Topological properties in tensor products of Banach spaces

TL;DR

This paper investigates the topological properties of tensor products of Banach spaces, establishing conditions under which these products have properties like Corson's property (C), WLD, or are subspaces of WCG or Hilbert generated spaces, with implications for measure theory.

Contribution

It provides new characterizations of when tensor products of Banach spaces possess certain topological properties and explores their stability under different tensor product types, extending previous results.

Findings

$X \u2208 ext{WLD}$ and $Y \u2208 ext{WLD}$ with operators of separable range imply $X \u2286 ext{WLD}$ in tensor product.

Tensor products of $\u2113_p(\u0393)$ and $\u2113_q(\u0393)$ are subspaces of Hilbert generated spaces under certain conditions.

The stability of property (C) and WLD in injective tensor products relates to measures having countable Maharam type.

Abstract

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 subspace of a Hilbert generated space. For instance, we show that: (i) $X\widehat{\otimes}_\pi Y$ is WLD if and only if both $X$ and $Y$ are WLD and all operators from $X$ to $Y^*$ and from $Y$ to $X^*$ have separable range; (ii) $X\widehat{\otimes}_\pi Y$ is subspace of a WCG space if the same holds for both $X$ and $Y$ under the assumption that every operator from $X$ to $Y^*$ is compact; (iii) $\ell_p(\Gamma)\widehat{\otimes}_\pi \ell_q(\Gamma)$ is subspace of a Hilbert generated space for any $1< p,q<\infty$ such that $1/p+1/q<1$ and for any infinite set $\Gamma$. We also pay attention to the injective tensor product $X\widehat{\otimes}_\varepsilon Y$. In this case, the stability of property (C) and the property of being WLD turn out to be closely related to the condition that all regular Borel probability measures on the dual ball have countable Maharam type. Along this way, we generalize a result of Plebanek and Sobota that if $K$ is a compact space such that $C(K\times K)$ has property (C), then all regular Borel probability measures on $K$ have countable Maharam type. This generalization provides a consistent negative answer to a question of Ruess and Werner about the preservation of the $w^*$-angelicity of the dual unit ball under injective tensor products.