Szymon Draga, Tomasz Kochanek
We prove a formula for the Szlenk power type of the injective tensor product of Banach spaces with Szlenk index at most $ω$. We also show that the Szlenk power type as well as summability of the Szlenk index are separably determined, and we extend some of our recent results concerning direct sums.
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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 $ω^α$.
We discuss pruning and coloring lemmas on regular families. We discuss several applications of these lemmas to computing the Szlenk index of certain $w^*$ compact subsets of the dual of a separable Banach space. Applications include estimates of the Szlenk index of Minkowski sums, infinite direct sums of separable Banach spaces, constant reduction, and three space properties. We also consider using regular families to construct Banach spaces with prescribed Szlenk index. As a consequence, we give a characterization of which countable ordinals occur as the Szlenk index of a Banach space, prove …
For each ordinal $α<ω_1$, we prove the existence of a space with a basis and Szlenk index $ω^{α+1}$ which is universal for the class of spaces with Szlenk index not exceeding $ω^α$. Our proof involves developing a characterization of which Banach spaces embed into spaces with an FDD with upper Schreier space estimates.
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 $ω^{1+γ+1}$.
For a countable ordinal a we denote by C_a the class of separable, reflexive Banach spaces whose Szlenk index and the Szlenk index of their dual are bounded by a. We show that each C_a admits a separable, reflexive universal space. We also show that spaces in the class C_{omega^{a*omega}} embed into spaces of the same class with a basis. As a consequence we deduce that each C_a is analytic in the Effros-Borel structure of subspaces of C[0,1].
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 $α$, …
Given any Banach space $X$ and any weak*-compact subset $K$ of $X^*$, we compute the Szlenk index of the weak*-closed, convex hull of $K$ as a function of the Szlenk index of $K$. Also as an application, we compute the Szlenk index of any injective tensor product of two operators. In particular, we compute the Szlenk index of an injective tensor product $\tens$ in terms of $Sz(X)$ and $Sz(Y)$. As another application, we give a complete characterization of those ordinals which occur as the Szlenk index of a Banach space, as well as those ordinals which occur as the Bourgain $\ell_1$ or $c_0$ in…
We generalize the notion of summable Szlenk index from a Banach space to an arbitrary weak$^*$-compact set. We prove that a weak$^*$-compact set has summable Szlenk index if and only if its weak$^*$-closed, absolutely convex hull does. As a consequence, we offer a new, short proof of a result from [Draga and Kochanek 2016] regarding the behavior of summability of Szlenk index under $c_0$ direct sums. We also use this result to prove that the injective tensor product of two Banach spaces has summable Szlenk index if both spaces do, which answers a question from [Draga and Kochanek 2017]. As a f…
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 a Banach space $X$, a $w^*$-compact subset of $X^*$, and $1<p<\infty$, we provide an optimal relationship between the Szlenk index of $K$ and the Szlenk index of an associated subset of $L_p(X)^*$. As an application, given a Banach space $X$, we prove an optimal estimate of the Szlenk index of $L_p(X)$ in terms of the Szlenk index of $X$. This extends a result of Hájek and Schlumprecht to uncountable ordinals. More generally, given an operator $A:X\to Y$, we provide an estimate of the Szlenk index of the "pointwise $A$" operator $A_p:L_p(X)\to L_p(Y)$ in terms of the Szlenk index of $A$.
For every $α<ω_1$ we establish the existence of a separable Banach space whose Szlenk index is $ω^{αω+1}$ and which is universal for all separable Banach spaces whose Szlenk-index does not exceed $ω^{αω}$. In order to prove that result we provide an intrinsic characterization of which Banach spaces embed into a space admitting an FDD with upper estimates.
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…
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.
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…
Enveloping balls of Szlenk derivations
本文研究具有收缩 FDD 的 Banach 空间，估计其对偶单位球在 ε-Szlenk 导运算下包络球的半径。
We give necessary and sufficient conditions for an operator $A:X\to Y$ on a Banach space having a shrinking FDD to factor through a Banach space $Z$ such that the Szlenk index of $Z$ is equal to the Szlenk index of $A$. We also prove that for every ordinal $ξ\in (0, ω_1)\setminus\{ω^η: η<ω_1\text{\ a limit ordinal}\}$, there exists a Banach space $\mathfrak{G}_ξ$ having a shrinking basis and Szlenk index $ω^ξ$ such that for any separable Banach space $X$ and any operator $A:X\to Y$ having Szlenk index less than $ω^ξ$, $A$ factors through a subspace and through a quotient of $\mathfrak{G}_ξ$, a…
We discuss an alternate method for computing the Szlenk index of an arbitrary $w^*$ compact subset of the dual of a Banach space. We discuss consequences of this method as well as offer simple, alternative proofs of a number of results already found in the literature.
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…
We prove that the $c_0$-sum of separable Banach spaces with uniformly summable Szlenk index has summable Szlenk index, whereas this result is no longer valid for more general direct sums. We also give a formula for the Szlenk power type of the $\mathfrak{E}$-direct sum of separable spaces provided that $\mathfrak{E}$ has a shrinking unconditional basis whose dual basis yields an asymptotic $\ell_p$ structure in $\mathfrak{E}^\ast$. As a corollary, we show that the Tsirelson direct sum of infinitely many copies of $c_0$ has power type $1$ but non-summable Szlenk index.
We find an optimal upper bound on the values of the weak$^*$-dentability index $Dz(X)$ in terms of the Szlenk index $Sz(X)$ of a Banach space $X$ with separable dual. Namely, if $\;Sz(X)=ω^α$, for some $α<ω_1$, and $p\in(1,\infty)$, then $$Sz(X)\le Dz(X)\le Sz(L_p(X))\le {cases} ω^{α+1} &\text{if $α$ is a finite ordinal,} ω^α &\text{if $α$ is an infinite ordinal.} {cases}$$
We extend a precise renorming result of Godefroy, Kalton, and Lancien regarding asymptotically uniformly smooth norms of separable Banach spaces with Szlenk index $ω$. For every ordinal $ξ$, we characterize the operators, and therefore the Banach spaces, which admit a $ξ$-asymptotically uniformly smooth norm with power type modulus and compute for those operators the best possible exponent in terms of the values of $Sz_ξ(\cdot, \varepsilon)$. We also introduce the $ξ$-Szlenk power type and investigate ideal and factorization properties of classes associated with the $ξ$-Szlenk power type.
When is the Szlenk derivation of a dual unit ball another ball?
本文研究对偶单位球的 Szlenk 导集何时仍为球，借助包络球半径估计与 Karush-Kuhn-Tucker 定理，区分了性质 $(M^\ast)$ 空间及 Baernstein、Tsirelson、Schlumprecht 和 Orlicz 空间的情形。
We compute the Szlenk index of an arbitrary projective tensor product $C(K)\widehat{\otimes}_πC(L)$ of spaces $C(K), C(L)$ of continuous functions on scattered, compact, Hausdorff spaces. In particular, we show that it is simply equal to the maximum of the Szlenk indices of the spaces $C(K), C(L)$. We deduce several results regarding non-isomorphism of $C(K)\widehat{\otimes}_πC(L)$ and $C(M)$ or $C(M)\widehat{\otimes}_πC(N)$ for particular choices of $K,L,M,N$.
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…
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.
We introduce asymptotic analogues of the Rademacher and martingale type and cotype of Banach spaces and operators acting on them. Some classical local theory results related, for example, to the `automatic-type' phenomenon, the type-cotype duality, or the Maurey-Pisier theorem, are extended to the asymptotic setting. We also investigate operator ideals corresponding to the asymptotic subtype/subcotype. As an application of this theory, we provide a sharp version of a result of Brooker and Lancien by showing that any twisted sum of Banach spaces with Szlenk power types $p$ and $q$ has Szlenk po…
In this note we extend to the quasi-reflexive setting the result of F. Baudier, N. Kalton and G. Lancien concerning the non-embeddability of the family of countably branching trees into reflexive Banach spaces whose Szlenk index and Szlenk index from the dual are both equal to the first infinite ordinal $ω$. In particular we show that the family of countably branching trees does neither embed into the James space $\mathcal{J}_p$ nor into its dual space $\mathcal{J}_p^*$ for $p\in(1,\infty)$.
We extend to the non separable setting many characterizations of the Banach spaces admitting an equivalent norm with the property $(β)$ of Rolewicz. These characterizations involve in particular the Szlenk index and asymptotically uniformly smooth or convex norms. This allows to extend easily to the non separable case some recent results from the non linear geometry of Banach spaces.
Banach-Saks Theorem for $L^1$ revisited
本文综述了 $L^p$ 空间的 Banach-Saks 性质，重点讨论 $p=1$ 情形，并在更一般背景下重新证明 $L^1$ 空间具有弱 Banach-Saks 性质。
We study the nonlinear embeddability of Banach spaces and the equi-embeddability of the family of Kalton's interlaced graphs $([\mathbb N]^k,d_{\mathbb K})_k$ into dual spaces. Notably, we define and study a modification of Kalton's property $\mathcal Q$ that we call property $\mathcal{Q}_p$ (with $p \in (1,+\infty]$). We show that if $([\mathbb N]^k,d_{\mathbb K})_k$ equi-coarse Lipschitzly embeds into $X^*$, then the Szlenk index of $X$ is greater than $ω$, and that this is optimal, i.e., there exists a separable dual space $Y^*$ that contains $([\mathbb N]^k,d_{\mathbb K})_k$ equi-Lipschitz…
In this paper we show the weak Banach-Saks property of the Banach vector space $(L_μ^p)^m$ generated by $m$ $L_μ^p$-spaces for $1\leq p<+\infty,$ where $m$ is any given natural number. When $m=1,$ this is the famous Banach-Saks-Szlenk theorem. By use of this property, we also present inequalities for integrals of functions that are the composition of nonnegative continuous convex functions on a convex set of a vector space ${\bf R}^m$ and vector-valued functions in a weakly compact subset of the space $(L_μ^p)^m$ for $1\leq p<+\infty$ and inequalities when these vector-valued functions are in …
We prove that if $Y$ is a closed subspace of a Banach space $X$ such that $Y$ and $X/Y$ admit an equivalent asymptotically uniformly smooth norm, then $X$ also admits an equivalent asymptotically uniformly smooth norm. The proof is based on the use of the Szlenk index and yields a few other applications to renorming theory.
It is shown that a script L_infty-space with separable dual constructed by Bourgain and Delbaen has small Szlenk index and thus does not have a quotient isomorphic to C(omega^omega). It follows that this is a script L_infty-space which is the same size as c_0 in the sense of the Szlenk index but does not contain c_0. This has some consequences in the theory of uniform homeomorphism of Banach spaces.
Asymptotic smoothness, concentration properties in Banach spaces and applications
本文证明了 Hamming 图上 Lipschitz 映射的集中性质在 $\ell_p$-和下具有最优稳定性，并由此构造出 Szlenk 指标任意高且具有集中性质但无渐近光滑性的 Banach 空间实例。
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。
We study four asymptotic smoothness properties of Banach spaces, denoted $\textsf{T}_p,\textsf{A}_p, \textsf{N}_p$ and $\textsf{P}_p$. We complete their description by proving the missing renorming theorem for $\textsf{A}_p$. We prove that asymptotic uniform flattenability (property $\textsf{T}_\infty$) and summable Szlenk index (property $\textsf{A}_\infty$) are three space properties. Combined with the positive results of the first named author, Draga, and Kochanek, and with the counterexamples we provide, this completely solves the three space problem for this family of properties. We also …
Denote by $[0,ω_1)$ the set of countable ordinals, equipped with the order topology, let $L_0$ be the disjoint union of the compact ordinal intervals $[0,α]$ for $α$ countable, and consider the Banach spaces $C_0[0,ω_1)$ and $C_0(L_0)$ consisting of all scalar-valued, continuous functions which are defined on the locally compact Hausdorff spaces $[0,ω_1)$ and $L_0$, respectively, and which vanish eventually. Our main result states that a bounded operator $T$ between any pair of these two Banach spaces fixes a copy of $C_0(L_0)$ if and only if the identity operator on $C_0(L_0)$ factors through…
We define two metrics $d_{1,α}$ and $d_{\infty,α}$ on each Schreier family $\mathcal{S}_α$, $α<ω_1$, with which we prove the following metric characterization of reflexivity of a Banach space $X$: $X$ is reflexive if and only if there is an $α<ω_1$, so that there is no mapping $Φ:\mathcal{S}_α\to X$ for which $$ cd_{\infty,α}(A,B)\le \|Φ(A)-Φ(B)\|\le C d_{1,α}(A,B) \text{ for all $A,B\in\mathcal{S}_α$.}$$ Secondly, we prove for separable and reflexive Banach spaces $X$, and certain countable ordinals $α$ that $\max(\text{ Sz}(X),\text{ Sz}(X^*))\le α$ if and only if $({\mathcal S}_α, d_{1,α})$…
The Banach space $\ell_1(\Z)$ admits many non-isomorphic preduals, for example, $C(K)$ for any compact countable space $K$, along with many more exotic Banach spaces. In this paper, we impose an extra condition: the predual must make the bilateral shift on $\ell_1(\Z)$ weak$^*$-continuous. This is equivalent to making the natural convolution multiplication on $\ell_1(\Z)$ separately weak*-continuous and so turning $\ell_1(\Z)$ into a dual Banach algebra. We call such preduals \emph{shift-invariant}. It is known that the only shift-invariant predual arising from the standard duality between $C_…
The aim of this note is to study some geometrical properties like diameter two properties, octahedrality and almost squareness in the setting of (symmetric) tensor product spaces. In particular, we show that the injective tensor product of two octahedral Banach spaces is always octahedral, the injective tensor product of an almost square Banach space with any Banach space is almost square, and the injective symmetric tensor product of an octahedral Banach space is octahedral.
We continue the investigation of the behaviour of octahedral norms in tensor products of Banach spaces. Firstly, we will prove the existence of a Banach space $Y$ such that the injective tensor products $l_1\widehat{\otimes}_\varepsilon Y$ and $L_1\widehat{\otimes}_\varepsilon Y$ both fail to have an octahedral norm, which solves two open problems from the literature. Secondly, we will show that in the presence of the metric approximation property octahedrality is preserved from a non-reflexive $L$-embedded Banach space taking projective tensor products with an arbitrary Banach space.
We continue the investigation of the behaviour of diameter two properties in tensor products of Banach spaces. Our main result shows that the symmetric strong diameter two property is stable by taking projective tensor products. We also prove a result for the symmetric strong diameter two property for the injective tensor product.
We prove a complex interpolation formula for the injective tensor product of vector-valued Banach function spaces satisfying certain geometric assumptions. This result unifies results of Kouba, and moreover, our approach offers an alternate proof of Kouba's interpolation formula for scalar-valued Banach function spaces.
We generalize results of Davie and Raeburn describing homotopy types of the group of invertible elements and of the set of idempotents of the projective tensor product of complex unital Banach algebras. We illustrate our results by specific examples.
In this article, the relationship between Birkhoff-James orthogonality of elementary tensors in certain tensor product spaces with the Birkhoff-James orthogonality of individual elements in their respective spaces is studied.
Let $A$ and $B$ be commutative semisimple Banach algebras. In this paper, the $BSE$ and $BED$- property of tensor Banach algebra $A{\otimes}_γ B$ with respect to the Banach algebras $A$ and $B$ are assesed. In particular, $BSE$ and $BED$- structure of vector-valued group algebra $L_{1}(G, A)$ and $C^{*}$- algebra $C_{0}(X, A)$ characterized.
We investigate the relationship between the diagonal of the Fremlin projective tensor product of a Banach lattice $E$ with itself and the 2-concavification of $E$.
In this paper, we show that four main diagonal spaces of injective tensor products are pairwise isometrically isomorphic. When E is a Banach lattice, we show that the tensor diagonal of E is a 1-unconditional basic sequence in both the n-fold injective tensor product of E and the n-fold symmetric injective tensor product of E.
Series representation in tensor products of Banach spaces
本文通过极其简单的论证证明，在向量空间张量积上赋予任意范数时，其完备化空间中的每个元素都能表示为初等张量的收敛级数。
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。
Projective and injective tensor products of Banach $L^0$-modules
本文研究$σ$-有限测度空间上Banach $L^0$-模的射影与内射张量积，并将商算子、可和族和Schauder基等技术工具推广到该框架。
Suppose that $E$ is a uniformly complete vector lattice and $p_1,..., p_n$ are positive reals. We prove that the diagonal of the Fremlin projective tensor product of $E_{(p_1)},..., E_{(p_n)}$ can be identified with $E_{(p)}$ where $p = p_1+...+p_n$ and $E_{(p)}$ stands for the $p$-concavification of $E$. We also provide a variant of this result for Banach lattices. This extends the main result of [BBPTT].
The tensor product of the Fremlin tensor product of Banach lattice algebras
该论文研究巴拿赫格代数的Fremlin投射张量积的Arens正则性，引入L-弱几乎周期泛函并刻画其性质，证明其Arens正则性的蕴含关系、反例及充分条件，还得出巴拿赫f-代数的Fremlin投射张量积的Arens正则性结论。
In this note, we study the Arens regularity of projective tensor product $A\hat{\otimes}B$ whenever $A$ and $B$ are Arens regular. We establish some new conditions for showing that the Banach algebras $A$ and $B$ are Arens regular if and only if $A\hat{\otimes}B$ is Arens regular. We also introduce some new concepts as left-weak$^*$-weak convergence property [$Lw^*wc-$property] and right-weak$^*$-weak convergence property [$Rw^*wc-$property] and for Banach algebra $A$, suppose that $A^*$ and $A^{**}$, respectively, have $Rw^*wc-$property and $Lw^*wc-$property. Then if $A^{**}$ is weakly amenab…
Von Neumann-Jordan Constants in Banach Tensor Products
研究Banach张量积在注入和射影范数下的von Neumann-Jordan常数，建立下界、刻画极值情形并确定重赋范包络，进而阐明张量范数、维数和重赋范对该几何常数的影响。
This work performs a study of the category of complete matrix-normed spaces, called matricial Banach spaces. Many of the usual constructions of Banach spaces extend in a natural way to matricial Banach spaces, including products, direct sums, and completions. Also, while the minimal matrix-norm on a Banach space is well-known, this work characterizes the maximal matrix-norm on a Banach space from the work of Effros and Ruan as a dual operator space. Moreover, building from the work of Blecher, Ruan, and Sinclair, the Haagerup tensor product is merged with the direct sum to form a Haagerup tens…
Given unital Banach algebras $A$ and $B$ and elements $a\in A$ and $b\in B$, the Drazin spectrun of $a\otimes b\in A\overline{\otimes} B$ will be fully characterized, where $A\overline{\otimes} B$ is a Banach algebra that is the completion of $A\otimes B$ with respect to a uniform crossnorm. To this end, however, first the isolated points of the spectrum of $a\otimes b\in A\overline{\otimes} B$ need to be characterized. On the other hand, given Banach spaces $X$ and $Y$ and Banach space operators $S\in L(X)$ and $T\in L(Y)$, using similar arguments the Drazin spectrum of $τ_{ST}\in L(L(Y,X))$,…
Fremlin tensor product of Banach lattices and unbounded convergences
本文研究了在局部固矢量格上的Fremlin张量积拓扑，探讨了其在Banach格中保持无界收敛性的条件。
We show that the tensor product of approximately amenable algebras need not be approximately amenable, and investigate conditions under which $A$ and $B$ being approximately amenable implies, or is implied by, $A\hat{\otimes}B$ or $A^{\#}\hat{\otimes} B^{\#}$ being approximately amenable. Our methods also enable us to prove that if $A\hat{\otimes}B$ is amenable, then so are $A$ and $B$, a result proved by Barry Johnson in 1996 under an additional assumption.
Galego and Samuel showed that if $K,L$ are metrizable, compact, Hausdorff spaces, then $C(K)\widehat{\otimes}_πC(L)$ is $c_0$-saturated if and only if it is subprojective if and only if $K$ and $L$ are both scattered. We remove the hypothesis of metrizability from their result, and extend it from the case of the two-fold projective tensor product to the general $n$-fold projective tensor product to show that for any $n\in\mathbb{N}$ and compact, Hausdorff spaces $K_1, \ldots, K_n$, $\widehat{\otimes}_{π, i=1}^n C(K_i)$ is $c_0$-saturated if and only if it is subprojective if and only if each $…
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