AI 中文总结
该研究探讨射影张量积最优积分表示的可数性与拓扑问题，通过构造仿射嵌入证明存在仅具不可数最优积分表示的张量，并明确不同拓扑下范数可达张量类的等价性及相关空间性质。
AI 中文摘要
我们研究射影张量积中的最优积分表示，聚焦于两个问题：它们是否总能被可数最优分解替代，以及由此产生的概念是否依赖于单位球乘积上所用的博雷尔拓扑。我们证明，每个无限维巴拿赫空间$X$都存在一个等价范数，记所得空间为$Z$，则存在一个仿射同胚嵌入$\Psi: \mathcal{P}([0,1]) \to S_{Z \widehat\otimes_\pi Z}$，满足$\Psi(\mathcal{P}([0,1]) ) \subseteq \operatorname{INA}_\pi(Z\widehat\otimes_\pi Z)$，且$\Psi(\alpha) \in \operatorname{NA}_\pi(Z \widehat\otimes_\pi Z)$当且仅当$\alpha$具有可数支撑。特别地，这意味着存在张量$u \in \operatorname{INA}_\pi(Z\widehat\otimes_\pi Z) \setminus \operatorname{NA}_\pi(Z\widehat\otimes_\pi Z)$，且其见证测度可选为非原子拉东概率测度。该构造将$\mathcal P([0,1])$的一个仿射副本实现为$B_Z$的一个暴露面，并将对角勒贝格耦合与乘积测度的可数混合进行比较。我们还证明，积分射影范数可达的范数版本、弱版本以及（在对偶空间上的）弱星版本定义了同一类张量。由此可得，每个具有逼近性质的无限维可分自反巴拿赫空间$X$都存在一个等价范数，使得对于所得空间$Z$，有$\operatorname{NA}_\pi(Z\widehat\otimes_\pi Z) \subsetneq \operatorname{INA}_\pi(Z\widehat\otimes_\pi Z) = Z\widehat\otimes_\pi Z$。
英文摘要
We study optimal integral representations in projective tensor products, focusing on two questions: whether they can always be replaced by countable optimal decompositions, and whether the resulting notion depends on the Borel topology used on the product of the unit balls. We show that every infinite-dimensional Banach space $X$ admits an equivalent norm for which, denoting the resulting space by $Z$, there exists an affine homeomorphic embedding \[ Ψ: \mathcal{P}([0,1]) \to S_{Z \widehat\otimes_π Z} \] such that $Ψ(\mathcal{P}([0,1]) ) \subseteq \operatorname{INA}_π(Z\widehat\otimes_π Z)$ and \[ Ψ(α) \in \operatorname{NA}_π(Z \widehat\otimes_π Z) \iff \text{$α$ is countably supported}. \] In particular, this implies that there exists a tensor \[ u \in \operatorname{INA}_π(Z\widehat\otimes_π Z) \setminus \operatorname{NA}_π(Z\widehat\otimes_π Z) \] and the witnessing measure may be chosen to be a nonatomic Radon probability measure. The construction realizes an affine copy of $\mathcal P([0,1])$ as an exposed face of $B_Z$ and compares the diagonal Lebesgue coupling with countable mixtures of product measures. We also prove that the norm, weak, and--on dual spaces--weak-star versions of integral projective norm attainment define the same class of tensors. Consequently, every infinite-dimensional separable reflexive Banach space $X$ with the approximation property admits an equivalent norm such that, for the resulting space $Z$, $\operatorname{NA}_π(Z\widehat\otimes_π Z) \subsetneq \operatorname{INA}_π(Z\widehat\otimes_π Z) = Z\widehat\otimes_π Z$.
Comments18 pages
