G. Godefroy, N. J. Kalton, G. Lancien
We prove some rather precise renorming theorems for Banach spaces with Szlenk index $ω_0$. We use these theorems to show the invariance of certain quantitative Szlenk-type indices under uniform homeomorphisms.
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We prove the optimal estimate between the Szlenk and $w^*$-dentability indices of an arbitrary $w^*$-compact subset of the dual of a Banach space. For a given $w^*$-compact, convex subset $K$ of the dual of a Banach space, we introduce a two player game the winning strategies of which determine the Szlenk index of $K$. We give applications to the $w^*$-dentability index of a Banach space and of an operator.
Given any compact, Hausdorff space $K$ and $1<p<\infty$, we compute the Szlenk and $w^*$-dentability indices of the spaces $C(K)$ and $L_p(C(K))$. We show that if $K$ is compact, Hausdorff, scattered, $CB(K)$ is the Cantor-Bendixson index of $K$, and $ξ$ is the minimum ordinal such that $CB(K)\leqslant ω^ξ$, then $Sz(C(K))=ω^ξ$ and $Dz(C(K))=Sz(L_p(C(K)))= ω^{1+ξ}.$
Let $α$ be an infinite ordinal and $γ$ the unique ordinal satisfying $ω^{ω^γ}\leq α< ω^{ω^{γ+1}}$. We show that the Banach space $C([0,\,α])$ of all continuous scalar-valued functions on the compact ordinal interval $[0,\,α]$ has Szlenk index equal to $ω^{γ+1}$ and $w^\ast$-dentability index equal to $ω^{1+γ+1}$.
We study the general measures of non-compactness defined on subsets of a dual Banach space, their associated derivations and their $ω$-iterates. We introduce the notions of convexifiable and sublinear measure of non-compactness and investigate the properties of its associated fragment and slice derivations. We apply our results to the Kuratowski measure of non-compactness and to the study of the Szlenk index of a Banach space. As a consequence, we obtain that the Szlenk index and the convex Szlenk index of a separable Banach space are always equal. We also give, for any countable ordinal $α$, …
We introduce two new local l_1-indices of the same type as the Bourgain l_1 index; the l_1^+-index and the l_1^+-weakly null index. We show that the l_1^+-weakly null index of a Banach space X is the same as the Szlenk index of X, provided X does not contain l_1. The l_1^+-weakly null index has the same form as the Bourgain l_1 index: if it is countable it must take values omega^alpha for some alpha<omega_1. The different l_1-indices are closely related and so knowing the Szlenk index of a Banach space helps us calculate its local l_1-index, via the l_1^+-weakly null index. We show that I(C(om…
Let $X$ be a Banach space and $K$ an absolutely convex, weak$^\ast$-compact subset of $X^\ast$. We study consequences of $K$ having a large or undefined Szlenk index and subsequently derive a number of related results concerning basic sequences and universal operators. We show that if $X$ has a countable Szlenk index then $X$ admits a subspace $Y$ such that $Y$ has a basis and the Szlenk indices of $Y$ are comparable to the Szlenk indices of $X$. If $X$ is separable, then $X$ also admits subspace $Z$ such that the quotient $X/Z$ has a basis and the Szlenk indices of $X/Z$ are comparable to the…
For each ordinal $α< ω_1$, we prove the existence of a separable, reflexive Banach space with a basis and Szlenk index $ω^{α+1}$ which is universal for the class of separable, reflexive Banach spaces $X$ such that the Szlenk indices $Sz(X), Sz(X^*)$ do not exceed $ω^α$.
In a previous work, the first named author described the set $\cal P$ of all values of the Szlenk indices of separable Banach spaces. We complete this result by showing that for any integer $n$ and any ordinal $α$ in $\cal P$, there exists a separable Banach space $X$ such that the Szlenk of the dual of order $k$ of $X$ is equal to the first infinite ordinal $ω$ for all $k$ in $\{0,..,n-1\}$ and equal to $α$ for $k=n$. One of the ingredients is to show that the Lindenstrauss space and its dual both have a Szlenk index equal to $ω$. We also show that any element of $\cal P$ can be realized as a…
For each ordinal $ξ$ and each $1\leqslant q<\infty$, we define the notion of $ξ$-$q$-summable Szlenk index. When $ξ=0$ and $q=1$, this recovers the usual notion of summable Szlenk index. We define for an arbitrary weak$^*$-compact set a transfinite, asymptotic analogue $α_{ξ,p}$ of the martingale type norm of an operator. We prove that this quantity is determined by norming sets and determines $ξ$-Szlenk power type and $ξ$-$q$-summability of Szlenk index. This fact allows us to prove that the behavior of operators under the $α_{ξ,p}$ seminorms passes in the strongest way to injective tensor pr…
Szlenk and $w^\ast$-dentability indices of C$^\ast$-algebras
本文计算无限维C*-代数及其 $L_p$ 空间的 Szlenk 指数和 $w^\ast$-可凹性指数，并给出张量积公式及可分情形下由单个自伴元生成子代数实现指数的结论。
We show that if the Szlenk index of a Banach space $X$ is larger than the first infinite ordinal $ω$ or if the Szlenk index of its dual is larger than $ω$, then the tree of all finite sequences of integers equipped with the hyperbolic distance metrically embeds into $X$. We show that the converse is true when $X$ is assumed to be reflexive. As an application, we exhibit new classes of Banach spaces that are stable under coarse-Lipschitz embeddings and therefore under uniform homeomorphisms.
On metric characterizations of tree and fragmentability indices of Banach spaces
本文引入两个序数指标（二叉树指标和蔓延树指标）作为Banach空间的线性不变量，证明它们也是双Lipschitz不变量，并通过序数高度的二叉或可数分支钻石图的子Lipschitz嵌入性刻画其值，进而应用于完备可分度量空间和双Lipschitz嵌入的泛函Banach空间，并讨论这些树指标与经典的可碎片化指标（如齿形性、弱可碎片化和Szlenk指标）的联系。
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