# On the embeddability of countably branching bundle graphs into dual spaces

Y. Perreau  Yoël Perreau, Laboratoire de Mathématiques de Besançon, Université
Bourgogne Franche-Comté, CNRS UMR-6623, 16 route de Gray, 25030 Besançon
Cédex, Besançon, France<EMAIL_ADDRESS>
###### Abstract.

In this note the result from [20] by A. Swift concerning the embeddability of
countably branching bundle graphs into Banach spaces is extended from the
context of reflexive spaces with an unconditional asymptotic structure to the
context of dual spaces with a weak∗ unconditional asymptotic structure. We
also investigate weak∗ asymptotic midpoint uniform convexity in dual spaces
which turns out to be equivalent to its weak version in general and to the
standard weak∗ asymptotic uniform convexity up to renorming in dual spaces
with a weak∗ unconditional asymptotic structure.

###### 2010 Mathematics Subject Classification:

46B20, 46B80, 46B85, 46T99

The author is supported by the French “Investissements d’Avenir” program,
project ISITE-BFC (contract ANR-15-IDEX-03).

###### Contents

  1. 1 Preliminaries: trees and asymptotic structure in dual spaces
    1. 1.1 Trees in dual spaces
    2. 1.2 Asymptotic structure of a dual space
    3. 1.3 Tree reformulation for duals of separable spaces
  2. 2 Szlenk index and asymptotic structure
    1. 2.1 Szlenk index and bi-orthogonal systems indexed by trees
    2. 2.2 Weak∗ asymptotic structure and Szlenk index
    3. 2.3 Asymptotic structure and Szlenk index
  3. 3 Weak∗ asymptotic midpoint convexity in dual spaces
    1. 3.1 Properties AUC∗ and AMUC∗
    2. 3.2 Size of approximate midpoints in dual spaces
    3. 3.3 A few remarks on the equivalence AUC/AMUC up to renorming
  4. 4 Embeddability of countably branching bundle graphs into dual spaces
    1. 4.1 Good $\ell_{p}$-trees and weak∗ asymptotic structure
    2. 4.2 Bundle graphs and embeddability results
    3. 4.3 Bundle graphs and $c_{0}$-spreading models
  5. 5 Questions and comments
    1. 5.1 Szlenk index of WCG spaces is determined by separable quotients
    2. 5.2 About asymptotic midpoint convexity
    3. 5.3 About James tree-spaces

## Introduction

The main result in the present paper in the following.

######  Theorem A.

Let $X$ be a separable Banach space and let $G$ be a non-trivial countably
branching bundle graph. Let us assume that the Slzenk index $S_{Z}(X)$ is
strictly greater than the first infinite ordinal $\omega$ and that the dual
space $X^{*}$ has a weak∗ unconditional asymptotic structure. Then every non-
trivial countably branching bundle graph embeds Lipschitz in $X^{*}$ with
distortion less than $6+\varepsilon$.

Combined with previous results from [2] by F. Baudier and his co-authors this
yields the following new characterization of weak∗ asymptotic uniform
convexity up to renorming in dual spaces with a weak∗ unconditional asymptotic
structure.

######  Theorem B.

Let $X$ be a separable Banach space and let $G$ be a non-trivial countably
branching bundle graph. Let us assume that $X^{*}$ has a weak∗ unconditional
asymptotic structure. Then the following properties are equivalent.

  1. (1)

The space $X$ admits an equivalent norm whose dual norm is AUC∗

  2. (2)

The family $(G^{\otimes n})_{n\geq 1}$ of all countably branching bundle
graphs generated by $G$ does not equi-Lipschitz embed into $X^{*}$.

Those results fit in the asymptotic counterpart of the so called Ribe program
which intends to characterize asymptotic properties of Banach spaces by purely
metric ones. Interested readers are encouraged to look at the introduction in
[2] for a detailed background around this kind of results.

The proof of the main result consists in pushing forward methods from [2] in
order to give informations about the weak∗ asymptotic structure of dual spaces
satisfying the conditions stated in the theorem and then in appealing to known
embeddability results. More precisely we show that such a dual space must
contain $\ell_{\infty}^{n}$ in its $n^{\text{th}}$ weak∗ asymptotic structure
for every $n$ and this leads to the construction of a certain family of trees
(the so called good-$\ell_{\infty}$ trees of arbitrary height) which is known
by Swift’s work in [20] to allow for the embeddability of the graphs.

Taking a closer look at the proof it appears that one can forget the
unconditionality condition by focusing on duals of separable spaces for which
the unit ball contains sort of arbitrarily high multi-directional separated
weak∗-convergent trees. This is the following property $(*)$.

$\displaystyle(*):\ \ $ $\displaystyle\exists\varepsilon>0:\ \forall n\geq 1,\
\exists(x_{s}^{*})_{s\in T_{n}}\subset X^{*}\text{ weak${}^{*}$-null}:\ \
x^{*}_{\emptyset}\in B_{X^{*}},\ \forall s\neq\emptyset,\ \left\lVert
x_{s}^{*}\right\rVert\geq\varepsilon,$ $\displaystyle\text{and }\forall s\in
T_{n},\ \left\lvert s\right\rvert=n,\
\forall\varepsilon_{1},\dots,\varepsilon_{n}\in\\{-1,1\\},\
x^{*}_{\emptyset}+\sum_{i=1}^{n}\varepsilon_{i}x^{*}_{s\lvert i}\in
B_{X^{*}}.$

The result is then that the dual of a separable space with property $(*)$ must
contain $\ell_{\infty}^{n}$ in its $n^{\text{th}}$ weak∗-asymptotic structure
for all $n$ and thus is also a target space for the embeddability of families
of bundle graphs. This observation lead us to introduce and to study a weak∗
asymptotic midpoint property on dual spaces (property AMUC∗). Our hope was
that we would be able to develop a renorming theory to relate this property to
the preceding one and then obtain a nice metric characterisation of it in
terms of embeddability of families of bundle graphs in the class of duals of
separable spaces. But although it is straightforward to show that a dual with
property AMUC∗ satisfies property $\neg(*)$ we are not able yet to obtain a
renorming result that would give us the converse implication in full
generality. Still we can restrict ourselves to the context of duals with a
weak∗ unconditional asymptotic structure in which we can use the AUC∗
renorming theory to get the following linear result.

######  Theorem C.

Let $X$ be a separable Banach space and let us assume that $X^{*}$ has a weak∗
unconditional asymptotic structure. Then the following properties are
equivalent.

  1. (1)

$S_{Z}(X)\leq\omega$

  2. (2)

$X^{*}$ admits an equivalent AUC∗ norm

  3. (3)

$X^{*}$ admits an equivalent AMUC∗ norm

  4. (4)

$X^{*}$ satisfies property $\neg(*)$.

Actually an obstruction to a more general renorming result for AMUC∗ is the
lack of a dual property (a kind of AUS/AUC∗ duality) and the fact that we are
not able to define a suitable derivation in order to find a replacement for
the Szlenk index in this context.

Another goal of this study was to investigate the question of the equivalence
of property AUC and property AMUC up to renorming. A corollary of the
preceding theorem is that it holds for duals of separable spaces with a weak∗
unconditional asymptotic structure. The question of whether this structural
condition is essential or not remains widely open (even for reflexive spaces).
By the way let us mention that is is not even known whether a Banach space $X$
with property AMUC has weak Szlenk index $w-S_{Z}(X)<\infty$ or not (in other
word whether AMUC implies PCP or not) except again in the case it has an
unconditional asymptotic structure. However we were able to get some
information on the subject. We observed that properties AMUC and AMUC∗ are in
fact equivalent on dual spaces. This is due to the tight relation between
midpoint properties and the size of approximate midpoints (as stated in [6] in
the asymptotic context). Let us mention that this was already exploited by
Kalton in his paper [14] where he tried to answer the question of the
stability of property AUC under coarse-Lispchitz maps and that it was a also a
key tool to implement a self-improvement argument in [2] in order to show non-
embeddability of countably branching bundle graphs in AUC spaces. This
observation has the following consequence: the dual of the James tree space
$JT$ is AMUC∗. So since $JT$ admits no equivalent norm whose dual norm is AUC∗
the two properties are not equivalent up to renorming.

Let us discuss briefly the organisation of the paper. In section $1$ we
introduce all the asymptotic structural properties we will need in the paper
and we translate them in terms of properties of trees. In section $2$ we
recall the definition of the Szlenk index $S_{Z}(X)$ of a Banach space $X$ and
we show that under some separability and unconditional asymptotic structural
assumptions, a Banach space $X$ such that $S_{Z}(X)>\omega$ satisfies
$\ell_{1}^{n}\in\\{X\\}_{n}$ and $\ell_{\infty}^{n}\in\\{X^{*}\\}^{w^{*}}_{n}$
for every $n\geq 1$. In section $3$ we introduce property AMUC∗ and write down
all the preceding comments concerning its properties. In section $4$ we recall
the definition of bundle graphs and we prove the main result of the paper and
his corollaries. We also give a small result on the embeddability of such
graphs into Banach spaces with a $c_{0}$-spreading model. We conclude the
paper with a section $5$ devoted to questions and comments. In particular we
say a few words about the separability assumption in theorems $B$ and $C$ and
we show that it is possible to get rid of it if we work with WCG Banach spaces
by passing to a separable quotient.

##  1\. Preliminaries: trees and asymptotic structure in dual spaces

In this section we introduce several properties of dual spaces around the
concept of weak∗ asymptotic structure and we express them in a suitable way
for our later study in terms of behaviour of weak∗-null normalized trees.
Although we might refer to it in the sequel we won’t give the definitions of
(weak) asymptotic structure properties and the corresponding expressions in
terms of weak-null trees in general Banach spaces to avoid repetitions.
Basically they are written in the same way as their weak∗ analogue but using
the weak topology instead (that is mainly by replacing weak∗-null objects by
weak null objects and weak∗-closed subspaces by closed subspaces) and
replacing separability assumptions on the space by separability assumptions on
the dual when needed. We refer to [2] for precise statements.

###  1.1. Trees in dual spaces

Let us start by introducing the notion of trees in a dual space.

For all $N\geq 1$, let
$T_{N}=\\{\emptyset\\}\cup\bigcup_{n=1}^{N}\mathbb{N}^{n}$ be the countably
branching tree of height $N$ (without any metric consideration). Also let
$T_{\infty}=\bigcup_{N\geq 1}T_{N}$. We will use the following notations.

  1. (1)

For all $s=(s_{1},\dots,s_{n})\in T_{\infty}$ non empty, $\left\lvert
s\right\rvert:=n$ is the length of the sequence $s$, $s_{\lvert
k}:=(s_{1},\dots s_{k})$ for every $1\leq k\leq n$ and $s_{\lvert
0}:=\emptyset$. Also, $s^{-}:=s_{\lvert n-1}$.

  2. (2)

For all $s=(s_{1},\dots s_{n})$ and $t=(t_{1},\dots t_{m})$ in $T_{\infty}$,
$s\smallfrown t:=(s_{1},\dots s_{n},t_{1},\dots,t_{m})$.

A subtree $T$ of $T_{N}$ is a non empty subset of $T_{N}$ containing all the
predecessors of its elements. A full subtree of $T_{N}$ is a subtree $T$ of
$T_{N}$ such that all the sets $T\cap\\{t\smallfrown n,\ n\in\mathbb{N}\\}$
with $t\in T\cap T_{N-1}$ are infinite.

Let $T$ be a full subtree of $T_{N}$. For all $s,t\in T$, we write $s\leq t$
whenever the sequence $t$ is an extension of the sequence $s$ and we say that
$s$ is an ancestor of $t$. This defines an ordering of $T$. We say that an
enumeration $(\sigma_{i})_{i\geq 1}$ of $T$ is a compatible linear ordering of
$T$ if it satisfies the following condition: for every $i\geq 1$ and for every
ancestor $s$ of $\sigma_{i}$, there is a $j<i$ such that $s=\sigma_{j}$.

######  Remark 1.

For all $s=(s_{1},\dots,s_{n})\in T$, let $f(s)=\sum_{i=1}^{n}s_{i}$ and let
$f(\emptyset)=0$. If an enumeration $(\sigma_{i})_{i\geq 1}$ of $T$ satisfies
the condition: $f(\sigma_{i})\leq f(\sigma_{j})$ whenever $i\leq j$, then it
is a compatible linear ordering of $T$.

In the same way, if we replace $T$ by a full subtree of $[\mathbb{N}]^{\leq
N}=\\{s=(s_{1},\dots,s_{n})\in T_{N}:\ s_{1}<\dots<s_{n}\\}$ and if we replace
the function $f$ by the maximum in the preceding condition, we also get a
compatible linear ordering of $T$.

It can be easily checked that all the results involving the notion of
compatible linear ordering in this paper can be proved by using one of the two
conditions above instead and thus that they are compatible with definitions
and results either from [20] or [2].

Let $X$ be a Banach space and let $X^{*}$ be its dual space. A tree (of height
$N$) in $X^{*}$ is a subset of $X^{*}$ indexed by a full subtree $T$ of
$T_{N}$. Let $(x^{*}_{s})_{s\in T}$ be a tree in $X^{*}$. We will use the
following denominations.

  1. (1)

The root is the element $x^{*}_{\emptyset}$. An unrooted tree is a tree
deprived of its root.

  2. (2)

A node is a set of the form $\\{x^{*}_{t\smallfrown n},\ n\in\mathbb{N},\
t\smallfrown n\in T\\}$ with $t\in T\cap T_{N-1}$.

  3. (3)

A branch is a set of the form $\\{x^{*}_{\emptyset},x^{*}_{s_{\lvert
1}},\dots,x^{*}_{s}\\}$ where $s\in T$ is a sequence of maximal length. If we
are working with an unrooted tree, we omit $x^{*}_{\emptyset}$ in this
definition.

  4. (4)

A leaf is an element $x^{*}_{s}$ with $s\in T$ of maximal length.

We say that the tree is bounded (respectively normalized) if the corresponding
subset is bounded (respectively is contained in the unit sphere of $X^{*}$).
We say that it is weak∗ null ($w^{*}$-null) if every node forms a weak∗ null
sequence in $X^{*}$, that is to say if $w^{*}-\lim_{n}x^{*}_{t\smallfrown
n}=0$ for all $t\in T$.

###  1.2. Asymptotic structure of a dual space

Now let us talk about the weak∗ asymptotic structure of a dual space.

Let $X$ be a Banach space, let $X^{*}$ be its dual space and let $E$ be an
$n$-dimensional Banach space with a normalized basis $(e_{i})_{i=1}^{n}$ for
some $n\geq 1$. Let us denote by $\text{cof}^{*}(X^{*})$ the set of all weak∗
closed subspaces of $X^{*}$ of finite co-dimension. We say that $E$ is in the
$n^{\text{th}}$ weak∗ asymptotic structure of $X^{*}$ up to a constant $C\geq
1$ if we have the following property:

$\displaystyle\forall\varepsilon>0,\ \forall X_{1}\in\text{cof}^{*}($
$\displaystyle X^{*}),\ \exists x^{*}_{1}\in S_{X_{1}},\ \dots,\ \forall
X_{n}\in\text{cof}^{*}(X^{*}),\ \exists x^{*}_{n}\in S_{X_{n}}:$
$(x^{*}_{i})_{i=1}^{n}$ is $C(1+\varepsilon)$-equivalent to
$(e_{i})_{i=1}^{n}$.

If $C=1$, we say that $E$ is in the $n^{\text{th}}$ weak∗ asymptotic structure
of $X^{*}$ and we write $E\in\\{X^{*}\\}^{w^{*}}_{n}$.

We say that $X^{*}$ has a weak∗ unconditional asymptotic structure with
constant $C\geq 1$ if it satisfies:

$\displaystyle\exists X_{1}\in\text{cof}^{*}(X^{*}),\ \forall x^{*}_{1}$
$\displaystyle\in S_{X_{1}},\ \dots,\ \exists X_{n}\in\text{cof}^{*}(X^{*}),\
\forall x^{*}_{n}\in S_{X_{n}}:$ $(x^{*}_{i})_{i=1}^{n}$ is $C$-unconditional.

######  Remark 2.

This properties can be reformulated by using the langage of games between two
players. Here one player picks weak∗ closed subspaces of $X^{*}$ of finite co-
dimension and the other player picks normalized vectors in those subspaces.
Detailed explanitions and references are given in [2] paragraph 3.2.

###  1.3. Tree reformulation for duals of separable spaces

Finally, let us explain how the weak∗ asymptotic structure properties
introduced before can be translated in duals of separable spaces in terms of
properties of weak∗ null normalized trees. We believe that the following
results are well known but we give the proof for completeness.

######  Lemma 1.1.

Let $X$ be a separable Banach space. For all $n\geq 1$ and for all
$n$-dimensional space $E$ with a normalized basis $(e_{i})_{i=1}^{n}$, $E$ is
in the $n^{\text{th}}$ weak∗ asymptotic structure of $X^{*}$ up to a constant
$C\geq 1$ if and only if for all $\varepsilon>0$, there is a weak∗-null
normalized unrooted tree of height $n$ whose branches are all
$C(1+\varepsilon)$-equivalent to $(e_{i})_{i=1}^{n}$.

###### Proof.

Let us fix some $n\geq 1$ and some $n$-dimensional space $E$ with a normalized
basis $(e_{i})_{i=1}^{n}$.

First let us assume that $E$ is in the $n^{\text{th}}$ weak∗ asymptotic
structure of $X^{*}$ up to a constant $C\geq 1$. Since $X$ is separable, we
can choose a dense sequence $(z_{k})_{k\geq 1}$ in $S_{X}$. For all $k\geq 1$,
let us define the weak∗ closed subspace of finite codimension
$Z_{k}=\bigcap_{i=1}^{k}\ker z_{i}$ of $X^{*}$ where $z_{i}$ is identified
with the corresponding element in $X^{**}$. Then we have the property that any
normalized sequence $(z_{k}^{*})_{k\geq 1}$ in $S_{X^{*}}$ such that
$z_{k}^{*}\in Z_{k}$ for all $k\geq 1$ is weak∗-null. Let us fix
$\varepsilon>0$. Using the assumption on $E$, we will build a normalized
unrooted tree $(z_{s}^{*})_{s\in T_{n}\backslash\\{\emptyset\\}}$ in
$S_{X^{*}}$ such that:

  1. (1)

For all $s\in T_{n-1}$, for all $k\in\mathbb{N}$, $z_{s\smallfrown k}^{*}\in
Z_{k}$.

  2. (2)

All the branches of $(z_{s}^{*})_{s\in T_{n}\backslash\\{\emptyset\\}}$ are
$C(1+\varepsilon)$-equivalent to $(e_{i})_{i=1}^{n}$.

This gives the desired tree by the choice of the subspaces $Z_{k}$. To do this
construction, it is convenient to introduce a compatible linear ordering
$(\sigma_{i})_{i\geq 1}$ of $T_{n}\backslash\\{\emptyset\\}$. Then a
straightforward induction allows us to get a sequence
$(z_{\sigma_{i}}^{*})_{i\geq 1}$ in $S_{X^{*}}$ such that:

  1. (1)

If $\sigma_{i}=\sigma_{i}^{-}\smallfrown k$ for some $k\in\mathbb{N}$ then
$z_{\sigma_{i}}\in Z_{k}$.

  2. (2)

If $l=\left\lvert\sigma_{i}\right\rvert=n$, then $(z_{\sigma}^{*},\
\sigma\leq\sigma_{i})$ is $C(1+\varepsilon)$-equivalent to
$(e_{i})_{i=1}^{n}$.

  3. (3)

If $l=\left\lvert\sigma_{i}\right\rvert<n$ it satisfies:

$\displaystyle\forall X_{l+1}\in\text{cof}^{*}($ $\displaystyle X^{*}),\
\exists x^{*}_{l+1}\in S_{X_{l+1}},\ \dots,\ \forall
X_{n}\in\text{cof}^{*}(X^{*}),\ \exists x^{*}_{n}\in S_{X_{n}}:$
$\displaystyle(z_{\sigma}^{*},\ \sigma$
$\displaystyle\leq\sigma_{i})\smallfrown(x^{*}_{i})_{i=l+1}^{n}\text{ is
$C(1+\varepsilon)$-equivalent to $(e_{i})_{i=1}^{n}$. }$

Second, let us fix $\varepsilon>0$ and let us take $\delta>0$ such that: if
$(v_{i})_{i=1}^{n}$ and $(w_{i})_{i=1}^{n}$ are normalized sequences in
$X^{*}$, if $(v_{i})_{i=1}^{n}$ is $C(1+\delta)$-equivalent to
$(e_{i})_{i=1}^{n}$ and if $\left\lVert v_{i}-w_{i}\right\rVert\leq\delta$ for
all $i\leq n$, then $(w_{i})_{i=1}^{n}$ is $C(1+\varepsilon)$-equivalent to
$(e_{i})_{i=1}^{n}$. Then let us assume that there is a weak∗-null normalized
unrooted tree $(z_{s}^{*})_{s\in T_{n}\backslash\\{\emptyset\\}}$ in
$S_{X^{*}}$ whose branches are all $C(1+\delta)$-equivalent to
$(e_{i})_{i=1}^{n}$. Let us recall the following well known property: if $Z$
is a weak∗ closed subspace of finite codimension of $X^{*}$ and if
$(z_{k}^{*})_{k\geq 1}$ is normalized and weak∗-null, then the distance
$\text{dist}(z_{k}^{*},S_{Z})$ goes to $0$. Using this, a straightforward
induction shows that the following property holds for all $1\leq j\leq n$:

$\displaystyle\forall X_{1}\in\text{cof}(X^{*}),\ \exists x^{*}_{1}\in
S_{X_{1}},\ $ $\displaystyle\dots,\ \forall X_{j}\in\text{cof}(X^{*}),\
\exists x^{*}_{j}\in S_{X_{j}},\ \exists s\in T_{n},\ \left\lvert
s\right\rvert=j:$ $\displaystyle\forall 1\leq i\leq j,\ \left\lVert
x_{i}^{*}-z_{s_{\lvert i}}^{*}\right\rVert\leq\delta$

where the $X_{i}$ are weak∗ closed subspaces of finite co-dimension of $Y$. By
our choice of $\delta$ and by the properties of the branches of our tree, we
get the desired result.

∎

Let us recall two famous results in Banach space theory.

######  Theorem 1.2 (James, non-distordability of $c_{0}$).

For all $m\geq 1$, $C\geq 1$ and $\varepsilon>0$, there is an $n\geq 1$ such
that: any basic sequence $(e_{i})_{i=1}^{n}$ of length $n$ which is
$C$-equivalent to the unit vector basis of $\ell_{\infty}^{n}$ admits a block
basis of length $m$ which is $(1+\varepsilon)$-equivalent to the unit vector
basis of $\ell_{\infty}^{m}$.

######  Theorem 1.3 (Krivine/Rosenthal, non-distordability of $\ell_{p}$).

Let $1\leq p<\infty$. For all $m\geq 1$, $C\geq 1$ and $\varepsilon>0$, there
is an $n\geq 1$ such that: any basic sequence $(e_{i})_{i=1}^{n}$ of length
$n$ which is $C$-equivalent to the unit vector basis of $\ell_{p}^{n}$ admits
a block basis of length $m$ which is $(1+\varepsilon)$-equivalent to the unit
vector basis of $\ell_{p}^{m}$.

We will use these results and the preceding lemma to optimize constants in the
asymptotic structure.

######  Lemma 1.4.

Let $X$ be a separable Banach space and let $1\leq p\leq\infty$. If there is a
constant $C\geq 1$ such that the space $\ell_{p}^{n}$ is in the
$n^{\text{th}}$ weak∗ asymptotic structure of $X^{*}$ up to $C$ for all $n\geq
1$, then $\ell_{p}^{n}\in\\{X^{*}\\}^{w^{*}}_{n}$ for all $n\geq 1$.

###### Proof.

Let us assume that there is a constant $C>0$ such that for every $n\geq 1$,
$\ell_{p}^{n}$ is in the $n^{\text{th}}$ weak∗ asymptotic structure of $X^{*}$
up to $C$. First we will use a stabilisation argument to show that for every
$n\geq 1$ we can find an $n$-dimensional normed space $E_{n}$ which is $2C$
equivalent to $\ell_{p}^{n}$ and belongs to $\\{X^{*}\\}_{n}^{w^{*}}$.

Let us fix $n\geq 1$ and $\varepsilon\in(0,1)$. By compactness of the Banach-
Mazur distance on the set of $n$-dimensional normed spaces we can find a
finite collection $\mathcal{E}_{\varepsilon}$ of such spaces such that every
$n$-dimensional normed space which is $2C$ equivalent to $\ell_{p}^{n}$ is
actually $(1+\varepsilon)$ equivalent to some element of
$\mathcal{E}_{\varepsilon}$. Since $X$ is separable we can find by the
preceding lemma a weak∗ null normalized unrooted tree $(x_{s}^{*})_{s\in
T_{n}\backslash\\{\emptyset\\}}$ in $X^{*}$ whose branches are all $2C$
equivalent to the canonical basis of $\ell_{p}^{n}$. Thus for every branch
$\beta$ of the tree there is an element
$E_{\beta}\in\mathcal{E}_{\varepsilon}$ which is $(1+\varepsilon)$ equivalent
to the span of $\beta$. By Ramsey’s theorem we may assume up to the extraction
of a full subtree that there is an element
$E_{\varepsilon}\in\mathcal{E}_{\varepsilon}$ such that the span of every
branch $\beta$ of the tree is $(1+\varepsilon)$ equivalent to
$E_{\varepsilon}$. Finally letting $\varepsilon$ tends to $0$ and using
sequential compactness we can find a sequence $(\varepsilon_{i})_{i\geq 1}$ in
$(0,1)$ converging to $0$ and an $n$-dimensional normed space $E$ such that
the sequence $(E_{\varepsilon_{i}})_{i\geq 1}$ converges to $E$ in Banach-
Mazur distance. It is straightforward to check that $E_{n}=E$ has the two
desired properties.

Second we apply the non-distordability results to get the desired conclusion.
Fix $m\geq 1$ and $\varepsilon>0$. Then we can find an $n\geq 1$ such that
every normalized basic sequence of length $n$ which is $2C$ equivalent to the
canonical basis of $\ell_{p}^{n}$ admits a normalized block basis of length
$m$ which is $(1+\varepsilon)$ equivalent to the canonical basis of
$\ell_{p}^{m}$. Now use the first step to find an $n$-dimensional normed space
$E_{n}$ with a normalized basis $(e_{i})_{i=1}^{n}$ which is $2C$ equivalent
to $\ell_{p}^{n}$ and belongs to $\\{X^{*}\\}_{n}^{w^{*}}$ and use the
preceding statement to pick a normalized block basis $(f_{j})_{j=1}^{m}$ of
$(e_{i})_{i=1}^{n}$ which is $(1+\varepsilon)$ equivalent to the canonical
basis of $\ell_{p}^{m}$. Decomposing the $f_{j}$’s in the original basis and
following the idea that one can play the game with the $e_{i}$’s taking
successively the same subspace one can use a perturbation argument in order to
check that the following property holds:

$\displaystyle\forall X_{1}\in\text{cof}($ $\displaystyle X^{*}),\ \exists
x^{*}_{1}\in S_{X_{1}},\ \dots,\ \forall X_{n}\in\text{cof}(X^{*}),\ \exists
x^{*}_{m}\in S_{X_{n}}:$ $(x^{*}_{i})_{i=1}^{m}$ is
$(1+\varepsilon)$-equivalent to $(f_{i})_{i=1}^{m}$.

As a consequence all the sequences $(x^{*}_{i})_{i=1}^{m}$ are
$(1+\varepsilon)^{2}$ equivalent to the canonical basis of $\ell_{p}^{m}$ and
the conclusion follows by letting $\varepsilon$ go to $0$.

∎

Let $X$ be a Banach space and let $C\geq 1$. We say that $X^{*}$ has the
$C$-weak∗ unconditional finite tree property if any weak∗-null normalized
unrooted tree of finite height in $X^{*}$ has a branch which is
$C$-unconditional. Let us point out that in this case an application of
Ramsey’s combinatorial theorem yields that any weak∗-null normalized unrooted
tree of finite height in $X^{*}$ has a full subtree whose branches are all
$C$-unconditional. This property can be related to weak∗ asymptotic
unconditionally for duals of separable spaces.

######  Lemma 1.5.

Let $X$ be a separable Banach space. Then $X^{*}$ has a weak∗ unconditional
asymptotic structure if and only if it has the weak∗ unconditional finite tree
property.

###### Proof.

Clearly one can show using the same tools as in the proof of the preceding
lemma that:

  1. (1)

If $X^{*}$ does not have a $C$-weak∗ unconditional asymptotic structure then
there is a weak∗-null normalized unrooted tree of height $n$ in $X^{*}$ with
no $C$-unconditional branch.

  2. (2)

If $X^{*}$ has a $C$-weak∗ unconditional asymptotic structure then for all
$\varepsilon>0$, and for all weak∗-null normalized unrooted tree of height $n$
in $S_{X^{*}}$ there is a branch which is $(C+\varepsilon)$-unconditional.

∎

##  2\. Szlenk index and asymptotic structure

In this section we introduce a key object for our study, the Szlenk index of a
Banach space, and we investigate how this object affects the asymptotic
structure of Banach spaces and of their duals under some separability
assumptions and unconditionality conditions.

###  2.1. Szlenk index and bi-orthogonal systems indexed by trees

Let $X$ be a Banach space, let $K$ be a weak∗-compact subset of the dual space
$X^{*}$ and fix $\varepsilon>0$. Denote by $\mathcal{V}$ the set of all
weak∗-open subsets $V$ of $K$ satisfying $\text{diam}\ V\leq\varepsilon$ and
let $s_{\varepsilon}(K)$ be the weak∗ closed subset defined by
$s_{\varepsilon}(K)=K\backslash\left(\bigcup_{V\in\mathcal{V}}V\right)$. We
define inductively derivative subsets $s_{\varepsilon}^{\alpha}(K)$ of $K$ for
every ordinal $\alpha$ by $s_{\varepsilon}^{1}(K)=s_{\varepsilon}(K)$,
$s_{\varepsilon}^{\alpha+1}(K)=s_{\varepsilon}(s_{\varepsilon}^{\alpha}(K))$
if $\alpha\geq 1$ and
$s_{\varepsilon}^{\alpha}=\bigcap_{\beta<\alpha}s_{\varepsilon}^{\beta}$ if
$\alpha$ is a limit ordinal. Then we consider
$S_{Z}(K,\varepsilon)=\inf\left\\{\alpha\ :\
s_{\varepsilon}^{\alpha}(K)=\emptyset\right\\}$ if such an $\alpha$ exists and
$S_{Z}(K,\varepsilon)=\infty$ otherwise. Finally let
$S_{Z}(K)=\sup_{\varepsilon>0}S_{Z}(K,\varepsilon)$. The Szlenk index of $X$
is $S_{Z}(X)=S_{Z}\left(B_{X^{*}}\right)$ where $B_{X^{*}}$ denotes the closed
unit ball of $X^{*}$. We refer to the survey [18] for an extensive study of
the properties and applications of the Slzenk index.

In the separable case it is well known that the property $S_{Z}(X)>\omega$ can
be characterized by the existence in the unit ball of the dual space $X^{*}$
of arbitrarily high separated weak∗ null trees. This is the following lemma.

######  Lemma 2.1.

Let $X$ be a separable space. Then $S_{Z}(X)>\omega$ if and only if $X^{*}$
has the following property: there is an $\varepsilon>0$ such that for every
integer $n\geq 1$ there is a weak∗-null tree $(x_{s}^{*})_{s\in T_{n}}$ in
$X^{*}$ with $x_{\emptyset}^{*}\in B_{X^{*}}$ and $\left\lVert
x_{s}^{*}\right\rVert\geq\varepsilon$ for all $s\in T_{n}$ non-empty, such
that for every sequence $s\in T_{n}$ of maximal length
$x_{\emptyset}^{*}+\sum_{i=1}^{n}x_{s_{\lvert i}}^{*}$ is in $B_{X^{*}}$.

In [3], the authors used the preceding lemma to prove the following
proposition which we will exploit later.

######  Proposition 2.2 ([3], prop 2.2.).

Let $X$ be a separable Banach space with $S_{Z}(X)>\omega$. For all $n\geq 1$
and for all $\delta>0$, there exist a weak∗-null bounded tree
$(x_{s}^{*})_{s\in T_{n}}$ in $X^{*}$ and a normalized tree $(x_{s})_{s\in
T_{n}}$ in $X$ such that:

  1. (1)

For all $s\in T_{n}\backslash\\{\emptyset\\}$, $\left\lVert
x_{s}^{*}\right\rVert\geq 1$ and $\left\lVert\sum_{t\leq
s}x_{t}^{*}\right\rVert\leq 3$.

  2. (2)

For all $s\in T_{n}$, $x_{s}^{*}(x_{s})\geq\frac{1}{3}\left\lVert
x_{s}^{*}\right\rVert$.

  3. (3)

For all $s\neq t$ in $T_{n}$, $\left\lvert
x_{s}^{*}(x_{t})\right\rvert\leq\delta$.

The proof of this result rests on the following classical result from Mazur.

######  Lemma 2.3 (Mazur).

Let $(x_{k}^{*})_{k\geq 1}$ be a weak∗-null sequence in $X^{*}$ such that
$\left\lVert x_{k}^{*}\right\rVert\geq 1$ for all $k\geq 1$, and let $F$ be a
finite subset of $X^{*}$. Then there is a sequence $(x_{k})_{k\geq 1}$ in
$S_{X}$ such that $x_{k}\in\bigcap_{y^{*}\in F}\ker y^{*}$ for all $k\geq 1$
and $\liminf x^{*}_{k}(x_{k})\geq\frac{1}{2}$.

This proposition can be slightly improved when we assume that the dual space
is separable: one can ask the normalized trees in $S_{X}$ to be weakly-null.

######  Proposition 2.4.

Let $X$ be a Banach space with separable dual and with $S_{Z}(X)>\omega$. For
all $n\geq 1$ and for all $\delta$, there exist a weak∗-null bounded tree
$(x_{s}^{*})_{s\in T_{n}}$ in $X^{*}$ and a normalized weak-null tree
$(x_{s})_{s\in T_{n}}$ in $X$ such that:

  1. (1)

For all $s\in T_{n}\backslash\\{\emptyset\\}$, $\left\lVert
x_{s}^{*}\right\rVert\geq 1$ and $\left\lVert\sum_{t\leq
s}x_{t}^{*}\right\rVert\leq 3$.

  2. (2)

For all $s\in T_{n}$, $x_{s}^{*}(x_{s})\geq\frac{1}{3}\left\lVert
x_{s}^{*}\right\rVert$.

  3. (3)

For all $s\neq t$ in $T_{n}$, $\left\lvert
x_{s}^{*}(x_{t})\right\rvert\leq\delta$.

###### Proof.

Let us pick a dense sequence $(y_{k}^{*})_{n\geq 1}$ in $S_{X^{*}}$ and let us
define $Y_{k}=\bigcap_{i=1}^{k}\ker y_{i}^{*}$. Then any normalized sequence
$(x_{k})_{k\geq 1}$ in $X$ such that $x_{k}\in Y_{k}$ for all $k\geq 1$ is
weakly-null. Now let us fix $n\geq 1$ and $\delta>0$. Since $S_{Z}(X)>\omega$
and $X$ is separable we can find a weak∗-null bounded tree $(x_{s}^{*})_{s\in
T_{n}}$ in $X^{*}$ such that for all $s\in T_{n}\backslash\\{\emptyset\\}$,
$\left\lVert x_{s}^{*}\right\rVert\geq 1$ and $\left\lVert\sum_{t\leq
s}x_{t}^{*}\right\rVert\leq 3$. Let us pick a compatible linear ordering
$(\sigma_{i})_{i\geq 1}$ of $T_{N}$ and let us assume as we may that this
ordering also satisfies the following property: if
$\sigma_{i}=\sigma_{i_{0}}\smallfrown m$ and
$\sigma_{j}=\sigma_{i_{0}}\smallfrown m^{\prime}$ with $m<m^{\prime}$ then
$i<j$. By induction, we can build a sequence $(\theta_{i})_{i\geq 1}$ in
$T_{n}$ and a sequence $(x_{\theta_{i}})_{i\geq 1}$ in $S_{X}$ such that:

  1. (1)

$\theta_{1}=\emptyset$

  2. (2)

If $\sigma_{i}^{-}=\sigma_{i_{0}}$ then $\theta_{i}=\theta_{i_{0}}\smallfrown
m^{\prime}$ for some $m^{\prime}\in\mathbb{N}$ bigger than $\max\\{l:\ \exists
j<i:\ \theta_{j}=\theta_{i_{0}}\smallfrown l\\}$ and $x_{\theta_{i}}\in
Y_{m}$.

  3. (3)

For all $i<j$, $\left\lvert
x_{\theta_{j}}^{*}(x_{\theta_{i}})\right\rvert\leq\delta$ and
$x_{\theta_{i}}^{*}(x_{\theta_{j}})=0$.

  4. (4)

For all $i\geq 1$,
$x_{\theta_{i}}^{*}(x_{\theta_{i}})\geq\frac{1}{3}\left\lVert
x_{\theta_{i}}^{*}\right\rVert$.

For $i=1$, just let $\theta_{1}=\emptyset$ and pick any $x_{\emptyset}$ in
$S_{X}$ such that $x_{\emptyset}^{*}(x_{\emptyset})\geq\frac{1}{3}\left\lVert
x_{\emptyset}^{*}\right\rVert$. Now let us assume that
$\theta_{1},\dots\theta_{i}$ and $x_{\theta_{1}},\dots,x_{\theta_{i}}$ have
been chosen with the required properties. Since we are working with a
compatible linear ordering, there is some $i_{0}\leq i$ and such that
$\sigma_{i+1}^{-}=\sigma_{i_{0}}$. Since $(x_{\theta_{i_{0}}\smallfrown
k}^{*})_{k\geq 1}$ is weak∗-null there is a $K\in\mathbb{N}$ such that
$\left\lvert x_{\theta_{i_{0}}\smallfrown
k}^{*}(x_{\theta_{j}})\right\rvert\leq\delta$ for all $j\leq i$ and for all
$k\geq K$. Moreover, we can apply Mazur’s lemma to this sequence with
$F=\\{y_{1}^{*},\dots,y_{m}^{*},x_{\theta_{1}}^{*},\dots,x_{\theta_{i}}^{*}\\}$
and take any $m^{\prime}$ big enough in order to get a element
$x_{\theta_{i_{0}}\smallfrown m^{\prime}}\in S_{Y_{m}}$ such that
$x_{\theta_{i_{0}}\smallfrown m^{\prime}}^{*}(x_{\theta_{i_{0}}\smallfrown
m^{\prime}})\geq\frac{1}{3}\left\lVert x_{\theta_{i_{0}}\smallfrown
m^{\prime}}^{*}\right\rVert$ and $x_{j}^{*}(x_{\theta_{i_{0}}\smallfrown
m^{\prime}})=0$ for all $j\leq i$ and also $m^{\prime}$ bigger than $K$ and
bigger than $\max\\{l:\ \exists j<i:\ \theta_{j}=\theta_{i_{0}}\smallfrown
l\\}$. We conclude by putting $\theta_{i+1}=\theta_{i_{0}}\smallfrown
m^{\prime}$. Then $T=\\{\theta_{i},\ i\geq 1\\}$ defines a full subtree of
$T_{n}$ and for every node $(x_{\theta_{i_{0}}\smallfrown
m^{\prime}_{k}})_{k\geq 1}$ of $T$ there is a strictly increasing sequence
$(m_{k})_{k\geq 1}$ in $\mathbb{N}$ such that $x_{\theta_{i_{0}}\smallfrown
m^{\prime}_{k}}\in S_{Y_{m_{k}}}$. Consequently, $(x_{s})_{s\in T}$ is a
weakly-null tree in $X$ and satisfies all the required properties together
with $(x_{s}^{*})_{s\in T}$.

∎

###  2.2. Weak∗ asymptotic structure and Szlenk index

Using the almost bi-orthogonal system from [3], one can prove the following
result, which is an extension of the reflexive version proved in [2].

######  Theorem 2.5.

Let $X^{*}$ be the dual of a separable Banach space $X$ with
$S_{Z}(X)>\omega$, and assume that $X^{*}$ has a weak∗ unconditional
asymptotic structure. Then $\ell_{\infty}^{n}$ is in the $n^{\text{th}}$ weak∗
asymptotic structure of $X^{*}$ for every $n\geq 1$.

###### Proof.

By the results of the first section there is a constant $C\geq 1$ such that
any weak∗-null normalized tree of finite height in $X^{*}$ has a full subtree
whose branches are all $C$ unconditional. Moreover it is sufficient to show
that there is a constant $D\geq 1$ such that for all $n\geq 1$, there is a
weak∗ null normalized tree of height $n$ in $Y$ whose branches are all
$D$-equivalent to the unit vector basis of $\ell_{\infty}^{n}$. Let us fix
some $n\geq 1$ and let us fix $\delta>0$ to be chosen later. By proposition
2.2, there exist a weak∗-null bounded tree $(x_{s}^{*})_{s\in T_{n}}$ in
$X^{*}$ and a normalized tree $(x_{s})_{s\in T_{n}}$ in $X$ such that:

  1. (1)

For all $s\in T_{n}\backslash\\{\emptyset\\}$, $\left\lVert
x_{s}^{*}\right\rVert\geq 1$ and $\left\lVert\sum_{t\leq
s}x_{t}^{*}\right\rVert\leq 3$.

  2. (2)

For all $s\in T_{n}$, $x_{s}^{*}(x_{s})\geq\frac{1}{3}\left\lVert
x_{s}^{*}\right\rVert$.

  3. (3)

For all $s\neq t$ in $T_{n}$, $\left\lvert
x_{s}^{*}(x_{t})\right\rvert\leq\delta$.

As mentioned one can assume, up to passing to some full subtree, that all
banches of $(x_{s}^{*})_{s\in T_{n}}$ are $C$-unconditional. For all $s\in
T_{n}$, let $y_{s}=\frac{x_{s}^{*}}{\left\lVert x_{s}^{*}\right\rVert}$. Since
the function $(a_{0},\dots,a_{\left\lvert s\right\rvert})\in[-1,\
1]^{\left\lvert s\right\rvert}\mapsto\left\lVert\sum_{t\leq
s}a_{t}x_{t}^{*}\right\rVert$ is continuous and convex we have:

$\displaystyle\max\left\\{\left\lVert\sum_{t\leq s}a_{t}y_{t}\right\rVert,\
a_{t}\in[-1,\ 1]\right\\}$
$\displaystyle\leq\max\left\\{\left\lVert\sum_{t\leq
s}a_{t}x_{t}^{*}\right\rVert,\ a_{t}\in[-1,\ 1]\right\\}$
$\displaystyle=\max\left\\{\left\lVert\sum_{t\leq
s}a_{t}x_{t}^{*}\right\rVert,\ a_{t}\in\\{-1,\ 1\\}\right\\}$
$\displaystyle\leq C\left\lVert\sum_{t\leq s}x_{t}^{*}\right\rVert$
$\displaystyle\leq 3C.$

Thus, for all $s\in T_{n}$ with $\left\lvert s\right\rvert=n$ and for all
$a_{0},\dots,a_{n}\in\mathbb{R}$, we have:

$\left\lVert\sum_{t\leq s}a_{\left\lvert t\right\rvert}y_{t}\right\rVert\leq
3C\max_{0\leq i\leq n}\left\lvert a_{i}\right\rvert.$

Now if $j$ is such that $\left\lvert a_{j}\right\rvert=\max_{0\leq i\leq
n}\left\lvert a_{i}\right\rvert$, we have, using the properties of the almost
bi-orthogonal system:

$\displaystyle\left\lVert\sum_{t\leq s}a_{\left\lvert
t\right\rvert}y_{t}\right\rVert$
$\displaystyle\geq\left\langle{\text{sign}\left(a_{j}\right)x_{s_{\lvert
j}}},\ {\sum_{t\leq s}a_{\left\lvert t\right\rvert}y_{t}}\right\rangle$
$\displaystyle\geq\frac{1}{3}\left\lvert a_{j}\right\rvert-\delta\sum_{i\neq
j}\left\lvert a_{i}\right\rvert$
$\displaystyle\geq\left(\frac{1}{3}-n\delta\right)\max_{0\leq i\leq
n}\left\lvert a_{i}\right\rvert.$

Thus if $\delta$ was chosen say smaller than $\frac{1}{6n}$, the tree
$(y_{s})_{s\in T_{n}}$ satisfies the required conditions (with constant
$D=18C$).

∎

To conclude this paragraph let us make a brief comment about the property
$(*)$ mentioned in the introduction. Let us first recall the definition of
this property:

$\displaystyle(*):\ \ $ $\displaystyle\exists\varepsilon>0:\ \forall n\geq 1,\
\exists(x_{s}^{*})_{s\in T_{n}}\subset X^{*}\text{ weak${}^{*}$-null}:\
x^{*}_{\emptyset}\in B_{X^{*}},\ \left\lVert
x_{s}^{*}\right\rVert\geq\varepsilon\ \forall s\neq\emptyset,\ $
$\displaystyle\forall s\in T_{n},\ \left\lvert s\right\rvert=n,\
\forall\varepsilon_{1},\dots,\varepsilon_{n}\in\\{-1,1\\},\
x^{*}_{\emptyset}+\sum_{i=1}^{n}\varepsilon_{i}x^{*}_{s\lvert i}\in
B_{X^{*}}.$

By the expression of the Szlenk index in terms of containment of trees, is is
clear that if $X$ is a separable space, then $(*)$ $\implies$
$S_{Z}(X)>\omega$. Moreover one easily sees that the converse is true if one
adds the condition that $X^{*}$ has a weak∗unconditional asymptotic structure.
In fact it seems that property $(*)$ is the right property to make the
preceding proof work so we have the following result.

######  Theorem 2.6.

Let $X$ be a separable Banach space and let us assume that $X^{*}$ has
property $(*)$. Then $\ell_{\infty}^{n}$ is in the $n^{\text{th}}$ weak∗
asymptotic structure of $X^{*}$ for every $n\geq 1$.

This property $(*)$ remains quite mysterious and although having a formally
stronger result we don’t known wether a dual space satisfying this property
has a weak∗ unconditional asymptotic structure or not. We will see some more
things related to it in the following section.

###  2.3. Asymptotic structure and Szlenk index

To conclude this section we show how the small improvement of the result of
[3] from the first paragraph concerning almost bi-orthogonal systems allows to
prove the pendant of the structural result of the preceding paragraph in the
space $X$ with an $\ell_{1}$ flavour.

######  Theorem 2.7.

Let $X$ be a Banach space with a separable dual, with an unconditional
asymptotic structure and with $S_{Z}(X)>\omega$. Then $\ell_{1}^{n}$ is in the
$n^{\text{th}}$ asymptotic structure of $X$ for every $n\geq 1$.

###### Proof.

By the weak version from [2] of the results of the first section there is a
constant $C\geq 1$ such that any weak-null normalized tree of finite height in
$X$ has a full subtree whose branches are all $C$ unconditional. Moreover it
is sufficient to show that there is a constant $D\geq 1$ such that for all
$n\geq 1$, there is a weak-null normalized tree of height $n$ in $X$ whose
branches are all $D$-equivalent to the unit vector basis of $\ell_{1}^{n}$.
Let us fix some $n\geq 1$ and let us fix $\delta>0$ to be chosen latter. By
proposition 2.4, there is a weak∗-null bounded tree $(x_{s}^{*})_{s\in T_{n}}$
in $X^{*}$ and a normalized weak-null tree $(x_{s})_{s\in T_{n}}$ in $X$ such
that:

  1. (1)

For all $s\in T_{n}\backslash\\{\emptyset\\}$, $\left\lVert
x_{s}^{*}\right\rVert\geq 1$ and $\left\lVert\sum_{t\leq
s}x_{t}^{*}\right\rVert\leq 3$.

  2. (2)

For all $s\in T_{n}$, $x_{s}^{*}(x_{s})\geq\frac{1}{3}\left\lVert
x_{s}^{*}\right\rVert$.

  3. (3)

For all $s\neq t$ in $T_{n}$, $\left\lvert
x_{s}^{*}(x_{t})\right\rvert\leq\delta$.

As mentioned one can assume, up to passing to some full subtree, that all
branches of $(x_{s})_{s\in T_{n}}$ are $C$-unconditional. Since $(x_{s})_{s\in
T_{N}}$ is normalized, a simple triangular inequality yields:

$\left\lVert\sum_{t\leq s}a_{\left\lvert
t\right\rvert}x_{t}\right\rVert\leq\sum_{i=0}^{n}\left\lvert
a_{i}\right\rvert$

for all $a_{0},\dots,a_{n}\in\mathbb{R}$ and for all $s\in T_{n}$ of length
$n$.

Moreover,

$\displaystyle\left\lVert\sum_{t\leq s}a_{\left\lvert
t\right\rvert}x_{t}\right\rVert$
$\displaystyle\geq\frac{1}{C}\left\lVert\sum_{t\leq
s}\text{sign}\left(a_{\left\lvert t\right\rvert}\right)a_{\left\lvert
t\right\rvert}x_{t}\right\rVert$
$\displaystyle\geq\frac{1}{3C}\left\langle{\sum_{t\leq s}\left\lvert
a_{\left\lvert t\right\rvert}\right\rvert x_{t}},\ {\sum_{t\leq
s}x_{t}^{*}}\right\rangle$
$\displaystyle\geq\left(\frac{1}{9C}-n\delta\right)\sum_{i=0}^{n}\left\lvert
a_{i}\right\rvert$ $\displaystyle\geq\frac{1}{D}\sum_{i=0}^{n}\left\lvert
a_{i}\right\rvert$

for any chosen constant $D>9C$ if $\delta$ was chosen small enough.

∎

##  3\. Weak∗ asymptotic midpoint convexity in dual spaces

In this section we introduce a weak∗ midpoint asymptotic convexity property
(property AMUC∗) and begin a systematic study of it in comparison with the
well studied property AUC∗.

###  3.1. Properties AUC∗ and AMUC∗

Let $X$ be a Banach space and let $X^{*}$ be its dual space. For all $t>0$ and
for all $x^{*}\in S_{X^{*}}$, let

$\bar{\delta}^{*}_{X}\left(t,x^{*}\right)=\sup_{Y\in\text{cof}^{*}(X^{*})}\inf_{y^{*}\in
S_{Y}}(\left\lVert x^{*}+ty^{*}\right\rVert-1)$

and let

$\widehat{\delta}^{*}_{X}\left(t,x^{*}\right)=\sup_{Y\in\text{cof}^{*}(X^{*})}\inf_{y^{*}\in
S_{Y}}(\max\\{\left\lVert x^{*}+ty^{*}\right\rVert,\left\lVert
x^{*}-ty^{*}\right\rVert\\}-1).$

Then let $\bar{\delta}^{*}_{X}\left(t\right)=\inf_{x^{*}\in
S_{X^{*}}}\bar{\delta}^{*}_{X}\left(t,x^{*}\right)$ and let
$\widehat{\delta}_{X}\left(t\right)=\inf_{x^{*}\in
S_{X^{*}}}\widehat{\delta}^{*}_{X}\left(t,x^{*}\right)$. We say that $X^{*}$
is AUC∗ (weak∗ asymptoticaly uniformly convex) if
$\bar{\delta}^{*}_{X}\left(t\right)>0$ for all $t>0$ and we say that $X^{*}$
is AMUC∗ (weak∗ asymptotically midpoint uniformly convex) if
$\widehat{\delta}^{*}_{X}\left(t\right)>0$ for all $t>0$. Note that we clearly
have $\widehat{\delta}^{*}\geq\bar{\delta}^{*}$ so that any AUC∗ norm is
AMUC∗. The notation comes from [12] for the first modulus and the second is a
weak∗ version of the one introduced in [6]. Again we do not give the weak
version of those moduli and the corresponding results to avoid repetitions.

######  Remark 3.

By a straightforward convexity argument, one can show that

$\bar{\delta}^{*}_{X}\left(t,x^{*}\right)=\sup_{Y\in\text{cof}^{*}(X^{*})}\inf_{y^{*}\in
Y,\ \left\lVert y^{*}\right\rVert\geq 1}(\left\lVert
x^{*}+ty^{*}\right\rVert-1)$

and

$\widehat{\delta}^{*}_{X}\left(t,x^{*}\right)=\sup_{Y\in\text{cof}^{*}(X^{*})}\inf_{y^{*}\in
Y,\ \left\lVert y^{*}\right\rVert\geq 1}(\max\\{\left\lVert
x^{*}+ty^{*}\right\rVert,\left\lVert x^{*}-ty^{*}\right\rVert\\}-1).$

The following result is well known for AUC∗.

######  Lemma 3.1.

Let $X^{*}$ be a dual space with property AUC∗. Then for all $t>0$, there
exists $\delta>0$ such that for all $x^{*}\in S_{X^{*}}$ and for every
weak∗-null sequence $(x^{*}_{k})_{k\geq 1}$ with $\left\lVert
x^{*}_{k}\right\rVert=t$, we have $\limsup\left\lVert
x^{*}+x^{*}_{k}\right\rVert>1+\delta.$

The proof rests on the well known fact that if Y is a weak∗ closed subspace of
finite codimension of $X^{*}$ and if $(x_{k}^{*})_{k\geq 1}$ is a bounded
weak∗-null sequence in $X^{*}$ then the distance $\text{dist}(x_{k}^{*},Y)$
goes to 0. In the same way one can prove the following result.

######  Lemma 3.2.

Let $X^{*}$ be a dual space with property AMUC∗. Then for all $t>0$, there
exists $\delta>0$ such that for all $x^{*}\in S_{X^{*}}$ and for all
weak∗-null sequence $(x^{*}_{k})_{k\geq 1}$ with $\left\lVert
x^{*}_{k}\right\rVert=t$, we have $\limsup\max\\{\left\lVert
x^{*}+x^{*}_{k}\right\rVert,\left\lVert
x^{*}-x^{*}_{k}\right\rVert\\}>1+\delta.$

######  Remark 4.

As before, one can put $\left\lVert x^{*}_{k}\right\rVert\geq t$ in the two
preceding results and we can simply take
$\delta=\bar{\delta}^{*}_{X}\left(t\right)$ respectively
$\delta=\widehat{\delta}^{*}_{X}\left(t\right)$. Moreover the converse of the
implication holds in both cases under the assumption that $X$ is separable and
the moduli are then the best possible $\delta$’s.

It is also well known that property AUC∗ is equivalent to the so called UKK∗
(weak∗ uniform Kadec Klee) property in the form introduced in [17]. This is
the following result.

######  Lemma 3.3.

A dual space $X^{*}$ is AUC∗ if and only if it satisfies the following
property: for all $\varepsilon>0$ there is a $\Delta>0$ such that for all
$x^{*}\in B_{X^{*}}$, if $\left\lVert x\right\rVert>1-\Delta$ then there is a
weak∗-neighborhood of $x^{*}$ such that $\text{diam}{(V\cap
B_{X^{*}})}\leq\varepsilon$.

A direct consequence of this result is that if the dual of a Banach space $X$
is AUC∗ then the space $X$ has Slzenk index $S_{Z}(X)\leq\omega$. In fact
there is a huge renorming theory around property AUC∗ and this is resumed in
the following theorem.

######  Theorem 3.4.

Let $X$ be a Banach space. The following assertions are equivalent.

  1. (1)

The space $X$ satisfies $S_{Z}(X)\leq\omega$.

  2. (2)

The space $X$ admits an equivalent norm whose dual norm is AUC∗.

  3. (3)

The space $X$ admits an equivalent norm whose dual norm is $p$-AUC∗ for some
$1\leq p<\infty$ that is to say there is a constant $c>0$ such that for all
$t>0$ we have $\bar{\delta}^{*}_{X}\left(t\right)\geq ct^{p}$.

This result has a long history and we refer to [18] or [12] for more
informations and references. It was conjectured by R. Huff in [11] and then
first proved in [16] by H. Knaust, E. Odell and Th. Schlumprecht in the
separable case with the terminology UKK∗.

We have a corresponding result to the Lemma 3.3 above for property AMUC∗ with
a sort of symmetric UKK∗ property.

######  Lemma 3.5.

A dual space $X^{*}$ is AMUC∗ if and only if it satisfies the following
property: for all $\varepsilon>0$ there is a $\Delta>0$ such that for all
$x^{*}\in B_{X^{*}}$, if $\left\lVert x^{*}\right\rVert>1-\Delta$ then there
is a weak∗-neighbourhood $V$ of $x^{*}$ such that for all $y^{*}\in X^{*}$ of
norm $\left\lVert y^{*}\right\rVert\geq\varepsilon$ we have $x^{*}+y^{*}\notin
V\cap B_{X^{*}}*$ or $x^{*}-y^{*}\notin V\cap B_{X^{*}}*$.

###### Proof.

Before starting the proof let us mention the following facts which are key to
pass from weak∗ closed subspaces of finite codimension to weak∗ neighbourhoods
of $0$ in a dual space and vice versa.

  1. (1)

Every weak∗ neighbourhood of $0$ in $X^{*}$ contains a weak∗ closed subspace
of finite codimension of $X^{*}$.

  2. (2)

Every weak∗ closed subspace of finite codimension of $X^{*}$ can be seen as a
sort of limit of weak∗ neighbourhoods of $0$. More precisely we can find weak∗
neighbourhoods of $0$ whose elements are all at the same arbitrarily small
distance from the subspace.

This essentially comes from the fact that every weak∗ closed subspacs of
finite codimension of a dual can be written as the finite intersection of
kernels of elements of the predual.

Now let us prove the proposition. First let us assume that $X^{*}$ is not
AMUC∗. Using the first fact above we immediately get that there is a $t>0$
such that for every $\delta>0$ we can find some $x^{*}=x^{*}_{\delta}\in
S_{X^{*}}$ satisfying:

$\forall W\in\mathcal{V}_{w^{*}}(0),\ \exists w^{*}\in W,\ \left\lVert
w^{*}\right\rVert\geq t:\ \left\lVert x^{*}\pm w^{*}\right\rVert<1+\delta.$

Let $\varepsilon=t$, fix $\Delta>0$, pick $\delta>0$ small enough for our
later purpose and take $x^{*}=x^{*}_{\delta}\in S_{X^{*}}$ as above. Then let
$u^{*}=\frac{1}{1+\delta}x^{*}$ and fix some weak∗ neighbourhood $V$ of
$u^{*}$. Take $W\in\mathcal{V}_{w^{*}}(0)$ such that $V=u^{*}+W$ and apply the
preceding property to find some $w^{*}$ in $W$ of norm $\left\lVert
w^{*}\right\rVert\geq t$ such that $\left\lVert x^{*}\pm
w^{*}\right\rVert<1+\delta$. Finally let $v^{*}=\frac{1}{1+\delta}w^{*}$. Then
$u^{*}+v^{*}\in V\cap B_{X^{*}}$ and $u^{*}-v^{*}\in V\cap B_{X^{*}}$.
Moreover if $\delta$ is small enough then we have $\left\lVert
u^{*}\right\rVert=\frac{1}{1+\delta}>1-\Delta$ and $\left\lVert
v^{*}\right\rVert\geq\frac{\varepsilon}{1+\delta}\geq\frac{\varepsilon}{2}$.
Thus $X^{*}$ does not satisfy the symmetric UKK∗ property.

Second let us assume that there is an $\varepsilon>0$ such that for every
$\Delta>0$ there is some $x^{*}=x^{*}_{\Delta}$ of norm $\left\lVert
x^{*}\right\rVert>1-\Delta$ satisfying:

$\forall V\in\mathcal{V}_{w^{*}}(x^{*}):\ \exists y^{*}\in X^{*},\ \left\lVert
y^{*}\right\rVert\geq\varepsilon,:\ x^{*}+y^{*}\in V\cap B_{X^{*}}\text{ and
}x^{*}-y^{*}\in V\cap B_{X^{*}}.$

Let $t=\varepsilon$, fix $\delta>0$, pick $\Delta>0$ small enough for our
later purpose and take $x^{*}=x^{*}_{\Delta}$ as above. Then take
$Z\in\text{cof}^{*}(X^{*})$ and pick $\mu>0$ small enough for our later
purpose. Using the second fact above we can choose a weak∗ neighbourhood of
$0$ whose elements are all at distance less than $\mu$ from $Z$. Translating,
applying the preceding property, and finally projecting onto $Z$ yields a
$z^{*}\in Z$ of norm $\left\lVert z^{*}\right\rVert\geq t-\mu$ such that
$\left\lVert x^{*}\pm z^{*}\right\rVert\leq 1+\mu$. So if $\Delta$ and $\mu$
are small enough we have $\left\lVert\frac{x^{*}}{\left\lVert
x^{*}\right\rVert}\pm\frac{z^{*}}{\left\lVert
x^{*}\right\rVert}\right\rVert\leq\frac{1+\mu}{1-\Delta}<1+\delta$ and
$\left\lVert\frac{z^{*}}{\left\lVert
x^{*}\right\rVert}\right\rVert\geq\frac{t-\mu}{1-\Delta}\geq\frac{t}{2}$. Thus
$X^{*}$ is not AMUC∗.

∎

Again there is a sequential version of this result in the separable case.

######  Lemma 3.6.

Let $X$ be a separable Banach space. Then $X^{*}$ is AMUC∗ if and only it
satisfies the following property: for all $\varepsilon>0$, there exists
$\delta\in(0,1)$ such that for all $x^{*}\in X^{*}$, if $\left\lVert
x^{*}\right\rVert>1-\delta$ then for all weak∗-null sequence
$(x_{k}^{*})_{k\geq 1}$ in $X^{*}$ such that $\left\lVert
x_{k}^{*}\right\rVert\geq\varepsilon$ for all $k\geq 1$ there is a $k_{0}\geq
1$ such that $\left\lVert x^{*}+x_{k_{0}}^{*}\right\rVert>1$ or $\left\lVert
x^{*}-x_{k_{0}}^{*}\right\rVert>1$.

As we already mentioned in the preceding section the Szlenk index has a tight
relation with some tree behaviour in the dual of the considered space. In
particular, in a dual space $X^{*}$ with property AUC∗, all the weak∗-null
separated trees of a large enough height must get out of the unit ball. We can
show the same kind of results for AMUC∗.

######  Proposition 3.7.

Let $X^{*}$ be AMUC∗. Then $X^{*}$ has property $\neg(*)$ that is to say: for
all $\varepsilon>0$ there is an integer $N\geq 1$ such that: for all $n\geq N$
and for all weak∗-null tree $(x_{s}^{*})_{s\in T_{n}}$ in $X^{*}$ such that
$x_{\emptyset}^{*}\in B_{X}^{*}$ and $\left\lVert
x_{s}^{*}\right\rVert\geq\varepsilon$ for all $s\in T_{n}$ non-empty, there is
a sequence $s\in T_{n}$ of maximal length and a sequence of signs
$(\varepsilon_{i})_{i=1}^{n}$ in $\\{-1,1\\}$ such that $\left\lVert
x_{\emptyset}^{*}+\sum_{i=1}^{n}\varepsilon_{i}x_{s_{\lvert
i}}^{*}\right\rVert>1$.

###### Proof.

Let us assume that $X^{*}$ fails this property. Then there exists
$\varepsilon>0$ and infinitely many $n$’s for which there exists a weak∗-null
tree $(x^{*}_{s})_{s\in T_{n}}$ in $X^{*}$ such that $\left\lVert
x^{*}_{s}\right\rVert\geq\varepsilon$ for all $s\in T_{n}$ non-empty and
$\left\lVert x^{*}_{\emptyset}+\sum_{i=1}^{n}\varepsilon_{i}x^{*}_{s_{\lvert
i}}\right\rVert\leq 1$ for all $s$ in $T_{n}$ of maximal length and for all
choices of signs. Fix such an $n$ big enough for our later purpose and fix
such a weak∗ null tree. Note that by weak∗ lower continuity of the norm, one
has in fact $\left\lVert
x^{*}_{\emptyset}+\sum_{i=1}^{k}\varepsilon_{i}x^{*}_{s_{\lvert
i}}\right\rVert\leq 1$ for all $1\leq k\leq n$, $s$ of length $k$ and choices
of signs. Now let us assume that $X^{*}$ is AMUC∗. Then there is a
$\delta=\delta(\varepsilon)>0$ such that for all $x\in S_{X^{*}}$ and for all
$(x^{*}_{k})_{k\geq 1}$ weak∗ null with $\left\lVert
x^{*}_{k}\right\rVert\geq\varepsilon$ one has $\limsup\max\\{\left\lVert
x^{*}+x^{*}_{k}\right\rVert,\left\lVert
x^{*}-x^{*}_{k}\right\rVert\\}>1+\delta.$ In particular, for all $x^{*}\neq
0\in B_{X}$ and for all $(x^{*}_{k})_{k\geq 1}$ weak∗-null with $\left\lVert
x^{*}_{k}\right\rVert\geq\varepsilon$ one has $\limsup\max\\{\left\lVert
x^{*}+x^{*}_{k}\right\rVert,\left\lVert
x^{*}-x^{*}_{k}\right\rVert\\}>\left\lVert x^{*}\right\rVert(1+\delta).$ First
assume that $\left\lVert x^{*}_{\emptyset}\right\rVert\geq\frac{1}{2}$.
Applying successively this property in our weak∗-null tree we can get a
sequence $s\in T_{n}$ of maximal length and a sequence of signs
$(\varepsilon_{i})_{i=1}^{n}$ in $\\{-1,1\\}$ such that $\left\lVert
x^{*}_{\emptyset}+\sum_{i=1}^{n}\varepsilon_{i}x^{*}_{s_{\lvert
i}}\right\rVert>\left\lVert
x^{*}_{\emptyset}\right\rVert(1+\delta)^{n}\geq\frac{1}{2}(1+\delta)^{n}$.
Second if $\left\lVert x^{*}_{\emptyset}\right\rVert<\frac{1}{2}$ then one can
apply the preceding result with $\tilde{x^{*}_{\emptyset}}=-(1-\left\lVert
x^{*}_{\emptyset}\right\rVert)\frac{x^{*}_{\emptyset}}{\left\lVert
x^{*}_{\emptyset}\right\rVert}$ and get a sequence $s\in T_{N}$ and a sequence
of signs such that

$\left\lVert x^{*}_{\emptyset}+\sum_{i=1}^{n}\varepsilon_{i}x^{*}_{s_{\lvert
i}}\right\rVert\geq\left\lVert\tilde{x^{*}_{\emptyset}}+\sum_{i=1}^{n}\varepsilon_{i}x^{*}_{s_{\lvert
i}}\right\rVert-\frac{1}{\left\lVert
x^{*}_{\emptyset}\right\rVert}\geq\frac{1}{2}\left((1+\delta)^{n}-1\right).$

Clearly this is bigger than $1$ if $n$ was chosen big enough. A contradiction.

∎

In view of this result it is quite clear that if one restricts to the case of
duals of separable spaces with a weak∗ unconditional asymptotic structure one
gets the following renorming theorem.

######  Theorem 3.8.

Let $X$ be a separable space and let us assume that $X^{*}$ has a weak∗
unconditional asymptotic structure. Then the following properties are
equivalent.

  1. (1)

$S_{Z}(X)\leq\omega$

  2. (2)

The space $X$ admits an equivalent norm whose dual norm is AUC∗

  3. (3)

The space $X$ admits an equivalent norm whose dual norm is AMUC∗.

###### Proof.

$(1)\implies(2)$ comes from the renorming theorem for AUC∗ and
$(2)\implies(3)$ is straightforward in view of the AUC∗ and AMUC∗ moduli.

For $(3)\implies(1)$ recall that the dual $X^{*}$ of a separable space with a
weak∗ unconditional asymptotic structure has the weak∗ unconditional finite
tree property: there is a constant $C\geq 1$ such that one can extract a full
subtree whose branches are all $C$-unconditional of any weak∗-null normalized
tree in $X^{*}$. So if $X^{*}$ is AMUC∗, this combined with the preceding
proposition 3.7 yields the following property: for all $\varepsilon>0$ there
is an integer $N\geq 1$ such that: for all $n\geq N$ and for all weak∗-null
tree $(x_{s}^{*})_{s\in T_{n}}$ in $X^{*}$ such that $x_{\emptyset}^{*}\in
B_{X}^{*}$ and $\left\lVert x_{s}^{*}\right\rVert\geq\varepsilon$ for all
$s\in T_{n}$ non-empty, there is a sequence $s\in T_{n}$ of maximal length and
a choice of signs $\varepsilon_{i}$ such that $\left\lVert
x_{\emptyset}^{*}+\sum_{i=1}^{n}\varepsilon_{i}x_{s_{\lvert
i}}^{*}\right\rVert>1$ and consequently

$\left\lVert x_{\emptyset}^{*}+\sum_{i=1}^{n}x_{s_{\lvert
i}}^{*}\right\rVert\geq C^{-1}\left\lVert
x_{\emptyset}^{*}+\sum_{i=1}^{n}\varepsilon_{i}x_{s_{\lvert
i}}^{*}\right\rVert>C^{-1}.$

So by normalizing and by using the characterization 2.1 mentioned in the
preceding section we have $S_{Z}(X)\leq\omega$.

∎

###  3.2. Size of approximate midpoints in dual spaces

Let $M$ be a metric space. For all $x,y\in M$ and for all $\delta>0$ let

$\text{Mid}\left(x,y;\delta\right)=\\{z\in M:\
\max\\{d(x,z),d(y,z)\\}\leq\frac{1+\delta}{2}d(x,y)\\}$

be the set of $\delta$-approximate midpoints between $x$ and $y$. It appears
that the study of the size of those sets yields simple arguments for
preventing coarse-Lipschitz embeddability of certain Banach spaces into
others. The first result of this type is due to Enflo (in an unpublished
paper) and can be stated as follows: $L_{1}$ does not coarse-Lipschitz embed
into $\ell_{1}$. Let us mention that this can also be used to distinguish
between $\ell_{p}$ spaces under coarse-Lipschitz embeddings and that this
argument was pushed even further for example in [15] or [14].

In [6], property AMUC was related to the Kuratowski measure of non compactness
of approximate midpoint sets in the following way.

######  Lemma 3.9 ([6] theorem 2.1).

A Banach space $X$ is AMUC if and only if
$\alpha\left(\text{Mid}\left(x,-x;\delta\right)\right)$ tends to $0$ as
$\delta$ tends to $0$ uniformly on $S_{X}$.

Let us recall that the Kuratowski measure of non-compactness $\alpha$ of a set
$A$ is the infimum of all $\varepsilon>0$ such that $A$ can be covered by a
finite number of sets of diameter less than $\varepsilon$. Let us also point
out that this criteria is sort of an asymptotic version of a characterisation
of uniform convexity in terms of diameter of approximate midpoint sets (more
about that is described in the same paper).

One can show that the same characterization holds for property AMUC∗ on a dual
space and this yields the following result.

######  Proposition 3.10.

Let $X^{*}$ be a dual space. Then $\left\lVert.\right\rVert_{X^{*}}$ is AMUC
if and only if it is AMUC∗.

###### Proof.

It is clear from the definition of the moduli that if
$\left\lVert.\right\rVert_{X^{*}}$ is AMUC∗ then it is also AMUC.

Now let us assume that $\left\lVert.\right\rVert_{X^{*}}$ is not AMUC∗.
Following [DKR&co theorem 2.1] we will prove that
$\alpha\left(\text{Mid}\left(x^{*},-x^{*};\delta\right)\right)$ does not tend
uniformly to $0$ with $\delta$ on $S_{X^{*}}$. Since
$\left\lVert.\right\rVert_{X^{*}}$ is not AMUC∗ we can find some $t\in(0,1)$
such that:

$\forall\delta>0,\ \exists x_{\delta}^{*}\in S_{Y}:\ \forall
Y\in\text{cof}^{*}(X^{*}),\ \exists y^{*}\in S_{Y}:\ \max\\{\left\lVert
x_{\delta}^{*}+ty^{*}\right\rVert,\left\lVert
x_{\delta}^{*}-ty^{*}\right\rVert\\}\leq 1+\delta,$

that is to say
$ty^{*}\in\text{Mid}\left(x_{\delta}^{*},-x_{\delta}^{*};\delta\right)$.

Fix $\delta>0$ and pick $x^{*}=x^{*}_{\delta}$ satisfying this property. Then
one can choose inductively a sequence $(y_{k}^{*})_{k\geq 1}$ in $S_{X^{*}}$,
and a sequence $(y_{k})_{k\geq 1}$ in $S_{X}$ such that

  1. (1)

$\left\langle{y_{k}},\ {y_{k}^{*}}\right\rangle\geq\frac{1}{2}$ and
$y_{k}^{*}\in\bigcap_{i=1}^{k-1}\ker y_{i}$

  2. (2)

$ty_{k}^{*}\in\text{Mid}\left(x^{*},-x^{*};\delta\right)$.

Then we have $\left\lVert
ty_{k}^{*}-ty_{l}^{*}\right\rVert\geq\left\langle{y_{k}},\
{ty_{k}^{*}-ty_{l}^{*}}\right\rangle\geq\frac{t}{2}$ for all $k>l$ and thus
$\alpha\left(\text{Mid}\left(x,-x;\delta\right)\right)\geq\alpha\left(\\{ty_{k}^{*}\\}_{k\geq
1}\right)\geq\frac{t}{2}$. By the preceding lemma,
$\left\lVert.\right\rVert_{X^{*}}$ is not AMUC.

∎

As a consequence of this simple result we can show that properties AMUC∗ and
AUC∗ are not equivalent up to renorming.

######  Corollary 3.11.

There exists a separable Banach space whose dual norm is AMUC∗ but which
admits no equivalent norm whose dual norm is AUC∗.

###### Proof.

Let $JT$ be the James tree-space. Maria Girardi proved in [9] that the dual
$JT^{*}$ of $JT$ is AUC. Thus it is AMUC and by the preceding result it is
also AMUC∗. But since $JT^{*}$ is not separable while $JT$ is separable, $JT$
does not have any equivalent norm whose dual norm is AUC∗.

∎

######  Remark 5.

Also note that $JT^{*}$ being AMUC∗ it has property $\neg(*)$ that is to say
every sufficiently high weak∗-convergent tree in the unit ball of $JT^{*}$
must get out of it if we choose the right direction. This really enlightens
the importance of this choice of direction because the same unit ball contains
arbitrarily high separated weak∗-convergent trees since $JT$ is separable and
has a non-separable dual (and so has infinite Szlenk index).

###  3.3. A few remarks on the equivalence AUC/AMUC up to renorming

Let us just point out some results about property AMUC that can be obtained in
the same way as there weak∗ analogues.

######  Proposition 3.12.

Let $X$ be AMUC. Then for all $\varepsilon>0$ there is an integer $N\geq 1$
such that: for all $n\geq N$ and for every weakly-null tree $(x_{s})_{s\in
T_{n}}$ in $X$ such that $x_{\emptyset}\in B_{X}$ and $\left\lVert
x_{s}\right\rVert\geq\varepsilon$ for all $s\in T_{n}$ non-empty, there is a
sequence $s\in T_{n}$ of maximal length and a sequence of signs
$(\varepsilon_{i})_{i=1}^{n}$ in $\\{-1,1\\}$ such that $\left\lVert
x_{\emptyset}+\sum_{i=1}^{n}\varepsilon_{i}x_{s_{\lvert i}}\right\rVert>1$.

This immediately yields a result concerning the weak-Szlenk index of Banach
spaces with property AMUC and an unconditional (weak) asymptotic structure.

######  Corollary 3.13.

Let $X$ be a Banach space with separable dual and with an unconditional
asymptotic structure. If $X$ is AMUC, then $w-S_{Z}(X)\leq\omega$. In
particular, X has PCP.

Let us recall that the weak Szlenk index $w-S_{Z}(X)$ of a Banach space $X$ is
obtained by applying the same procedure as for the Szlenk index in the space
itself that is by taking away all the small weakly open subsets of the unit
ball of $X$ and then iterating. Unfortunately it is not possible to use this
result to get a renorming result for AMUC in this context since there is no
renorming theory for AUC in full generality. Still let us just mention that in
view of the results from the preceding paragraph one has the following.

######  Proposition 3.14.

Properties AMUC and AUC are equivalent up to renorming in the class of duals
of separable spaces with a weak∗ unconditional asymptotic structure.

##  4\. Embeddability of countably branching bundle graphs into dual spaces

In this section we introduce the so called bundle graphs and we prove the main
result of the paper by using the notion of good $\ell_{p}$-trees.

###  4.1. Good $\ell_{p}$-trees and weak∗ asymptotic structure

Since Mazur’s work it is a well known fact that it is possible to extract a
basic subsequence from every weak or weak∗-null normalized sequence. A simple
extension of Mazur’s proof using the concept of linear compatible ordering
allows to prove the tree version of this result whose weak analogue is used
and proved for example in [2].

######  Lemma 4.1.

Let $X$ be a Banach space and let $(x^{*}_{s})_{s\in T_{n}}$ be a weak∗-null
normalized tree in $X^{*}$. For all $\delta\in(0,1)$ there is a full subtree
$T$ of $T_{n}$ and a compatible linear ordering $(\sigma_{i})_{i\geq 1}$ of
$T$ such that the sequence $(x^{*}_{\sigma_{i}})_{i\geq 1}$ is
$(1+\delta)$-basic.

######  Remark 6.

Although not required in this paper let us mention that it is in fact possible
to do a bit better in the weak∗ case: we can extract a weak∗ basic full
subtree from every weak∗ null normalized tree in the sense of [13] .

Now let $X$ be a Banach space, let $(\sigma_{i})_{i\geq 1}$ be a linear
compatible ordering of $T_{n}$, let $p\in[1,\ \infty]$ and let $C,D\geq 1$. A
normalized tree $(x_{s})_{s\in T_{n}}$ in $X$ is called a $(C,D)$-good
$\ell_{p}$-tree of height $n$ if if satisfies the two following properties:

  1. (1)

All branches are $C^{2}$-equivalent to the unit vector basis of
$\ell_{p}^{n+1}$.

  2. (2)

The sequence $(x_{\sigma_{i}})_{i\geq 1}$ is $D$-basic.

We say that $X$ contains good $\ell_{p}$-trees of arbitrary height almost
isometrically if for every $n\geq 1$ and for every $\varepsilon>0$, $X$
contains a $(1+\varepsilon,1+\varepsilon)$-good $\ell_{p}$-tree of height $n$.

Following [2] (in the weak case) we can use the tree version of Mazur’s lemma
and the tree reformulations from the first section to get the following
result.

######  Lemma 4.2.

Let $X^{*}$ be the dual of a separable Banach space $X$ such that
$\ell_{p}^{n}\in\\{X^{*}\\}_{n}^{w^{*}}$ for every $n\geq 1$. Then $X^{*}$
contains good $\ell_{p}$-trees of arbitrary height almost isometrically.

Thus, as a consequence of the results of the third section and using the weak
analogue from [2], we get the following results.

######  Corollary 4.3.

Let $X^{*}$ be the dual of a separable Banach space $X$ with
$S_{Z}(X)>\omega$, and assume that $X^{*}$ has a weak∗ unconditional
asymptotic structure. Then $X^{*}$ contains good $\ell_{\infty}$-trees of
arbitrary height almost isometrically.

######  Corollary 4.4.

Let $X$ be a Banach space with separable dual, with an unconditional
asymptotic structure and with $S_{Z}(X)>\omega$. Then $X$ contains good
$\ell_{1}$-trees of arbitrary height almost isometrically.

###  4.2. Bundle graphs and embeddability results

A top-bottom graph is a graph with two distinguished vertices, one designated
as the top and the other as the bottom. A countably branching bundle graph is
a top-bottom graph which can be formed, starting by a path of length $1$, by a
finite sequence of the following operations:

  1. (1)

Parallel composition: given two countably branching bundle graphs , identify
the top of one with the bottom of the other and let the bottom of the first
(respectively top of the second) be the bottom (respectively the top) of the
new graph.

  2. (2)

Series composition: take countably many copies of a countably branching bundle
graph and identify all the bottoms (respectively all the tops) with each
other.

A non-trivial bundle countably branching bundle graph is a bundle graph
obtained by such a sequence with at least one series composition. We endow
every bundle graph with its graph distance.

One can have a look at [20] for a very detailed survey of properties of bundle
graphs (in particular an explicit formula for the distance in such a graph is
given and some results of embeddability into Banach spaces are proved). In
particular, it was proved in this paper that if one consider the operation
$\oslash$ which consist in replacing every edge of some countably branching
bundle graph by another countably branching bundle graph, one gets a new
countably branching bundle graph. Thus, for every countably branching bundle
graph $G$, one can consider the family $(G^{\oslash n})_{n\geq 1}$ of
countably branching bundle graphs (with growing height) generated by $G$
defined recursively by $G^{\oslash 1}=G$ and $G^{\oslash(n+1)}=G\oslash
G^{\oslash n}$.

In [2], the authors used a self improvement argument and an approximate
midpoint argument coming from the study of AMUC property to show the following
result (with an estimate of the distortion).

######  Theorem 4.5 (theorem 4.1 [2]).

Let $G$ be a non-trivial countably branching bundle graph and let $X$ be an
AMUC Banach space. Then the family $(G^{\oslash n})_{n\geq 1}$ does not equi-
Lipschitz embed into $X$.

Generalizing a result from [2] concerning the countably branching diamond
graphs, Swift proved the following result.

######  Theorem 4.6.

Let $X$ be a Banach space which contains good $\ell_{\infty}$-trees of
arbitrary height almost isometrically. Then every non-trivial countably
branching bundle graph embeds Lipschitz in $X$ with distortion less than
$6+\varepsilon$.

Thus, combining this result with the containment of good $\ell_{\infty}$-trees
coming from [2], he got ([20] corollary 3.4) the embeddability of any
countably branching bundle graph into a separable reflexive Banach space with
an unconditional asymptotic structure and with $S_{Z}(X^{*})>\omega$. By our
result from the last section, we can extend this to dual spaces in the
following.

######  Corollary 4.7.

Let $X^{*}$ be the dual of a separable Banach space $X$ with
$S_{Z}(X)>\omega$, and assume that $X^{*}$ has a weak∗ asymptotic
unconditional structure. Then every non-trivial countably branching bundle
graph embeds Lipschitz in $X^{*}$ with distortion less than $6+\varepsilon$.

Then combining this and the non-embeddability result from [2] one obtains the
following metric characterization.

######  Theorem 4.8.

Let $X^{*}$ be the dual of a separable Banach space $X$ and assume that
$X^{*}$ has a weak∗ asymptotic unconditional structure. Also let $G$ be any
non-trivial countably branching bundle graph. Then the following properties
are equivalent.

  1. (1)

$S_{Z}(X)\leq\omega$

  2. (2)

$X$ admits an equivalent norm whose dual norm is AUC∗

  3. (3)

$X$ admits an equivalent norm whose dual norm is AMUC∗

  4. (4)

The family $(G^{\oslash n})_{n\geq 1}$ does not equi-Lipschitz embed into
$X^{*}$.

As a corollary, we get the following stability result.

######  Corollary 4.9.

The class of $AUC$ renormable spaces is stable under coarse-Lipschitz
embeddings inside the class of duals of separable Banach spaces with a weak∗
asymptotic unconditional structure.

Also note that following the same procedure the remark from section $2$
concerning property $(*)$ yields the following result.

######  Corollary 4.10.

Let $X^{*}$ separable dual with property $(*)$. Then every non-trivial
countably branching bundle graph embeds Lipschitz in $X^{*}$ with distortion
less than $6+\varepsilon$.

###  4.3. Bundle graphs and $c_{0}$-spreading models

Let us first recall the definition of a spreading model. Let $X$ be a Banach
space. By using Ramsey’s theorem, one can show that for every bounded sequence
$(x_{n})_{n\geq 1}\subset X$ there a subsequence $(y_{n})_{n\geq 1}$ such that
for all $k\geq 1$ and for all $a_{1},\dots,a_{k}\in\mathbb{R}$ the limit
$\lim_{n_{1}<\dots<n_{k}}\left\lVert\sum_{i=1}^{k}a_{i}y_{n_{i}}\right\rVert$
exists. Let $(e_{i})_{i\geq 1}$ be the canonical basis of $c_{00}$. If the
sequence $(y_{n})_{n\geq 1}$ is not convergent the quantity

$\left\lVert\sum_{i=1}^{k}a_{i}e_{i}\right\rVert=\lim_{n_{1}<\dots<n_{k}}\left\lVert\sum_{i=1}^{k}a_{i}y_{n_{i}}\right\rVert$

defines a norm on $c_{00}$. The completion of the space
$(c_{00},\left\lVert.\right\rVert)$ is called _spreading model_ associated
with the fundamental sequence $(e_{i})_{i\geq 1}$ and generated by the
sequence $(y_{n})_{n\geq 1}$. Note that the fundamental sequence is spreading
in the sense that for all $k\geq 1$, for all $a_{1},\dots,a_{k}\in\mathbb{R}$
and for all $1\leq n_{1}<\dots<n_{k}$ we have the norm equality
$\left\lVert\sum_{i=1}^{k}a_{i}e_{i}\right\rVert=\left\lVert\sum_{i=1}^{k}a_{i}e_{n_{i}}\right\rVert$.
If $E$ is another Banach space we shall say that $X$ has an $E$-spreading
model if $X$ has a spreading model isomorphic to $E$. We refer to [4] for a
presentation of the theory of spreading models.

Let us recall the following well known result concerning $c_{0}$-spreading
models.

######  Proposition 4.11.

If $X$ has a $c_{0}$-spreading model then for every $\varepsilon>0$ there is a
normalized sequence $(x_{k})_{k\geq 1}$ in $X$ such that:

$\displaystyle\forall k\geq 1,\ \forall a_{1},\dots,a_{k}\in\mathbb{R},\
\forall n_{k}>...>n_{1}\geq k,$
$\displaystyle\frac{1}{1+\varepsilon}\max_{1\leq i\leq n}\left\lvert
a_{i}\right\rvert\leq\left\lVert\sum_{i=1}^{n}a_{i}x_{n_{i}}\right\rVert\leq(1+\varepsilon)\max_{1\leq
i\leq n}\left\lvert a_{i}\right\rvert\ \ (\Box)$

Moreover, let us recall that a $c_{0}$-spreading model is always generated by
a weak-null sequence in $X$. As a consequence it is clear that a Banach space
$X$ with property AMUC does not have $c_{0}$-spreading models. In fact we can
even prove the following result.

######  Theorem 4.12.

If $X$ has a $c_{0}$-spreading model, then $\ell_{\infty}^{n}\in\\{X\\}_{n}$
for all $n\geq 1$. Consequently, every non-trivial countably branching bundle
graph embeds Lipschitz in $X$ with distortion less than $6+\varepsilon$.

###### Proof.

Fix $n\geq 1$ and $\varepsilon>0$. We can apply the preceding proposition to
get a normalized sequence $(x_{k})_{k\geq 1}$ satisfying the property $(\Box)$
above. This sequence generates a $c_{0}$-spreading model and thus it is
necessarily weak-null.

For all $s=(s_{1},\dots,s_{k})\in T_{n}$ non empty, let $y_{s}=x_{\phi(s)}$
where $\phi(s)=k+\sum_{i=1}^{k}s_{i}$. Then $(x_{s})_{s\in
T_{n}\backslash\\{\emptyset\\}}$ is a weak-null normalized unrooted tree and
since $n\leq\phi(s_{\lvert 1})<\dots<\phi(s)$ for every $s\in T_{n}$ of length
$n$, property $(\Box)$ above yields that every branch of this unrooted tree is
$(1+\varepsilon)^{2}$ equivalent to the unit vector basis of
$\ell_{\infty}^{n}$.

From this we deduce that $\ell_{\infty}^{n}\in\\{X\\}_{n}$ for every $n\geq 1$
and as before we get that $X$ contains good $\ell_{\infty}$ trees of arbitrary
height almost isometrically. The conclusion follows from 4.6.

∎

##  5\. Questions and comments

###  5.1. Szlenk index of WCG spaces is determined by separable quotients

It is well known that the Szlenk index is separably determined and this
usually allows to get rid of separability assumptions in results concerning
it. Unfortunately passing to a separable subspace won’t be sufficient to do so
in theorem B and C since we might lose the asymptotic unconditional property
of the dual space in the process. What is required is to pass to a separable
quotient while conserving the same information on the Szlenk index and this is
obviously way more difficult since it is not even known if a separable
quotient exists. Still this can be done if we restrict ourselves to weakly
compactly generated (WCG) Banach spaces. Let us recall that a Banach space $X$
is WCG if there is a weakly compact subset $K$ of $X$ such that
$\overline{\text{span}}(K)=X$.

######  Theorem 5.1.

Let $X$ be a WCG Banach space and let $\alpha$ be a countable ordinal. If
$S_{Z}(X)>\alpha$ then there is a subspace $Y$ of $X$ such that the quotient
$Z=X\backslash Y$ is separable and satisfies $S_{Z}(Z)>\alpha$.

As we shall see the main reason for working with WCG Banach spaces is that
this property passes to quotients and that the density of such a space is
equal to the weak∗ density of its dual. This can be found in [7] in the first
section of chapter $13$.

We will prove this result by using the $\ell_{1}^{+}$ index introduced by D.
Alspach, R. Judd and E. Odell in [1] which is known to be equal to the Szlenk
index for Banach spaces not containing $\ell_{1}$. Let us recall that a space
$X$ has $\ell_{1}^{+}$ index $I^{+}(X)>\alpha$ for some countable ordinal
$\alpha$ if and only if there is a normalized tree $(x_{s})_{s\in T}$ of order
$\alpha$ in $X$ and a constant $K\geq 1$ such that every branch
$(x_{s})_{s\in\beta}$ of the tree is a $K$-$\ell_{1}^{+}$ sequence that is to
say it is $K$-basic and satisfies for all sequence $(a_{s})_{s\in\beta}$ in
$\mathbb{R}_{+}$ the following inequality

$\frac{1}{K}\sum_{s\in\beta}a_{s}\leq\left\lVert\sum_{s\in\beta}a_{s}x_{s}\right\rVert.$

We refer to their paper for precise definitions and for the related result.
This tool in hands let us prove the theorem.

###### Proof.

Let us start by assuming that $X$ does not contain $\ell_{1}$. Then
$I^{+}(X)=S_{Z}(X)>\alpha$ and we can find a normalized tree $(x_{s})_{s\in
T}$ of order $\alpha$ in $X$ witnessing this as recalled above. For every
element $s$ in $T$ pick a norming functional $f_{s}$ of $x_{s}$ and for every
branch $\beta$ of $T$ and every sequence $a=(a_{s})_{s\in\beta}$ in
$\mathbb{Q}_{+}$ pick a norming functional $g_{a,\beta}$ of
$\sum_{s\in\beta}a_{s}x_{s}$. Then let $F$ be the set of all functionals
$f_{s}$ and $g_{a,\beta}$ and let $Y=F^{\top}$ be the pre-orthogonal of $F$.
It is a well known fact that the dual of the quotient $Z=X\backslash Y$ is
isometric to the weak∗ closure $\overline{F}^{w^{*}}$ of $F$ in $X^{*}$ which
is weak∗ separable by construction. Now $Z$ is WCG as quotient of a WCG space
and thus it is separable since its density is the same as the weak∗ density of
its dual. Moreover the tree $(\bar{x}_{s})_{s\in T}$ will be normalized in $Z$
thanks to the functionals $f_{s}$ and its branches will all be
$K-\ell_{1}^{+}$ sequences over $\mathbb{Q}$ thus also over $\mathbb{R}$
thanks to the fonctionals $g_{a,\beta}$. Consequently the $\ell_{1}^{+}$ index
of $Z$ will also be strictly greater than $\alpha$. Now $Z$ cannot contain
$\ell_{1}$ as a quotient of a space not containing it (it is well known that
$\ell_{1}$ can be lifted from quotients) so $S_{Z}(Z)=I^{+}(Z)>\alpha$ and we
are done.

Now if $X$ contains $\ell_{1}$ the same idea allows to build a separable
quotient of $X$ which also contains $\ell_{1}$ and this is enough to conclude.

∎

This allows to give a non-separable version of theorems B and C combined for
WCG Banach spaces.

######  Theorem 5.2.

Let $X^{*}$ be the dual of a WCG Banach space $X$ and assume that $X^{*}$ has
a weak∗ asymptotic unconditional structure. Also let $G$ be any non-trivial
countably branching bundle graph. Then the following properties are
equivalent.

  1. (1)

$S_{Z}(X)\leq\omega$

  2. (2)

$X^{*}\in\langle AUC^{*}\rangle$

  3. (3)

$X^{*}\in\langle AMUC^{*}\rangle$

  4. (4)

The family $(G^{\oslash n})_{n\geq 1}$ does not equi-Lipschitz embed into
$X^{*}$.

In particular since every reflexive Banach space is WCG the separability
assumption from the original theorem from [20] can be removed.

######  Theorem 5.3.

Let $X$ be a reflexive Banach space with an unconditional asymptotic
structure. Then the following properties are equivalent.

  1. (1)

$S_{Z}(X^{*})\leq\omega$

  2. (2)

$X\in\langle AUC\rangle$

  3. (3)

The family $(G^{\oslash n})_{n\geq 1}$ does not equi-Lipschitz embed into $X$.

###  5.2. About asymptotic midpoint convexity

######  Question 1.

Is an equivalent AMUC∗ norm on a dual space a dual norm?

It is known to be true for AUC∗ norms because such a norm has to be weak∗
lower semicontinuous and thus has to be a dual norm (for example see [5]
chapter $1$, fact 5.4 ).

######  Question 2.

Is it possible to develop a specific renorming theory for AMUC∗?

As mentioned in the introduction a dual property for AMUC∗ or for property
$(*)$ is lacking in order to do a renorming à la [16] or [10]. Moreover it is
not clear whether one can find a suitable derivation index to go either with
the symmetric UKK∗ property we saw in section $3$ or with property $(*)$ or
not (or in an other way to find a suitable measure of non-compacteness to go
in the direction of [19]). It is also not known if it is possible to find a
correct equivalent expression of property $(*)$ in order to get a partial
result of the type [17] based on an Enflo-like renorming to get a symmetric
UKK∗ écart.

######  Question 3.

If a dual space is AMUC∗ then can we find an equivalent $p$-AMUC∗ norm for
some $1\leq p<\infty$?

The same question can be asked for property AMUC. Such a result would in
particular yield good estimates of the distortion of the families generated by
a countably branching bundle graph into an AMUC space as mentioned in [2]

######  Question 4.

If a space is AMUC does it have weak-Slzenk index $w-S_{Z}(X)<\infty$? Does it
have weak-Szlenk index $\omega$?

Again we saw that this is true when the space has a separable dual and an
unconditional asymptotic structure. The first question can be reformulated in
the following way: if a space is AMUC does it have PCP? In relation with the
comments of the following paragraph one can also ask whether AMUC implies
weaker versions of PCP like convex PCP or not. A negative answer to any of
these question would provide a counter example to the question of the
equivalence between AMUC and AUC up to renorming since since an AUC space has
finite weak-Szlenk index. Let us point out that [6] gave partial positive
answers to this question in the case of Banach spaces with an unconditional
basis and that they settled the question of isometric equivalence by
constructing an equivalent norm on $\ell_{2}$ which is AMUC but not AUC. Let
us also point out that there is no renorming theory around weak-Szlenk index
yet (the obstruction is always a lack of weak compactness).

###  5.3. About James tree-spaces

######  Question 5.

Is it possible to improve the exponent for AUC given in [9] (3-AUC) for the
predual of the James tree-space $JT$ on a dyadic tree? What exponent can we
get for its dual? In view of the definition of this James tree-space one
expects to get an exponent $2$.

######  Question 6.

Is the predual of the James-tree space $JT_{\infty}$ on a countably branching
tree AMUC? Is its dual AMUC∗? This would provide a counter-example to several
questions of the preceding paragraph since the predual of $JT_{\infty}$ does
not have PCP. Let us point out that this space still has convex PCP (see [8]).

## References

  * [1] D. Alspach, R. Judd, and E. Odell.  The Szlenk index and local $l_{1}$-indices.  Positivity, 9(1):1–44, 2005.
  * [2] F. Baudier, R. Causey, S. Dilworth, D. Kutzarova, N. L. Randrianarivony, T. Schlumprecht, and S. Zhang.  On the geometry of the countably branching diamond graphs.  J. Funct. Anal., 273(10):3150–3199, 2017.
  * [3] F. Baudier, N. J. Kalton, and G. Lancien.  A new metric invariant for Banach spaces.  Studia Math., 199(1):73–94, 2010.
  * [4] B. Beauzamy and J.-T. Lapresté.  Modèles étalés des espaces de Banach.  Travaux en Cours. [Works in Progress]. Hermann, Paris, 1984.
  * [5] R. Deville, G. Godefroy, and V. Zizler.  Smoothness and renormings in Banach spaces, volume 64 of Pitman Monographs and Surveys in Pure and Applied Mathematics.  Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993.
  * [6] S. J. Dilworth, D. Kutzarova, N. L. Randrianarivony, J. P. Revalski, and N. V. Zhivkov.  Lenses and asymptotic midpoint uniform convexity.  J. Math. Anal. Appl., 436(2):810–821, 2016.
  * [7] M. Fabian, P. Habala, P. Hájek, V. Montesinos, and V. Zizler.  Banach space theory.  CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, New York, 2011.  The basis for linear and nonlinear analysis.
  * [8] N. Ghoussoub, B. Maurey, and W. Schachermayer.  A counterexample to a problem on points of continuity in Banach spaces.  Proc. Amer. Math. Soc., 99(2):278–282, 1987.
  * [9] M. Girardi.  The dual of the James tree space is asymptotically uniformly convex.  Studia Math., 147(2):119–130, 2001.
  * [10] G. Godefroy, N. J. Kalton, and G. Lancien.  Szlenk indices and uniform homeomorphisms.  Trans. Amer. Math. Soc., 353(10):3895–3918, 2001.
  * [11] R. Huff.  Banach spaces which are nearly uniformly convex.  Rocky Mountain J. Math., 10(4):743–749, 1980.
  * [12] W. B. Johnson, J. Lindenstrauss, D. Preiss, and G. Schechtman.  Almost Fréchet differentiability of Lipschitz mappings between infinite-dimensional Banach spaces.  Proc. London Math. Soc. (3), 84(3):711–746, 2002.
  * [13] W. B. Johnson and H. P. Rosenthal.  On $\omega^{\ast}$-basic sequences and their applications to the study of Banach spaces.  Studia Math., 43:77–92, 1972.
  * [14] N. J. Kalton.  Uniform homeomorphisms of Banach spaces and asymptotic structure.  Trans. Amer. Math. Soc., 365(2):1051–1079, 2013.
  * [15] N. J. Kalton and N. L. Randrianarivony.  The coarse Lipschitz geometry of $l_{p}\oplus l_{q}$.  Math. Ann., 341(1):223–237, 2008.
  * [16] H. Knaust, E. Odell, and Th. Schlumprecht.  On asymptotic structure, the Szlenk index and UKK properties in Banach spaces.  Positivity, 3(2):173–199, 1999.
  * [17] G. Lancien.  On uniformly convex and uniformly Kadec-Klee renormings.  Serdica Math. J., 21(1):1–18, 1995.
  * [18] G. Lancien.  A survey on the Szlenk index and some of its applications.  RACSAM. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat., 100(1-2):209–235, 2006.
  * [19] G. Lancien, A. Procházka, and M. Raja.  Szlenk indices of convex hulls.  J. Funct. Anal., 272(2):498–521, 2017.
  * [20] A. Swift.  A coding of bundle graphs and their embeddings into Banach spaces.  Mathematika, 64(3):847–874, 2018.


DIII; both require physical time-reversal symmetry Schnyder  _et al._ (2008);
Ryu  _et al._ (2010). Class CI, in addition, possesses a $\mathcal{C}^{2}=-1$
particle-hole symmetry. In the superconductor interpretation, this encodes
invariance under $\pi$-rotations along $x$ and $y$ axes in spin space, which
is tantamount to full SU(2) symmetry. Class DIII by contrast has no spin
symmetry, and is usually cast in terms of a real (Balian-Werthammer Coleman
(2015)) spinor that encodes both spin and particle-hole degrees of freedom.
Physical time-reversal also appears as a chiral condition, while
$\mathcal{C}^{2}=+1$ particle-hole symmetry is imposed on the BdG Hamiltonian
by the reality condition on the spinor.