This is an announcement for the paper “Polish spaces of Banach spaces. Complexity of isometry classes and generic properties” by Marek Cúthhttps://arxiv.org/search/math?searchtype=author&query=C%C3%BAth%2C+M, Martin Doležalhttps://arxiv.org/search/math?searchtype=author&query=Dole%C5%BEal%2C+M, Michal Douchahttps://arxiv.org/search/math?searchtype=author&query=Doucha%2C+M, Ondřej Kurkahttps://arxiv.org/search/math?searchtype=author&query=Kurka%2C+O.
Abstract: We present and thoroughly study natural Polish spaces of separable Banach spaces. These spaces are defined as spaces of norms, resp. pseudonorms on the countable infinite-dimensional rational vector space. We provide an exhaustive comparison of these spaces with the admissible topologies recently introduced by Godefroy and Saint-Raymond and show that Borel complexities differ little with respect to these two different topological approaches. We then focus mainly on the Borel complexities of isometry classes of the most classical Banach spaces. We prove that the infinite-dimensional Hilbert space is characterized as the unique separable infinite-dimensional Banach space whose isometry class is closed, and also as the unique separable infinite-dimensional Banach space whose isomorphism class is $F_\sigma$. For $p\in\left[1,2\right)\cup\left(2,\infty\right)$, we show that the isometry classes of $L_p[0,1]$ and $\ell_p$ are $G_\delta$-complete and $F_{\sigma\delta}$-complete, respectively. Then we show that the isometry class of the Gurari\uı space is $G_\delta$-complete and the isometry class of $c_0$ is $F_{\sigma\delta}$-complete. The isometry class of the former space is moreover proved to be dense $G_\delta$. Additionally, we compute the complexities of many other natural classes of Banach spaces; for instance, $\mathcal{L}_{p,\lambda+}$-spaces, for $p,\lambda\geq 1$, are shown to be $G_\delta$, superreflexive spaces are shown to be $F_{\sigma\delta}$, and spaces with local $\Pi$-basis structure are shown to be $\boldsymbol{\Sigma}^0_6$. The paper is concluded with many open problems and suggestions for a future research.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1912.03994