Optimal integral representations in projective tensor products: countability and topology

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 $$\Psi : \mathcal{P}([0,1]) \to S_{Z \widehat\otimes_{\pi} Z}$$ such that $\Psi (\mathcal{P}([0,1]) ) \subseteq \operatorname{INA}_\pi (Z\widehat\otimes_{\pi} Z)$ and $$\Psi (\alpha) \in \operatorname{NA}_\pi(Z \widehat\otimes_{\pi} Z) \iff \text{$\alpha$ is countably supported}.$$ In particular, this implies that there exists a tensor $$u \in \operatorname{INA}_{\pi}(Z\widehat\otimes_{\pi} Z) \setminus \operatorname{NA}_{\pi}(Z\widehat\otimes_{\pi} Z)$$ 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 $Z$, $\operatorname{NA}_{\pi}(Z\widehat\otimes_{\pi} Z) \subsetneq \operatorname{INA}_{\pi}(Z\widehat\otimes_{\pi} Z) = Z\widehat\otimes_{\pi} Z$.