\text {supp}(x)=\{n\in \mathbb {N}:x(n)\neq 0\}$ , and by $\operatorname {\mathrm {range}}(x)$ , the minimum interval of $\mathbb {N}$ containing $\text {supp}(x)$ . Moreover, for $x,y\in c_{00}(\mathbb {N})$ , we write $x<y$ to denote that $\operatorname {\mathrm {maxsupp}}(x)<\operatorname {\mathrm {minsupp}}(y)$ .

2.1 Schreier families

For a family $\mathcal {M}$ and a sequence $(E_i)_{i=1}^k$ of finite subsets of $\mathbb {N}$ , we say that $(E_i)_{i=1}^k$ is $\mathcal {M}$ -admissible if there is $\{m_1,\ldots ,m_k\}\in \mathcal {M}$ , such that $m_1\le E_1<m_2\le E_2<\cdots <m_k\le E_k$ . Moreover, a sequence $(x_i)_{i=1}^k$ in $c_{00}(\mathbb {N})$ is called $\mathcal {M}$ -admissible if $(\text {supp} (x_i))_{i=1}^k$ is $\mathcal {M}$ -admissible. In the case where $\mathcal {M}$ is a spreading family (i.e. whenever $E=\{m_1,\ldots ,m_k\}\in \mathcal {M}$ and $F=\{n_1<\ldots <n_k\}$ satisfy $m_i\le n_i$ , $i=1,\ldots ,k$ , then $F\in \mathcal {M}$ ), a sequence $(E_i)_{i=1}^k$ is $\mathcal {M}$ -admissible if $\{\min E_i:i=1,\ldots ,k\}\in \mathcal {M}$ , and thus a sequence of vectors $(x_i)_{i=1}^k$ in $c_{00}(\mathbb {N})$ is $\mathcal {M}$ -admissible if $\{\min \text {supp} (x_i):i=1,\ldots ,k\}\in \mathcal {M}$ .

For $\mathcal {M}$ , $\mathcal {N}$ families of finite subsets of $\mathbb {N}$ , we define the convolution of $\mathcal {M}$ and $\mathcal {N}$ as follows:

$$ \begin{align*} \mathcal{M}*\mathcal{N}=\Big\{ & E\subset\mathbb{N}:\text{ there exists an }\mathcal{M}\text{-admissible finite sequence }\\&(E_i)_{i=1}^k\text{ in }\mathcal{N},\text{ such that }E=\cup_{i=1}^kE_i\Big\}\cup\big\{\emptyset\big\}. \end{align*} $$

The Schreier families $(\mathcal {S}_n)_{n\in \mathbb {N}}$ are defined inductively as follows:

$$\begin{align*}\mathcal{S}_0=\big\{ \{k\}:k\in\mathbb{N} \big\}\cup\big\{\emptyset\big\}\quad\text{and}\quad \mathcal{S}_1=\big\{E\subset\mathbb{N}:\# E\le \min E\big\}\cup\{\emptyset\} \end{align*}$$

and if $\mathcal {S}_n$ , for some $n\in \mathbb {N}$ , has been defined, then

$$ \begin{align*} \mathcal{S}_{n+1}=\mathcal{S}_1*\mathcal{S}_n=\Big\{ &E\subset\mathbb{N} : E=\cup_{i=1}^kE_i\text{ where } E_1<\ldots<E_k\in\mathcal{S}_n\ \text{and }k\le\min E_1\Big\}\cup\{\emptyset\}. \end{align*} $$

It follows easily by induction that for every $n,m\in \mathbb {N}$ ,

$$\begin{align*}\mathcal{S}_n*\mathcal{S}_{m}=\mathcal{S}_{n+m}. \end{align*}$$

Furthermore, for each $n\in \mathbb {N}$ , the family $\mathcal {S}_n$ is regular. This means that it includes the singletons, it is hereditary, that is, if $E\in \mathcal {S}_n$ and $F\subset E$ , then $F\in \mathcal {S}_n$ , it is spreading and finally it is compact, identified as a subset of $\{0,1\}^{\mathbb {N}}$ .

For each $n\in \mathbb {N}$ , we also define the regular family

$$\begin{align*}\mathcal{A}_n= \big\{ E\subset\mathbb{N} : \#E\le n \big\}. \end{align*}$$

Then, for $n,m\in \mathbb {N}$ , we are interested in the family $\mathcal {S}_n*\mathcal {A}_m$ , that is, the family of all subsets of $\mathbb {N}$ of the form $E=\cup ^k_{i=1}E_i$ , where $E_1<\ldots <E_k$ , $\#E_i\le m $ for $i=1,\ldots ,k$ and $\{\min E_i:1\le i \le k\}\in \mathcal {S}_n$ . In fact, any such E is the union of at most m sets in $\mathcal {S}_n$ , and moreover, if $m\le E$ , then $E\in \mathcal {S}_{n+1}$ , as we show next.

Lemma 2.1. For every $n,m\in \mathbb {N}$ ,

  1. (i) $\mathcal {S}_n*\mathcal {A}_m\subset \mathcal {A}_m*\mathcal {S}_n$ and

  2. (ii) if $E\in \mathcal {S}_n*\mathcal {A}_m$ with $m\le E$ , then $E\in \mathcal {S}_{n+1}$ .

Remark 2.2. Let $k,m\in \mathbb {N}$ and F be a subset of $\mathbb {N}$ with $\#F\le km$ and $k\le F$ . Set $d=\max \{1,\lfloor \#F/m \rfloor \}$ , and define $F_j=\{F(n):n=(j-1)d+1,\ldots ,jd\}$ for each $j=1,\ldots ,m-1$ and $F_m=F\setminus \cup _{j=1}^{m-1}F_j$ . Then, it is immediate to check that $F_j\in \mathcal {S}_1$ for every $i=1,\ldots ,m$ .

Proof of Lemma 2.1.

Fix $n,m\in \mathbb {N}$ . We prove (i) by induction on $n\in \mathbb {N}$ . For $n=1$ , let $E\in \mathcal {S}_1*\mathcal {A}_m$ , that is, $E=\cup _{i=1}^kE_i$ with $k\le E_1<\ldots <E_k$ and $\#E_i\le m$ for every $i=1,\ldots ,k$ . Since $\#E\le km$ , Remark 2.2 yields a partition $E=\cup _{j=1}^m F_j$ with $F_j\in \mathcal {S}_1$ for every $j=1,\ldots ,m$ , and, hence, $E\in \mathcal {A}_m*\mathcal {S}_1$ .

Suppose that (i) holds for some $n\in \mathbb {N}$ and let $E\in \mathcal {S}_{n+1}*\mathcal {A}_m$ . Then $E=\cup _{i=1}^kE_i$ for an $\mathcal {S}_{n+1}$ -admissible sequence $(E_i)_{i=1}^k$ with $\#E_i\le m$ for every $i=1,\ldots ,m$ . Hence, $\{\min E_i:i=1,\ldots ,k\}=\cup _{j=1}^lF_j$ , where $F_j\in \mathcal {S}_n$ for every $j=1,\ldots ,l$ and $l\le F_1<\cdots <F_l$ . Define, for each $j=1,\ldots ,l$ ,

$$\begin{align*}G_j=\cup\{E_i:i=1,\ldots,k\text{ and } \min E_i\in F_j\},\end{align*}$$

and note that $G_j\in \mathcal {S}_{n}*\mathcal {A}_m$ since $F_j\in \mathcal {S}_n$ . Hence, for every $j=1,\ldots ,l$ , the inductive hypothesis implies that $G_j\in \mathcal {A}_m*\mathcal {S}_n$ , that is, $G_j=\cup _{i=1}^{m_j}G^j_i$ with $m_j\le m$ and $G^j_i\in \mathcal {S}_n$ for all $i=1,\ldots ,m_j$ . Define

$$\begin{align*}H=\{\min G^j_i:j=1,\ldots,l,\text{ and }i=1,\ldots,m_j\}.\end{align*}$$

Observe that $H\in \mathcal {S}_1*\mathcal {A}_m$ and apply Remark 2.2 to obtain a partition $H=\cup _{q=1}^mH_q$ , where $H_q\in \mathcal {S}_1$ for every $q=1,\ldots ,m$ . Finally, define

$$\begin{align*}\Delta_q=\cup\{G^j_i:j=1,\ldots,l,\;i=1,\ldots,m_j\text{ and } \min G^j_i\in H_q\},\end{align*}$$

for each $q=1,\ldots ,m$ , and observe that $E=\cup _{q=1}^m\Delta _q$ and that $\Delta _q\in \mathcal {S}_1*S_{n}=S_{n+1}$ since $H_q\in \mathcal {S}_1$ and $G^j_i\in \mathcal {S}_{n}$ . Thus, we conclude that $E\in \mathcal {A}_m*\mathcal {S}_{n+1}$ .

Finally, note that (ii) is an immediate consequence of (i).

2.2 Repeated averages

The notion of repeated averages was first defined in [Reference Argyros, Mercourakis and Tsarpalias11]. The notation we use below, however, is somewhat different, and we instead follow the one found in [Reference Argyros and Tolias13], namely, $\{a(n,L):n\in \mathbb {N},\; L\in [\mathbb {N}]^{\infty }\}$ . The $n-$ averages $a(n,L)$ are defined as elements of $c_{00}(\mathbb {N})$ in the following manner.

Let $(e_j)_j$ denote the unit vector basis of $c_{00}(\mathbb {N})$ and $L\in [\mathbb {N}]^{\infty }$ . For $n=0$ , we define $a(0,L)=e_{l_1}$ , where $l_1=\min L$ . Suppose that $a(n,M)$ has been defined for some $n\in \mathbb {N}$ and every $M\in [\mathbb {N}]^{\infty }$ . We define $a(n+1,L)$ in the following way: We set $L_1=L$ and $L_k=L_{k-1}\setminus \text {supp}( a(n,L_{k-1}))$ for $k=2,\ldots ,l_1$ and finally define

$$\begin{align*}a(n+1,L)=\frac{1}{l_1}\big( a(n,L_1) + \cdots + a(n,L_{l_1}) \big). \end{align*}$$

Remark 2.3. Let $n\in \mathbb {N}$ and $L\in [\mathbb {N}]^{\infty }$ . The following properties are easily established by induction.

  1. (i) $a(n,L)$ is a convex combination of the unit vector basis of $c_{00}(\mathbb {N})$ .

  2. (ii) $\|a(n,L)\|_{\ell _1}=1$ and $a(n,L)(k)\ge 0$ for all $k\in \mathbb {N}$ .

  3. (iii) $\text {supp} (a(n,L))$ is the maximal initial segment of L contained in $\mathcal {S}_n$ .

  4. (iv) $\|a(n,L)\|_{\infty }= l_1^{-n}$ , where $l_1=\min L$ .

  5. (v) If $\text {supp}(a(n,L))=\{i_1<\ldots <i_d\}$ and $a(n,L)=\sum _{k=1}^da_{i_k}e_{i_k}$ , then we have that $a_{i_1}\ge \ldots \ge a_{i_d}$ .

A proof of the following proposition can be found in [Reference Argyros and Tolias13].

Proposition 2.4. Let $n\in \mathbb {N}$ and $L\in [\mathbb {N}]^{\infty }$ . For every $F\in \mathcal {S}_{n-1}$ , we have that

$$\begin{align*}\sum_{k\in F}a(n,L)(k)<\frac{3}{\min L}. \end{align*}$$

2.3 Special convex combinations

Here, we recall the notion of $(n,\varepsilon )$ -special convex combinations, where $n\in \mathbb {N}$ and $\varepsilon>0$ (see [Reference Argyros and Deliyanni5] and [Reference Argyros and Tolias13]).

Definition 2.5. For $n\in \mathbb {N}$ and $\varepsilon>0$ , a convex combination $\sum _{i\in F}c_ie_i$ , of the unit vector basis $(e_i)_i$ of $c_{00}(\mathbb {N})$ is called an $(n,\varepsilon )$ -basic special convex combination (or an $(n,\varepsilon )$ -basic s.c.c.) if

  1. (i) $F\in \mathcal {S}_n$ and

  2. (ii) for any $G\subset F$ with $G\in S_{n-1}$ , we have that $\sum _{i\in G}c_i<\varepsilon $ .

We will also call $\sum _{i\in F}c_i^{1/2}e_i$ a $(2,n,\varepsilon )$ -basic special convex combination.

As follows from Proposition 2.4, every n-average $a(n,L)$ is an $(n,3/\min L)$ -basic s.c.c., and this yields the following.

Proposition 2.6. Let $M\in [\mathbb {N}]^{\infty }$ , $n\in \mathbb {N}$ and $\varepsilon>0$ . Then there is a $k\in \mathbb {N}$ , such that for any $F\subset M$ , such that F is maximal in $\mathcal {S}_n$ and $k\le \min F$ , there exists an $(n,\varepsilon )$ -basic s.c.c. $x\in c_{00}(\mathbb {N})$ with $\text {supp} (x)=F$ .

Clearly, this also implies the existence of $(2,n,\varepsilon )$ -basic special convex combinations by taking the square roots of the coefficients of an $(n,\varepsilon )$ -b.s.c.c.

Definition 2.7. Let $x_1<\ldots <x_d$ be vectors in $c_{00}(\mathbb {N})$ , and define $t_i=\min \text {supp} (x_i)$ , $i=1,\ldots ,d$ . We say that the vector $\sum _{i=1}^dc_ix_i$ is an $(n,\varepsilon )$ -special convex combination (or an $(n,\varepsilon $ )-s.c.c.) for some $n\in \mathbb {N}$ and $\varepsilon>0$ if $\sum _{i=1}^d c_ie_{t_i}$ is an $(n,\varepsilon )$ -basic s.c.c. and a $(2,n,\varepsilon )$ -special convex combination if $\sum _{i=1}^d c_ie_{t_i}$ is a $(2,n,\varepsilon )$ -basic s.c.c.

3 Asymptotic structures

Let us recall the definitions of the asymptotic notions that appear in the results of this paper and were mentioned in the Introduction. Namely, asymptotic models, joint spreading models and the notions of Asymptotic $\ell _p$ and Asymptotic $c_0$ spaces. For a more thorough discussion, including several open problems and known results, we refer the reader to [Reference Argyros and Motakis9, Section 3].

Definition 3.1 [Reference Halbeisen and Odell20].

An infinite array of sequences $(x^{i}_j)_j$ , $i\in \mathbb {N}$ , in a Banach space X, is said to generate a sequence $(e_i)_i$ , in a seminormed space E, as an asymptotic model if for every $\varepsilon>0$ and $n\in \mathbb {N}$ , there is a $k_0\in \mathbb {N}$ , such that for any natural numbers $k_0\leq k_1<\cdots <k_n$ and any scalars $a_1,\ldots ,a_n$ in $[-1,1]$ , we have

$$\begin{align*}\Big|\big\|\sum_{i=1}^na_ix_{k_i}^{i}\big\| - \big\|\sum_{i=1}^na_ie_{i}\big\|\Big| < \varepsilon.\end{align*}$$

A Banach space X is said to admit a unique asymptotic model with respect to a family $\mathscr {F}$ of normalised sequences in X if whenever two infinite arrays, consisting of sequences from $\mathscr {F}$ , generate asymptotic models, then those must be equivalent. Typical families under consideration are those of normalised weakly null sequences, denoted $\mathscr {F}_0(X)$ , normalised Schauder basis sequences, denoted $\mathscr {F}(X)$ , or the family of all normalised block sequences of a fixed basis of X, if it has one, denoted $\mathscr {F}_b(X)$ .

Definition 3.2 [Reference Argyros, Georgiou, Lagos and Motakis6].

Let $M\in [\mathbb {N}]^{\infty }$ and $k\in \mathbb {N}$ . A plegma (respectively, strict plegma) family in $[M]^k$ is a finite sequence $(s_i)_{i=1}^l$ in $[M]^k$ satisfying the following.

  1. (i) $s_{i_1}(j_1)<s_{i_2}(j_2)$ for every $1\le j_1<j_2\le k$ and $1\le i_1,i_2\le l$ .

  2. (ii) $s_{i_1}(j)\le s_{i_2}(j)$ (respectively, $s_{i_1}(j)< s_{i_2}(j)$ ) for all $1\le i_1<i_2\le l$ and $1\le j\le k$ .

For each $l\in \mathbb {N}$ , the set of all sequences $(s_i)^l_{i=1}$ which are plegma families in $[M]^k$ will be denoted by $Plm_l([M]^k)$ and that of the strict plegma ones by S- $Plm_l([M]^k)$ .

Definition 3.3 [Reference Argyros, Georgiou, Lagos and Motakis6].

A finite array of sequences $(x^{i}_j)_j$ , $1\leq i\leq l$ , in a Banach space X, is said to generate another array of sequences $(e_j^{i})_j$ , $1\leq i\leq l$ , in a seminormed space E, as a joint spreading model if for every $\varepsilon>0$ and $n\in \mathbb {N}$ , there is a $k_0\in \mathbb {N}$ , such that for any $(s_i)_{i=1}^l\in S$ - $Plm_{l}([\mathbb {N}]^n)$ with $k_0\le s_1(1)$ and for any $l\times n$ matrix $A=(a_{ij})$ with entries in $[-1,1]$ , we have that

$$\begin{align*}\Big|\big\|\sum_{i=1}^l\sum_{j=1}^na_{ij}x_{s_i(j)}^{i}\big\| - \big\|\sum_{i=1}^l\sum_{j=1}^na_{ij}e_j^{i}\big\|\Big|<\varepsilon.\end{align*}$$

A Banach space X is said to admit a uniformly unique joint spreading model with respect to a family of normalised sequences $\mathscr {F}$ in X if there exists a constant C, such that whenever two arrays $(x_j^{i})_j$ and $(y_j^{i})_j$ , $1\leq i\leq l$ , of sequences from $\mathscr {F}$ generate joint spreading models, then those must be C-equivalent. Moreover, a Banach space admits a uniformly unique joint spreading model with respect to a family $\mathscr {F}$ if and only if it admits a unique asymptotic model with respect to $\mathscr {F}$ (see, e.g. [Reference Argyros, Georgiou, Lagos and Motakis6, Remark 4.21] or [Reference Argyros and Motakis9, Proposition 3.12]). In particular, if a space admits a uniformly unique joint spreading model with respect to some family $\mathscr {F}$ satisfying certain conditions described in [Reference Argyros, Georgiou, Lagos and Motakis6, Proposition 4.9], then this is equivalent to some $\ell _p$ . In order to show that a space admits some $\ell _p$ as a uniformly unique joint spreading model, it may be more convenient to prove (ii) of the following lemma, thereby avoiding the use of plegma families.

Lemma 3.4. Let X be a Banach space and $\mathscr {F}$ be a family of normalised sequences in X. Let also $1\le p<\infty $ . The following are equivalent.

  1. (i) X admits $\ell _p$ as a uniformly unique joint spreading model with respect to the family $\mathscr {F}$ .

  2. (ii) There exist constants $c,C>0$ , such that for every array $(x^i_j)_j$ , $1\le i\le l$ , of sequences from $\mathscr {F}$ , there is $M=\{m_1<m_2<\ldots \}$ , an infinite subset of the naturals, such that for any choice of $1\le i_j\le l$ , $j\in M$ , every $F\subset M$ with $m_k\le F$ and $|F|\le k$ and any choice of scalars $a_j$ , $j\in F$ ,

    $$\begin{align*}c \|(a_j)_{j\in F}\|_p \le \big\| \sum_{j\in F} a_j x^{i_j}_{j} \big\| \le C \|(a_j)_{j\in F}\|_p. \end{align*}$$

Proof. Note that (i) implies that there are constants $c,C>0$ , such that for every array $(x^i_j)_j$ , $1\le i\le l$ , of sequences from $\mathscr {F}$ , there is $N=\{n_1<n_2<\ldots \}$ , an infinite subset of the naturals, such that for any k, any strict plegma family $(s_i)_{i=1}^l\in S\text {-}Plm_l([N]^k)$ with $n_k\le s_1(1)$ and any $l\times k$ matrix $A=(a_{ij})$ of scalars, we have that

$$\begin{align*}c \|(a_{ij})_{i=1,j=1}^{l,k}\|_p \le \big\| \sum_{i=1}^l\sum_{j=1}^k a_{ij} x^{i}_{s_i(j)} \big\| \le C \|(a_{ij})_{i=1,j=1}^{l,k}\|_p. \end{align*}$$

Let $N'=\{ n_{2kl}:k\in \mathbb {N} \}$ and observe that for $k_1,\ldots ,k_d\in \mathbb {N}$ , there is a strict plegma family $(s_i)_{i=1}^l\in S\text {-}Plm_l([N]^d )$ , such that $n_{2k_jl}\in \{s_i(j):i=1,\ldots ,l\}$ for all $j=1,\ldots , d$ . Hence, we may find $M\subset N'$ satisfying (ii) with constants $c,C$ . Finally, by repeating the sequences in the array, it follows easily that (ii) yields (i).

We recall the main result from [Reference Argyros, Georgiou, Lagos and Motakis6], stating that whenever a Banach space admits a uniformly unique joint spreading model with respect to some family satisfying certain stability conditions, then it satisfies a property concerning its bounded linear operators called the Uniform Approximation on Large Subspaces property (see [Reference Argyros, Georgiou, Lagos and Motakis6, Theorem 5.17] and [Reference Argyros, Georgiou, Lagos and Motakis6, Theorem 5.23]).

Definition 3.5 [Reference Maurey, Milman and Tomczak-Jaegermann24].

A Banach space X is called Asymptotic $\ell _p$ , $1\leq p<\infty $ , (respectively, Asymptotic $c_0$ ) if there exists a constant C, such that in a two-player n-turn game $G(n,p,C)$ , where in each turn $k=1,\ldots ,n$ , player (S) picks a finite codimensional subspace $Y_k$ of X, and then player (V) picks a normalised vector $x_k\in Y_k$ , player (S) has a winning strategy to force player (V) to pick a sequence $(x_k)_{k=1}^n$ that is C-equivalent to the unit vector basis of $\ell ^n_p$ (respectively, $\ell _{\infty }^n)$ .

Although this is not the initial formulation, it is equivalent and follows from [Reference Maurey, Milman and Tomczak-Jaegermann24, Subsection 1.5]. The typical example of a nonclassical Asymptotic $\ell _p$ space is the Tsirelson space from [Reference Figiel and Johnson17]. This is a reflexive Asymptotic $\ell _1$ space, and it is the dual of Tsirelson’s original space from [Reference Tsirelson27] which is Asymptotic $c_0$ . Finally, whenever a Banach space is Asymptotic $\ell _p$ or Asymptotic $c_0$ , it admits a uniformly unique joint spreading model with respect to $\mathscr {F}_0(X)$ (see, e.g. [Reference Argyros, Georgiou, Lagos and Motakis6, Corollary 4.12]).

The above definition is the coordinate-free version of the notion of an asymptotic $\ell _p$ Banach space with a basis introduced by Milman and Tomczak-Jaegermann in [Reference Milman and Tomczak-Jaegermann25].

Definition 3.6 [Reference Milman and Tomczak-Jaegermann25].

Let X be a Banach space with a Schauder basis $(e_i)_i$ and $1\le p< \infty $ . We say that the Schauder basis $(e_i)_i$ of X is asymptotic $\ell _p$ if there exist positive constants $D_1$ and $D_2$ , such that for all $n\in \mathbb {N}$ , there exists $N(n)\in \mathbb {N}$ with the property that whenever $N(n)\le x_1< \cdots < x_n$ are vectors in X, then

$$\begin{align*}\frac{1}{D_1}(\sum_{i=1}^n\|x_i\|^p)^{\frac{1}{p}}\le \|\sum_{i=1}^nx_i\|\le D_2 (\sum_{i=1}^n\|x_i\|^p)^{\frac{1}{p}}. \end{align*}$$

Specifically, we say that $(e_i)_i$ is D-asymptotic $\ell _p$ for $D=D_1D_2$ . The definition of an asymptotic $c_0$ space is given similarly.

It is easy to show that if X has a Schauder basis that is asymptotic $\ell _p$ , then X is Asymptotic $\ell _p$ . Moreover, if X is Asymptotic $\ell _p$ , then it contains an asymptotic $\ell _p$ sequence. In particular, note that if X has a Schauder basis and Y is an Asymptotic $\ell _p$ subspace of X, then Y contains a further subspace that is isomorphic to an asymptotic $\ell _p$ block subspace.

A noteworthy remark is that sequential asymptotic properties, array asymptotic properties and global asymptotic properties of a Banach space X can alternatively be interpreted as properties of special weakly null trees. A collection $\{x_A:A\in [\mathbb {N}]^{\leq n}\}$ in X is said to be a normalised weakly null tree of height n, if for every $A\in [\mathbb {N}]^{\leq n-1}$ , $(x_{A\cup \{j\}})_{j>\max (A)}$ is a normalised weakly null sequence. Such a tree is said to originate from a sequence $(x_j)_j$ if for all $A = \{a_1,\ldots ,a_i\}$ , we have $x_A = x_{a_i}$ . Similarly, a tree $\{x_A:A\in [\mathbb {N}]^{\leq n}\}$ is said to originate from an array of sequences $(x^{(i)}_j)_j$ , $1\leq i\leq n$ if for all $A = \{a_1,\ldots ,a_i\}$ , we have $x_A = x^{(i)}_{a_i}$ . Then, X has a uniformly unique $\ell _p$ spreading model if and only if there exists $C>0$ , so that every tree $\{x_A:A\in [\mathbb {N}]^{\leq n}\}$ originating from a normalised weakly null sequence $(x_j)_j$ in X has a maximal branch that is C-equivalent to the unit vector basis of $\ell _p^n$ . Similarly, X has a unique $\ell _p$ asymptotic model if the same can be said about all trees originating from normalised weakly null arrays in X. Finally, a Banach space X is an Asymptotic $\ell _p$ space (or an Asymptotic $c_0$ space if $p=\infty $ ) if there exists $C>0$ , so that every normalised weakly null tree of height n has a maximal branch $x_{\{a_1\}}, x_{\{a_1,a_2\}},\ldots ,x_{\{a_1,a_2,\ldots ,a_n\}}$ that is C-equivalent to the unit vector basis of $\ell _p^n$ . For more details, see [Reference Baudier, Lancien, Motakis and Schlumprecht14, Remark 3.11].

4 Measures on countably branching well-founded trees

In this section, we recall certain results from [Reference Argyros, Georgiou and Motakis8] concerning measures on countably branching well-founded trees. These will be used to prove that for all $1\le p<\infty $ , the space $\mathfrak {X}^{(p)}_{\text {awi}}$ admits $\ell _p$ as a unique asymptotic model. In particular, Proposition 4.1 and Lemma 4.6 will be used to prove Lemma 7.2, which is one of the key ingredients in the proof of the main result, Theorem 1.4.

Let $\mathcal {T}=(A,<_{\mathcal {T}})$ , where A is a countably infinite set equipped with a partial order $<_{\mathcal {T}}$ . In the sequel, we use $t\in \mathcal {T}$ instead of $t\in A$ . We assume that $<_{\mathcal {T}}$ is such that there is a unique minimal element in $\mathcal {T}$ , and for each $t\in \mathcal {T}$ , the set $S_t=\{s\in \mathcal {T}: s\le _{\mathcal {T}} t \}$ is finite and totally ordered, that is, $\mathcal {T}$ is a rooted tree. We also assume that $\mathcal {T}$ is well founded, that is, it contains no infinite totally ordered sets, and countably branching, that is, every nonmaximal node has countably infinite immediate successors.

Observe that ${\widetilde {\mathcal {T}}}=(\{S_t :{t\in \mathcal {T}}\},<_{\widetilde {\mathcal {T}}})$ , where $<_{\widetilde {\mathcal {T}}}$ denotes inclusion, is also a tree, and that $\mathcal {T}$ is in fact isomorphic to ${\widetilde {\mathcal {T}}}$ via the mapping $t\mapsto S_t$ . Given $t\in \mathcal {T}$ , we will denote $S_t$ by ${\tilde {t}}$ , identifying it as an element of ${\widetilde {\mathcal {T}}}$ . For each ${\tilde {t}}\in {\widetilde {\mathcal {T}}}$ , we denote by $S({\tilde {t}})$ the set of immediate successors of ${\tilde {t}}$ in ${\widetilde {\mathcal {T}}}$ . In particular, if ${\tilde {t}}$ is maximal, then $S({\tilde {t}})$ is empty. Moreover, for ${\tilde {t}}\in {\widetilde {\mathcal {T}}}$ , we denote $V_{\tilde {t}} = \{\tilde {s}\in {\widetilde {\mathcal {T}}}:{\tilde {t}}\leq _{\widetilde {\mathcal {T}}} \tilde {s}\}$ and view ${\widetilde {\mathcal {T}}}$ as a topological space with the topology generated by the sets $V_{\tilde {t}}$ and ${\widetilde {\mathcal {T}}}\setminus V_{\tilde {t}}$ , ${\tilde {t}}\in {\widetilde {\mathcal {T}}}$ , that is, the pointwise convergence topology. This is a compact metric topology, such that for each ${\tilde {t}}\in {\widetilde {\mathcal {T}}}$ , the sets of the form $V_{\tilde {t}}\setminus (\cup _{\tilde {s}\in F}V_{\tilde {s}})$ , $F\subset S({\tilde {t}})$ finite, form a neighbourhood base of clopen sets for $\tilde t$ . We denote by $\mathcal {M}_+({\widetilde {\mathcal {T}}})$ the cone of all bounded positive measures $\mu :\mathcal {P}({\widetilde {\mathcal {T}}})\to [0,+\infty )$ . For $\mu \in \mathcal {M}_+({\widetilde {\mathcal {T}}})$ , we define the support of $\mu $ to be the set $\mathrm {supp}(\mu ) = \{{\tilde {t}}\in {\widetilde {\mathcal {T}}}:\mu (\{{\tilde {t}}\})>0\}$ . Finally, we say that a subset $\mathcal {A}$ of $\mathcal {M}_+({\widetilde {\mathcal {T}}})$ is bounded if $\sup _{\mu \in \mathcal {A}}\mu ({\widetilde {\mathcal {T}}})<\infty $ .

Proposition 4.1. Let $(\mu _i)_i$ be a bounded and disjointly supported sequence in $\mathcal {M}_+({\widetilde {\mathcal {T}}})$ . Then for every $\varepsilon>0$ , there is an $L\in [\mathbb {N}]^{\infty }$ and a subset $G_i$ of $\text {supp}(\mu _{i})$ for each $i\in L$ , satisfying the following.

  1. (i) $\mu _{i}({\widetilde {\mathcal {T}}}\setminus G_i)\le \varepsilon $ for every $i\in L$ .

  2. (ii) The sets $G_i$ , $i\in L$ , are pairwise incomparable.

For the proof, we refer the reader to [Reference Argyros, Georgiou and Motakis8, Proposition 3.1].

Definition 4.2. Let $(\mu _i)_i$ be a sequence in $\mathcal {M}_+({\widetilde {\mathcal {T}}})$ and $\nu \in \mathcal {M}_+(\mathcal {{\widetilde {\mathcal {T}}}})$ . We say that $\nu $ is the successor-determined limit of $(\mu _i)_i$ if for all ${\tilde {t}}\in {\widetilde {\mathcal {T}}}$ , we have $\nu (\{{\tilde {t}}\}) = \lim _i\mu _i(S({\tilde {t}}))$ . In this case, we write $\nu = \mathrm {succ}\text {-}\!\lim _i\mu _i$ .

Recall that a bounded sequence $(\mu _i)_i$ in $\mathcal {M}_+({\widetilde {\mathcal {T}}})$ converges in the $w^*$ -topology to a $\mu \in \mathcal {M}_+({\widetilde {\mathcal {T}}})$ if and only if for all clopen sets $V\subset {\widetilde {\mathcal {T}}}$ , we have $\lim _i\mu _i(V) = \mu (V)$ if and only if for all ${\tilde {t}}\in {\widetilde {\mathcal {T}}}$ , we have $\lim _i\mu _i(V_{\tilde {t}}) = \mu (V_{\tilde {t}})$ . In this case, we write $\mu = w^*\text {-}\lim _i\mu _i$ .

Lemma 4.3. Let $(\mu _i)_i$ be a bounded sequence in $\mathcal {M}_+({\widetilde {\mathcal {T}}})$ . There exist a subsequence $(\mu _{i_n})_n$ of $(\mu _i)_i$ and $\nu \in \mathcal {M}_+({\widetilde {\mathcal {T}}})$ with $\nu = \mathrm {succ}\text {-}\!\lim _n\mu _{i_n}$ .

Remark 4.4. It is possible for a bounded sequence $(\mu _i)_i$ in $\mathcal {M}_+({\widetilde {\mathcal {T}}})$ to satisfy $w^*\text {-}\lim _i\mu _i\neq \mathrm {succ}\text {-}\!\lim _i\mu _i$ . Take, for example, ${\widetilde {\mathcal {T}}} = [\mathbb {N}]^{\leq 2}$ (all subsets of $\mathbb {N}$ with at most two elements with the partial order of initial segments), and define $\mu _i = \delta _{\{i,i+1\}}$ , $i\in \mathbb {N}$ . Then $w^*\text {-}\lim _i\mu _i = \delta _{\emptyset }$ , whereas $\mathrm {succ}\text {-}\!\lim _i\mu _i = 0$ .

Although these limits are not necessarily the same, there is an explicit formula relating $\mathrm {succ}\text {-}\!\lim _i\mu _i$ to $w^*\text {-}\lim _i\mu _i$ .

Lemma 4.5. Let $(\mu _i)_i$ be a bounded and disjointly supported sequence in $\mathcal {M}_+({\widetilde {\mathcal {T}}})$ , such that $w^*\text {-}\lim _i\mu _i = \mu $ exists, and for all ${\tilde {t}}\in {\widetilde {\mathcal {T}}}$ , the limit $\nu (\{{\tilde {t}}\}) = \lim _i\mu _i(S({\tilde {t}})) $ exists as well. Then for every ${\tilde {t}}\in {\widetilde {\mathcal {T}}}$ and enumeration $({\tilde {t}}_j)_j$ of $S({\tilde {t}})$ , we have

(4.1) $$ \begin{align} \mu(\{{\tilde{t}}\}) = \nu(\{{\tilde{t}}\}) + \lim_j\lim_i\mu_i\Big(\cup_{k\geq j}(V_{{\tilde{t}}_k}\setminus\{{\tilde{t}}_{k}\})\Big). \end{align} $$

In particular, $\mu (\{{\tilde {t}}\}) = \nu (\{{\tilde {t}}\})$ if and only if the double limit in (4.1) is zero.

Lemma 4.6. Let $(\mu _i)_i$ be a bounded and disjointly supported sequence in $\mathcal {M}_+({\widetilde {\mathcal {T}}})$ , such that $\mathrm {succ}\text {-}\!\lim _i\mu _i = \nu $ exists. Then there exist an infinite $L\subset \mathbb {N}$ and partitions $A_i$ , $B_i$ of $\mathrm {supp}(\mu _i)$ , $i\in L$ , such that the following are satisfied.

  1. (i) If for all $i\in L$ , we define the measure $\mu _i^1$ by $\mu _i^1(C)= \mu _i(C\cap A_i)$ , then $\nu = w^*\text {-}\lim _{i\in L}\mu _i^1 = \mathrm {succ}\text {-}\!\lim _{i\in L}\mu _i^1$ .

  2. (ii) If for all $i\in L$ , we define the measure $\mu _i^2$ by $\mu _i^2(C)= \mu _i(C\cap B_i)$ , then for all ${\tilde {t}}\in {\widetilde {\mathcal {T}}}$ , the sequence $(\mu _i^2(S({\tilde {t}})))_i$ is eventually zero. In particular, $\mathrm {succ}\text {-}\!\lim _{i\in L}\mu _i^2 = 0$ .

For the proofs, we refer the reader to [Reference Argyros, Georgiou and Motakis8, Lemma 4.10] and [Reference Argyros, Georgiou and Motakis8, Lemma 4.12].

Remark 4.7. Although the results from [Reference Argyros, Georgiou and Motakis8] were formulated for trees $\mathcal {T}$ defined on infinite subsets of $\mathbb {N}$ , this is not a necessary restriction, and they can be naturally extended to the more general setting of countably branching well-founded trees.

PART I The case of $\boldsymbol {\ell _1}$

5 Definition of the space $\mathfrak {X}_{\text {awi}}^{(1)}$

The method of saturation with asymptotically weakly incomparable constraints, that is used in the construction of both spaces presented in this paper, was introduced in [Reference Argyros, Georgiou and Motakis8], where it was shown that (b) $_1\not \Rightarrow $ (a) $_1$ . There, it was also used to prove an even stronger result, namely, the existence of a Banach space with a basis admitting $\ell _1$ as a unique asymptotic model, and in which any infinite subsequence of the basis generates a non-Asymptotic $\ell _1$ subspace. This method requires the existence of a well-founded tree defined either on the basis of the space or on a family of functionals of its norming set. In this section, we define the space $\mathfrak {X}_{\text {awi}}^{(1)}$ by introducing its norm via a norming set, which is a subset of the norming set of a Mixed Tsirelson space $\mathcal {T}[(m_j,\mathcal {S}_{n_j})_j]$ for an appropriate choice of $(m_j)_j$ and $(n_j)_j$ described below. The key ingredient in the definition of this norming set is the notion of asymptotically weakly incomparable sequences of functionals, which is also introduced in this section. This notion will allow the space $\mathfrak {X}_{\text {awi}}^{(1)}$ to admit $\ell _1$ as a unique asymptotic model, while at the same time, it will force the norm to be small on the branches of a tree, in every subspace of $\mathfrak {X}_{\text {awi}}^{(1)}$ , showing that the space does not contain Asymptotic $\ell _1$ subspaces.

5.1 Definition of the space $\mathfrak {X}_{\text {awi}}^{(1)}$

Define a pair of strictly increasing sequences of natural numbers $(m_j)_j$ , $(n_j)_j$ as follows:

$$ \begin{align*} m_1 &= 2 & n_1 &= 1 \\ m_{j+1}&=m_j^{m_j} & n_{j+1}&=2^{2m_{j+1}}n_j .\end{align*} $$

Definition 5.1. Let $V_{(1)}$ denote the minimal subset of $c_{00}(\mathbb {N})$ that

  1. (i) contains 0 and all $\pm e_j^*$ , $j\in \mathbb {N}$ and

  2. (ii) is closed under the operations $(m_j,\mathcal {S}_{n_j})_j$ , that is, if $j\in \mathbb {N}$ and $f_1<\ldots <f_n$ is an $\mathcal {S}_{n_j}$ -admissible sequence (see Section 2.1) in $V_{(1)}\setminus \{0\}$ , then $m_j^{-1}\sum _{i=1}^nf_i $ is also in $V_{(1)}$ .

Remark 5.2.

  1. (i) If $f\in V_{(1)}\setminus \{0\}$ , then either $f\in \{\pm e^*_j:j\in \mathbb {N}\}$ , or it is of the form $f=m_j^{-1}\sum _{i=1}^nf_i$ with $f_1<\ldots <f_n$ an $\mathcal {S}_{n_j}$ -admissible sequence in $V_{(1)}$ for some $j\in \mathbb {N}$ .

  2. (ii) As usual, we view the elements of $V_{(1)}$ as functionals acting on $c_{00}(\mathbb {N})$ , inducing a norm $\|\cdot \|_{V_{(1)}}$ . The completion of $(c_{00}(\mathbb {N}),\|\cdot \|_{V_{(1)}})$ is the Mixed Tsirelson space $\mathcal {T}[(m_j,\mathcal {S}_{n_j})_j]$ introduced for the first time in [Reference Argyros and Deliyanni5]. The first space with a saturated norm defined by a countable family of operations is the Schlumprecht space [Reference Schlumprecht26], which is a fundamental discovery and was used by Gowers and Maurey [Reference Gowers and Maurey19] to define the first hereditarily indecomposable (HI) space.

We now recall the notion of tree analysis which appeared for the first time in [Reference Argyros and Deliyanni4]. This has become a standard tool in proving upper bounds for the estimations of functionals on certain vectors in Mixed Tsirelson spaces. However, it is the first time where the tree analysis has a significant role in the definition of the norming set $W_{(1)}$ . Additionally, it is also a key ingredient in the proof that $\mathfrak {X}^{(1)}_{\text {awi}}$ contains no Asymptotic $\ell _1$ subspaces.

Let $\mathcal {A}$ be a rooted tree. For a node $\alpha \in \mathcal {A}$ , we denote by $S(\alpha )$ the set of all immediate successors of $\alpha $ , by $|\alpha |$ the height of $\alpha $ , that is, $|\alpha |=\#\{\beta \in \mathcal {A}:\beta <_{\mathcal {A}}\alpha \}$ , and finally, we denote by $h(\mathcal {A})$ the height of $\mathcal {A}$ , that is, the maximum height over its nodes.

Definition 5.3. Let $f\in V_{(1)}\setminus \{0\}$ . For a finite tree $\mathcal {A}$ , a family $(f_{\alpha })_{\alpha \in \mathcal {A}}$ is called a tree analysis of f if the following are satisfied.

  1. (i) $\mathcal {A}$ has a unique root denoted by $0$ and $f_0=f$ .

  2. (ii) Each $f_{\alpha }$ is in $V_{(1)}$ , and if $\beta <\alpha $ in $\mathcal {A}$ , then $\text {range}(f_{\alpha })\subset \text {range}(f_{\beta })$ .

  3. (iii) For every maximal node $\alpha \in \mathcal {A}$ , we have that $|\alpha |=h(\mathcal {A})$ .

  4. (iv) For every nonmaximal node $\alpha \in \mathcal {A}$ , either $f_{\alpha }$ is the result of some $(m_j,\mathcal {S}_{n_j})$ operation of $(f_{\beta })_{\beta \in S(\alpha )}$ , i.e., $f_{\alpha }=m_j^{-1}\sum _{\beta \in S(\alpha )} f_{\beta }$ , or $f_{\alpha }\in \{\pm e_j^*:j\in \mathbb {N}\}$ and $S(\alpha )=\{\beta \}$ with $f_{\beta }=f_{\alpha }$ .

  5. (v) For every maximal node $\alpha \in \mathcal {A}$ , $f_{\alpha }\in \{\pm e^*_j:j\in \mathbb {N}\}.$

Remark 5.4.

  1. (i) It follows by minimality that every f in $V_{(1)}\setminus \{0\}$ admits a tree analysis, but it may not be unique. For example, $f=(m_{1}m_2)^{-1}e^*_1$ admits two distinct tree analyses.

  2. (ii) The standard definition of a tree analysis does not include 5.3 (iii). This property is included for technical reasons and is used below in the equality of Remark 5.8 (i).

Definition 5.5. Let $f\in V_{(1)}$ .

  1. (i) If $f=0$ or $f\in \{\pm e^*_j:j\in \mathbb {N}\}$ , then we define the weight $w(f)$ of f as $w(f)=0$ and $w(f)=1$ , respectively.

  2. (ii) If f is the result of an $(m_j,S_{n_j})$ -operation for some $j\in \mathbb {N}$ , then $w(f)=m_j$ .

Remark 5.6. It is not difficult to see that $w(f)$ , for $f\in V_{(1)}$ , is not uniquely determined, that is, f could be the result of more than one distinct $(m_j,\mathcal {S}_{n_j})$ -operation. However, if we fix a tree analysis $(f_{\alpha })_{\alpha \in \mathcal {A}}$ of f, then for $\alpha \in \mathcal {A}$ with $f_{\alpha }=(m_{j_{\alpha }})^{-1}\sum _{\beta \in S(\alpha )}f_{\beta }$ , the tree analysis determines the weight $w(f_{\alpha })$ , being equal to $m_{j_{\alpha }}$ . Thus, for $f\in V_{(1)}$ and a fixed tree analysis $(f_{\alpha })_{\alpha \in \mathcal {A}}$ of f, with $w(f_{\alpha })$ , we will denote the weight $m_{j_{\alpha }}$ determined by $(f_{\alpha })_{\alpha \in \mathcal {A}}$ , for every $\alpha \in \mathcal {A}$ . In addition, we will denote by $\bar {f}_{\alpha }$ the pair $(f_{\alpha },m_{j_{\alpha }})$ .

Definition 5.7. Let $f\in V_{(1)}$ and $(f_{\alpha })_{\alpha \in \mathcal {A}}$ be a tree analysis of f. Then for $\alpha \in \mathcal {A}$ , we define the relative weight $w_f(f_{\alpha })$ of $f_{\alpha }$ as

$$\begin{align*}w_f(f_{\alpha}) = \begin{cases} \prod_{\beta<\alpha}w(f_{\beta}) & \text{if }\alpha\neq0\\ 1 &\text{otherwise.} \end{cases} \end{align*}$$

Remark 5.8. Let $f\in V_{(1)}$ and $(f_{\alpha })_{\alpha \in \mathcal {A}}$ be a tree analysis of f.

  1. (i) For every $k=1,\ldots ,h(\mathcal {A})$

    $$\begin{align*}f=\sum_{|a|=k}w_f(f_{\alpha})^{-1}f_{\alpha}. \end{align*}$$
    This can be proved by induction and essentially relies on the fact that $(f_{\alpha })_{\alpha \in \mathcal {A}}$ satisfies 5.3 (iii).
  2. (ii) If $\mathcal {B}$ is a maximal pairwise incomparable subset of $\mathcal {A}$ , then

    $$\begin{align*}f=\sum_{\beta\in\mathcal{B}}w_f(f_{\beta})^{-1}f_{\beta}. \end{align*}$$
  3. (iii) For every $\alpha \in \mathcal {A}$ , whose immediate predecessor $\beta $ in $\mathcal {A}$ (if one exists) satisfies $f_{\beta }\notin \{\pm e_j^*:j\in \mathbb {N}\}$ , we have $w_f(f_{\alpha })\ge 2^{|\alpha |}.$

Fix an injection $\sigma $ that maps any pair $(f,w(f))$ , for $f\in V_{(1)}$ and $w(f)$ a weight of f, to some $m_j$ with $m_j>\max \text {supp} (f)\: w(f)$ whenever $f\neq 0$ .

Definition 5.9. Define a partial order $<_{\mathcal {T}}$ on the set of all pairs $(f,w(f))$ for $f\in V_{(1)}$ and $w(f)$ a weight of f, as follows: $(f,w(f))<_{\mathcal {T}} (g ,w(g))$ either if $f=0$ or if there exist $f_1<\ldots <f_n \in V_{(1)}$ and weights $w(f_1),\ldots ,w(f_n)$ , such that

  1. (i) $(f_i)_{i=1}^n$ is $\mathcal {S}_1$ -admissible,

  2. (ii) $w(f_1)=\sigma (0,0)$ and $w(f_i)=\sigma (f_{i-1},w(f_{i-1}))$ for every $i=2,\ldots ,n$ ,

  3. (iii) there are $1\le i_1<i_2\le n$ , such that $f=f_{i_1}$ and $g=f_{i_2}$ .

It is easy to see that $<_{\mathcal {T}}$ induces a tree structure rooted at $\bar {0}=(0,0)$ . Let us denote this tree by $\mathcal {T}$ , and observe that this is a countably branching well-founded tree, due to 5.9(i). For $t=(f,w(f))\in \mathcal {T}$ , we set $f_t=f$ and $w(t)=w(f)$ .

It is clear that unlike the case where the tree is defined on the basis of the space, here, incomparable segments need not necessarily have disjoint supports. This forces us to introduce the notion of essentially incomparable nodes, which was first defined in [Reference Argyros, Georgiou and Motakis8]. To this end, we first need to define an additional tree structure that is readily implied by $\mathcal {T}$ via the projection $(f,w(f))\mapsto w(f)$ .

Definition 5.10. Define a partial order $<_{\mathcal {W}}$ on $\{ m_j:j\in \mathbb {N} \}$ as follows: $m_i<_{\mathcal {W}} m_j$ if there exist $t_{1},t_{2}\in \mathcal {T}$ , such that $t_1<_{\mathcal {T}} t_2$ , $w(t_1)=m_i$ and $w(t_2)=m_j$ .

As an immediate consequence of the fact that $\mathcal {T}$ is a countably branching well-founded tree, we have that $<_{\mathcal {W}}$ also defines a tree structure. Let us denote this tree by $\mathcal {W}$ and note that it is also countably branching and well founded.

Remark 5.11. The above definition implies that if $t_1,t_2\in \mathcal {T}$ are such that $w(t_1)<_{\mathcal {W}} w(t_2)$ , then there exist $t_3,t_4\in \mathcal {T}$ , such that $t_3<_{\mathcal {T}} t_4$ , $w(t_3)=w(t_1)$ and $w(t_4)=w(t_2)$ . The tree structure of $\mathcal {T}$ implies that $t_3$ is uniquely defined, and we will say that $t_3$ generates $w(t_2)$ . This is not the case, however, for $t_4$ , and, moreover, it is not necessary that $t_3<_{\mathcal {T}} t_2$ .

Definition 5.12.

  1. (i) A subset A of $\mathcal {T}\setminus \{\bar {0}\}$ is called essentially incomparable if whenever $t_1,t_2\in A$ are such that $w(t_1)<_{\mathcal {W}} w(t_2)$ , then for the unique $t_3\in \mathcal {T}$ with $w(t_3)=w(t_1)$ that generates $w(t_2)$ , we have that $f_{t_3}<f_{t_1}$ .

  2. (ii) A subset A of $\mathcal {T}$ is called weight incomparable if for any $t_1\neq t_2$ in A, $w(t_1)\neq w(t_2)$ and the weights $w(t_1)$ and $w(t_2)$ are incomparable in $\mathcal {W}$ .

  3. (iii) A sequence $(A_j)_j$ of subsets of $\mathcal {T}$ is called pairwise weight incomparable if for every $j_1\neq j_2$ in $\mathbb {N}$ , $t_1\in A_{j_1}$ and $t_2\in A_{j_2}$ , $w(t_1)\neq w(t_2)$ and the weights $w(t_1)$ and $w(t_2)$ are incomparable in $\mathcal {W}$ .

Remark 5.13.

  1. (i) If A is an essentially (respectively, weight) incomparable subset of $\mathcal {T}$ , then every $B\subset A$ is also essentially (respectively, weight) incomparable.

  2. (ii) Any subsequence of a pairwise weight incomparable sequence in $\mathcal {T}$ is also pairwise weight incomparable.

  3. (iii) Any weight incomparable subset of $\mathcal {T}$ is essentially incomparable.

  4. (iv) Let $A=\{(f,1):f\in \{\pm e^*_j:j\in \mathbb {N}\} \}$ . Then A is essentialy incomparable, and, additionally, if $B\subset \mathcal {T}$ is essentially incomparable, then the same holds for $A\cup B$ .

We can finally describe the rule used to define the norming set $W_{(1)}$ of $\mathfrak {X}_{\text {awi}}^{(1)}$ , namely, asymptotically weakly incomparable constraints.

Definition 5.14. Let J be an initial segment of $\mathbb {N}$ or $ J =\mathbb {N}$ . Then a sequence $(f_j)_{j\in J}$ of functionals with successive supports in $V_{(1)}\setminus \{0\}$ is called asymptotically weakly incomparable (AWI) if each $f_j$ admits a tree analysis $(f_{j,\alpha })_{\alpha \in \mathcal {A}_j}$ , $j\in J$ , such that the following are satisfied.

  1. (i) There is a partition $\{\bar {f}_j:j\in J\}= C_1^0\cup C_2^0$ , such that $C^0_1$ is essentially incomparable and $C^0_2$ is weight incomparable.

  2. (ii) For every $k,j\in J$ with $j\ge k+1$ , there exists a partition

    $$\begin{align*}\{\bar{f}_{j,\alpha}:\alpha\in \mathcal{A}_j\text{ and }|\alpha|=k\}=C^{k}_{1,j}\cup C^{k}_{2,j}, \end{align*}$$
    such that $\cup _{j=k+1}^{\infty } C^k_{1,j}$ is essentially incomparable and $(C^{k}_{2,j})_{j=k+1}^{\infty }$ is pairwise weight incomparable.

Figure 1 The collection of nodes of a fixed level in rectangles across all tree analyses forms an essentially incomparable subset, while circles across a fixed level form a family of pairwise weight incomparable subsets.

Before defining the space $\mathfrak {X}_{\text {awi}}^{(1)}$ , we prove that AWI sequences are stable under taking subsequences and under taking restrictions of functionals to subsets. This fact will imply the unconditionality of the basis of $\mathfrak {X}_{\text {awi}}^{(1)}$ .

Remark 5.15. Let $f\in V_{(1)}$ and $(f_{\alpha })_{\alpha \in \mathcal {A}}$ be a tree analysis of f. Let $\Delta $ be a nonempty subset of $\text {supp}(f)$ , and set $g=f|_{\Delta }$ . First, note that $g\in V_{(1)}$ . Moreover, $(f_{\alpha })_{\alpha \in \mathcal {A}}$ naturally induces a tree analysis $(g_{\alpha })_{\alpha \in \mathcal {B}}$ for g as follows: $\mathcal {B}=\{\alpha \in \mathcal {A}:\text {supp}(f_{\alpha })\cap \Delta \neq \emptyset \}$ and $g_{\alpha }=f_{\alpha }|_{\Delta }$ , $\alpha \in \mathcal {B}$ . Finally, it is easy to see that $w(g)=w(f)$ .

Proposition 5.16. Let J be an initial segment of $\mathbb {N}$ or $ J =\mathbb {N}$ and $(f_j)_{j\in J}$ be an AWI sequence in $V_{(1)}$ .

  1. (i) Every subsequence of $(f_j)_{j\in J}$ is also an AWI sequence in $V_{(1)}$ .

  2. (ii) If $\Delta _j$ is a nonempty subset of $\text {supp} (f_j)$ and $g_j=f_j|_{\Delta _j}$ , $j\in J$ , then $(g_j)_{j\in J}$ is an AWI sequence in $V_{(1)}$ .

  3. (iii) If $(g_j)_{j\in J}$ is a sequence in $V_{(1)}$ , such that $|g_j|=|f_j|$ for all $j\in J$ , then $(g_j)_{j\in J}$ is also AWI.

Proof. Let for every $j\in J$ , $(f_{j,\alpha })_{\alpha \in \mathcal {A}_j}$ be a tree analysis of $f_j$ with

$$\begin{align*}\{ \bar{f}_j:j\in J \}= C^0_1\cup C^0_2 \end{align*}$$

and for every $k,j\in J$ with $j> k$

$$\begin{align*}\{\bar{f}_{j,\alpha}:\alpha\in \mathcal{A}_j\text{ and }|\alpha|=k\}=C^{k}_{1,j}\cup C^{k}_{2,j}, \end{align*}$$

witnessing that $(f_j)_{j\in J}$ is AWI. We will define the desired partitions proving the cases (i)–(iii).

To prove (i), let N be a subset of J and define

$$\begin{align*}F^0_i=\{\bar{f}_j:j\in N\}\cap C^0_i,\quad i=1,2. \end{align*}$$

Then $\{\bar {f}_j:j\in N \} = F^0_1\cup F^0_2$ , where $F^0_1$ is essentially incomparable and $F^0_2$ is weight incomparable. For the remaining part, let $k\in N$ , and note that for $N_k=\{j\in N:j\ge k\}$ , $\cup _{j\in N_k} C^{k}_{1,j}$ is essentially incomparable and $(C^{k}_{2,j})_{j\in N_k}$ is pairwise weight incomparable.

To prove (ii), Remark 5.15 implies that $g_j\in V_{(1)}$ , $w(g_j)=w(f_j)$ , and we let $(g_{j,\alpha })_{\alpha \in \mathcal {B}_j}$ be the tree analysis of g induced by $(f_{j,\alpha })_{\alpha \in \mathcal {A}_j}$ , $j\in J$ . Define

$$\begin{align*}F^0_i=\{\bar{g}_j:j\in J\text{ and }g_j=f_j|_{\Delta_j}\text{ with }\bar{f}_j\in C^0_i \}, \quad i=1,2, \end{align*}$$

and observe that $\{\bar {g}_j:j\in J \}=F^0_1\cup F^0_2$ . Moreover, for $j\in J$ , $\text {supp} (g_j)\subset \text {supp} (f_j)$ and $w(g_j)=w(f_j)$ , and, hence, whenever $g_i\neq g_j$ are in $F^0_1$ with $w(g_i)<_{\mathcal {W}} w(g_j)$ , we have $w(f_i)<_{\mathcal {W}} w(f_j)$ , implying that the generator $t_3\in \mathcal {T}$ of $w(f_j)=w(g_j)$ with $w(t_3)=w(f_i)=w(g_i)$ is such that $f_{t_3}<f_i$ , and thus $f_{t_3}<g_i$ . This yields that $F^0_1$ is essentially incomparable. Clearly, $F^0_2$ is weight incomparable. Next, for $k,j\in J$ with $j>k$ , define

$$\begin{align*}F^k_{i,j}=\{\bar{g}_{j,\alpha}:g_{j,\alpha}=f_{j,\alpha}|_{\Delta_j}\text{ with }\bar{f}_{j,\alpha}\in C^k_{i,j} \}, \quad i=1,2. \end{align*}$$

Note that for each $k\in J$ , $(F^{k}_{2,j})_{j=k+1}^{\infty }$ is pairwise weight incomparable, and the proof that $\cup _{j=k+1}^{\infty } F^k_{1,j}$ is essentially incomparable is identical to that for $F^1_0$ . Finally, the proof of (iii) is similar that of (ii).

Definition 5.17. Let $W_{(1)}$ be the smallest subset of $V_{(1)}$ that is symmetric, contains the singletons and for every $j\in \mathbb {N}$ and every $\mathcal {S}_{n_j}$ -admissible AWI sequence $(f_i)_{i=1}^n$ in $W_{(1)}$ , we have that $m_j^{-1}\sum _{i=1}^nf_i$ is in $W_{(1)}$ . Moreover, let $\mathfrak {X}_{\text {awi}}^{(1)}$ denote the completion of $c_{00}(\mathbb {N})$ with respect to the norm induced by $W_{(1)}$ .

Remark 5.18.

  1. (i) The norming set $W_{(1)}$ can be defined as the increasing union of a sequence $(W^n_{(1)})_{n=0}^{\infty }$ , where $W^0_{(1)}=\{\pm e^*_k:k\in \mathbb {N}\}\cup \{0\}$ and

    $$ \begin{align*} W^{n+1}_{(1)}=W^n_{(1)}\cup\Big\{\frac{1}{m_j}\sum_{l=1}^df_l:\;&j,d\in\mathbb{N}\text{ and }(f_l)_{l=1}^d \text{ is an } \mathcal{S}_{n_j}\text{-admissible AWI sequence in }W^n_{(1)}\Big\}. \end{align*} $$
  2. (ii) Note that Remark 5.13 (iv) implies that any sequence of singletons is AWI. Hence, we have that

    $$\begin{align*}W^{1}_{(1)}=W^0_{(1)}\cup\Big\{\frac{1}{m_j}\sum_{k\in E}\epsilon_ke^*_k:\;j\in\mathbb{N}, \; E\in\mathcal{S}_{n_j} \text{ and } \epsilon_k\in\{-1,1\} \text{ for }k\in E \Big\}. \end{align*}$$
  3. (iii) Proposition 5.16 yields that the standard unit vector basis of $c_{00}(\mathbb {N})$ forms an $1$ -unconditional Schauder basis for $\mathfrak {X}_{\text {awi}}^{(1)}$ .

6 Outline of proof

Although unconditionality of the basis of $\mathfrak {X}_{\text {awi}}^{(1)}$ is almost immediate, it is not, however, straightforward to show that $\mathfrak {X}_{\text {awi}}^{(1)}$ admits $\ell _1$ as an asymptotic model. Indeed, this requires Lemma 7.2, which is based on the combinatorial results concerning measures on well-founded trees of Section 4, which first appeared in [Reference Argyros, Georgiou and Motakis8]. This lemma yields that for any choice of successive families $(F_j)_j$ of normalised blocks in $\mathfrak {X}_{\text {awi}}^{(1)}$ and for any $\varepsilon>0$ , we may pass to a subsequence $(F_j)_{j\in M}$ and find a family $(G_j)_{j\in M}$ of subsets of $W_{(1)}$ , such that for any choice of $x_j\in F_j$ , $j\in M$ , there is a $g_j\in G_j$ with $g_j(x_j)>1-\varepsilon $ so that $(g_j)_{j\in M}$ is AWI. Thus, we are able to prove, employing Lemma 3.4, the aforementioned result.

To prove the nonexistence of Asymptotic $\ell _1$ subspaces in $\mathfrak {X}_{\text {awi}}^{(1)}$ , we start with the notion of exact pairs. This is a key ingredient in the study of Mixed Tsirelson spaces, used for the first time by Schlumprecht [Reference Schlumprecht26].

Definition 6.1. We call a pair $(x,f)$ , where $x\in \mathfrak {X}_{\text {awi}}^{(1)}$ and $f\in W_{(1)}$ , an $m_j$ -exact pair if the following hold.

  1. (i) $\|x\|\le 3$ , $f(x)=1$ and $w(f)=m_j$ .

  2. (ii) If $g\in W_{(1)}$ with $w(g)<w(f)$ , then $|g(x)|\le 18w(g)^{-1}$ .

  3. (iii) If $g\in W_{(1)}$ with $w(g)>w(f)$ , then $|g(x)|\le 6(m_{j}^{-1}+m_j w(g)^{-1}) $ .

If, additionally, for every $g\in W_{(1)}$ that has a tree analysis $(g_{\alpha })_{\alpha \in \mathcal {A}}$ , such that $w(g_{\alpha })\neq m_j$ for all $\alpha \in \mathcal {A}$ , we have $|g(x)|\le 18m^{-1}_j$ , then we call $(x,f)$ a strong exact pair.

That is, roughly speaking, for an exact pair $(x,f)$ , the evaluation of a functional g in $W_{(1)}$ , on x, admits an upper bound depending only on the weight of g. In the case of an $m_j$ -strong exact pair $(x,f)$ , any g in $W_{(1)}$ with a tree analysis $(g_{\alpha })_{\alpha \in \mathcal {A}}$ , such that $w(g_{\alpha })\neq m_j$ , has negligible evaluation on x. We will consider certain exact pairs which we call standard exact pairs (SEP) (see Definition 8.7) and which we prove to be strong exact pairs. It is the case that such pairs can be found in any block subspace of $\mathfrak {X}_{\text {awi}}^{(1)}$ , and this is used to prove the reflexivity of $\mathfrak {X}_{\text {awi}}^{(1)}$ as well as the following proposition which yields the nonexistence of Asymptotic $\ell _1$ subspaces.

Proposition 6.2. Given $0<c<1$ , there is $n\in \mathbb {N}$ so that in any block subspace Y there is a sequence $(x_1,f_1),\ldots ,(x_n,f_n)$ of SEPs, where $x_i\in Y$ , $i=1,\ldots ,n$ , with $\bar {f}_1<_{\mathcal {T}}\ldots <_{\mathcal {T}} \bar {f}_n$ , such that $\|x_1+\cdots +x_n\|<c\:n$ .

To this end, we first employ the following lemma that highlights the importance of the asymptotically weakly incomparable constraints.

Lemma 6.3. Let $(x_1,f_1),\ldots ,(x_n,f_n)$ be SEPs with $\bar {f}_1<_{\mathcal {T}}\ldots <_{\mathcal {T}} \bar {f}_n$ . Then, for any $f\in W_{(1)}$ with a tree analysis $(f_{\alpha })_{\alpha \in \mathcal {A}}$ and $k\in \mathbb {N}$ , the number of $f_i$ ’s, $i=1,\ldots ,n$ , such that there exists $\alpha \in \mathcal {A}$ with $|\alpha |=k$ , $w(f_i)=w(f_{\alpha })$ and $\text {supp}(x_i)\cap \text {supp}(f_{\alpha })\neq \emptyset $ , is at most $ek!$ , where e denotes Euler’s number.

Then, we consider a sequence of standard exact pairs $(x_1,f_1),\ldots ,(x_n,f_n)$ with $\bar {f}_1<_{\mathcal {T}}\ldots <_{\mathcal {T}} \bar {f}_n$ and fix $0<c<1$ . Pick an $m\in \mathbb {N}$ , such that $3/2^m<c$ . For $f\in W_{(1)}$ , with a tree analysis $(f_{\alpha })_{\alpha \in \mathcal {A}}$ , we consider partitions $f=g+h$ and $g=g_1+g_2$ as follows: First, set

$$\begin{align*}G=\cup\{\text{range} (x_k)\cap \text{range} (f_{\alpha}): k\in\{1,\ldots,n\}\text{ and }\alpha\in\mathcal{A}\text{ with }w(f_{\alpha})=w(f_k)\}, \end{align*}$$

and define $g=f|_G$ and $h=f|_{\mathbb {N}\setminus G}$ (see Figure 2).

Figure 2 The tree analysis of f and the induced tree analyses of g and h. The circled nodes $\alpha $ are such that $w(f_{\alpha })=w(f_k)$ and $\text {supp}(x_k)\cap \text {supp}(f_{\alpha })\neq \emptyset $ for some $k\in \{1,\ldots ,n\}$ .

To define $g_1$ , consider the tree analysis $(g_{\alpha })_{\alpha \in \mathcal {A}_g}$ of g that is induced by $(f_{\alpha })_{\alpha \in \mathcal {A}}$ , that is, $g_{\alpha }=f_{\alpha }|_G$ for $\alpha \in \mathcal {A}_g$ and $\mathcal {A}_g=\{\alpha \in \mathcal {A}:\text {supp}(f_{\alpha })\cap G\neq \emptyset \}$ . Then, we define

$$ \begin{align*} \mathcal{B}^1_k=\{\alpha\in\mathcal{A}_g:&\;|a|\le m,\; w(f_{\alpha})=w(f_k)\text{ and } w(f_{\beta})\neq w(f_k)\text{ for all }\beta<\alpha\text{ in }\mathcal{A}_g\} \end{align*} $$

for $k=1,\ldots ,n$ ,

$$\begin{align*}G_1=\cup_{k=1}^n\cup\left\{\text{supp}(g_{\alpha})\cap \text{supp}(x_k):\alpha\in\mathcal{B}^1_k\right\}, \end{align*}$$

and finally $g_1 = g|_{G_1}$ (see Figure 3). Observe that Lemma 6.3 implies that

(6.1) $$ \begin{align} \#\left\{ k\in\{1,\ldots,n\} : g_1(x_k)\neq 0\right\} \le \ell = e \sum_{k=1}^mk!. \end{align} $$

Moreover, the induced tree analysis $(h_{\alpha })_{\alpha \in \mathcal {A}_h}$ of h is such that $w(h_{\alpha })\neq w(f_k)$ for all $k=1,\ldots ,n$ , and, therefore, the fact that $(x_k,f_k)$ are strong exact pairs yields

(6.2) $$ \begin{align} |h\left(x_k\right)|\le \frac{18}{w(f_k)}, \quad k=1,\ldots,n. \end{align} $$

Considering a further partition of $g_2|_{\text {supp}(x_k)}$ , we show that

(6.3) $$ \begin{align} |g_2\left(x_k\right)|\le \frac{18}{w(f_k)}+\frac{3}{2^m}, \quad k=1,\ldots,n. \end{align} $$

Hence, (6.1), (6.2) and (6.3) imply

$$\begin{align*}\left|f\left( \frac{1}{n} \sum_{k=1}^n x_k \right)\right| \le \frac{36+\ell}{n}+\frac{3}{2^m}. \end{align*}$$

Then, our choice of m yields Proposition 6.2 for sufficiently large n and $w(f_1)$ , where $w(f_1)$ is chosen appropriately to deal with the case where $f\in \{\pm e_j^*:j\in \mathbb {N}\}$ .

Figure 3 We consider the m-th level of the induced tree analysis of g. Nodes $\alpha $ with $w(f_{\alpha })=w(f_k)$ and $|\alpha |\le m$ are used to define $g_1$ , while such nodes of height greater than m define $g_2$ , restricted on each $x_k$ for $k=1,\ldots ,n$ .

Assuming that Y is a C-asymptotic $\ell _1$ block subspace of $\mathfrak {X}^{(1)}_{\text {awi}}$ , we pick a sequence of standard exact pairs $(x_1,f_1),\ldots ,(x_n,f_n)$ in $Y\times W_{(1)}$ , satisfying the conclusion of Proposition 6.2 and derive a contradiction.

The remainder of this part of the paper is organised as follows. In Section 7, we prove that