Related papers: Higher projective tensor products of $c_0$

In the present paper we prove that the $3$-fold projective tensor product of $c_0$, $c_{0} \widehat{\otimes}_\pi c_{0}\widehat{\otimes}_\pi c_{0}$, is not isomorphic to a subspace of $c_{0} \widehat{\otimes}_\pi c_{0}$. In particular, this…

In this paper we prove that the $3$-fold injective tensor product $\ell_1 \widehat{\otimes}_\varepsilon \ell_1 \widehat{\otimes}_\varepsilon \ell_1 $ is not isomorphic to any subspace of $\ell_1 \widehat{\otimes}_\varepsilon \ell_1$. This…

We prove that for infinite, countable, compact, Hausdorff spaces $K,L$, $C(K)\widehat{\otimes}_\pi C(L)$ is isomorphic to exactly one of the spaces $C(\omega^{\omega^\xi})\widehat{\otimes}_\pi C(\omega^{\omega^\zeta})$, $0\leqslant…

The aim of this note is to prove that, given two superreflexive Banach spaces $X$ and $Y$, then $X\widehat{\otimes}_\pi Y$ is superreflexive if and only if either $X$ or $Y$ is finite-dimensional. In a similar way, we prove that…

We show that, in some cases, the projective and the injective tensor products of two Banach spaces do not have the Dunford-Pettis property (DPP). As a consequence, we obtain that $(c_0\hat{\otimes}_\pi c_0)^{**}$ fails the DPP. Since…

In this paper, we study the (uniform) strong subdifferentiability of the norms of the Banach spaces $\mathcal{P}(^N X, Y^*)$, $X \hat{\otimes}_\pi \cdots \hat{\otimes}_\pi X$ and $\hat{\otimes}_{\pi_s,N} X$. Among other results, we…

We prove that there exists a finite-dimensional Banach space $X$ such that $L_1^\mathbb C([0,1])\widehat{\otimes}_\varepsilon X$ fails the strong diameter two property and $L_\infty^\mathbb C([0,1])\widehat{\otimes}_\pi X^*$ fails to have…

We give a sufficient condition for a pair of Banach spaces $(X,Y)$ to have the following property: whenever $W_1 \subseteq X$ and $W_2 \subseteq Y$ are sets such that $\{x\otimes y: \, x\in W_1, \, y\in W_2\}$ is weakly precompact in the…

We prove that, given two Banach spaces $X$ and $Y$ and bounded, closed convex sets $C\subseteq X$ and $D\subseteq Y$, if a nonzero element $z\in \overline{\mathrm{co}}(C\otimes D)\subseteq X\widehat{\otimes}_\pi Y$ is a preserved extreme…

Galego and Samuel showed that if $K,L$ are metrizable, compact, Hausdorff spaces, then $C(K)\widehat{\otimes}_\pi C(L)$ is $c_0$-saturated if and only if it is subprojective if and only if $K$ and $L$ are both scattered. We remove the…

Given two Banach spaces $X$ and $Y$, we analyze when the projective tensor product $X\widehat{\otimes}_\pi Y$ has Corson's property (C) or is weakly Lindel\"of determined (WLD), subspace of a weakly compactly generated (WCG) space or…

We prove that the norm of $X\widehat{\otimes}_\pi Y$ is SSD if either $X=\ell_p(I)$ for $p>2$ and $Y$ is a finite-dimensional Banach space such that the modulus of convexity is of power type $q<p$ (e.g. if $Y^*$ is a subspace of $L_q$) or…

In this paper we analyse when every element of $X\widehat{\otimes}_\pi Y$ attains its projective norm. We prove that this is the case if $X$ is the dual of a subspace of a predual of an $\ell_1(I)$ space and $Y$ is $1$-complemented in its…

We study the property of being strongly weakly compactly generated (and some relatives) in projective tensor products of Banach spaces. Our main result is as follows. Let $1<p,q<\infty$ be such that $1/p+1/q\geq 1$. Let $X$ (resp., $Y$) be…

Let $X$ be a complex Banach space and $x,y\in X$. By definition, we say that $x$ is Birkhoff-James orthogonal to $y$ if $ \|x+\lambda y\|_{X} \geq \|x\|_{X}$ for all $\lambda \in \mathbb{C}$. We prove that $x$ is Birkhoff-James orthogonal…

In this paper, we introduce a concept of norm-attainment in the projective symmetric tensor product $\widehat{\otimes}_{\pi,s,N} X$ of a Banach space $X$, which turns out to be naturally related to the classical norm-attainment of…

The aim of this note is to obtain results about when the norm of a projective tensor product is strongly subdifferentiable. We prove that if $X\hat{\otimes}_\pi Y$ is strongly subdifferentiable and either $X$ or $Y$ has the metric…

For $C^{*}$-algebras $A$ and $B$, the operator space projective tensor product $A\hat{\otimes}B$ and the Banach space projective tensor product $A\otimes_{\gamma}B$ are shown to be symmetric. We also show that $A\hat{\otimes}B$ is weakly…

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 $l_1\widehat{\otimes}_\varepsilon Y$…

In this short note, we prove that the set of elementary tensors is weakly closed in the projective tensor product of two Banach spaces. As a result, we are able to answer a question from the literature proving that if $(x_n) \subset X$ and…