Hahn--Banach Theorem and Duality Theory on non-Archimedean Locally Convex Spaces

非阿基米德巴纳赫空间的雷德类结构层次

Structural Hierarchy of Reid Class of non-Archimedean Banach Spaces

本文研究非阿基米德巴纳赫空间的雷德类结构层次，提出类比阿贝尔群雷德类的巴纳赫空间类，并验证其分类定理的对应形式，同时证明某些空间无法通过迭代有界直积和完成直和表达。

Stein Domains in Banach Algebraic Geometry

In this article we give a homological characterization of the topology of Stein spaces over any valued base field. In particular, when working over the field of complex numbers, we obtain a characterization of the usual Euclidean (transcendental) topology of complex analytic spaces. For non-Archimedean base fields the topology we characterize coincides with the topology of the Berkovich analytic space associated to a non-Archimedean Stein algebra. Because the characterization we used is borrowed from a definition in derived geometry, this work should be read as a contribution towards the found…

Non-Archimedean Unitary Operators

We describe a subclass of the class of normal operators on Banach spaces over non-Archimedean fields (A. N. Kochubei, J. Math. Phys. 51 (2010), article 023526) consisting of operators whose properties resemble those of unitary operators. In particular, an analog of Stone's theorem about one-parameter groups of unitary operators is proved.

Non-Archimedean analytic geometry as relative algebraic geometry

We show that Berkovich analytic geometry can be viewed as relative algebraic geometry in the sense of Toën--Vaquié--Vezzosi over the category of non-Archimedean Banach spaces. For any closed symmetric monoidal quasi-abelian category we can define a topology on certain subcategories of the of the category of affine schemes with respect to this category. By examining this topology for the category of Banach spaces we recover the G-topology or the topology of admissible subsets on affinoids which is used in analytic geometry. This gives a functor of points approach to non-Archimedean analytic geo…

非阿基米德算子值域的Borel复杂度

The Borel complexity of non-Archimedean operator ranges

该研究分类了波兰非阿基米德非平凡赋值域上可分巴拿赫空间非闭算子值域的Borel复杂度类，还补充了弗雷歇空间对应的额外复杂度类。

Non-Archimedean and p-adic Functional Welch Bounds

We prove the non-Archimedean (resp. p-adic) Banach space version of non-Archimedean (resp. p-adic) Welch bounds recently obtained by M. Krishna. More precisely, we prove following results. 1. Let $\mathbb{K}$ be a non-Archimedean (complete) valued field satisfying $\left|\sum_{j=1}^{n}λ_j^2\right|=\max_{1\leq j \leq n}|λ_j|^2$ for all $ λ_j \in \mathbb{K}, 1\leq j \leq n$, for all $n \in \mathbb{N}.$ Let $\mathcal{X}$ be a $d$-dimensional non-Archimedean Banach space over $\mathbb{K}$. If $\{τ_j\}_{j=1}^n$ is any collection in $\mathcal{X}$ and $\{f_j\}_{j=1}^n$ is any collection in $\mathcal{…

Characterisation of the Berkovich Spectrum of the Banach Algebra of Bounded Continuous Functions

For a complete valuation field k and a topological space X, we prove the universality of the underlying topological space of the Berkovich spectrum of the Banach k-algebra Cbd(X,k) of bounded continuous k-valued functions on X. This result yields three applications: a partial solution to an analogue of Kaplansky conjecture for the automatic continuity problem over a local field, comparison of two ground field extensions of Cbd(X,k), and non-Archimedean Gel'fand theory.

Global analytic geometry

We define a new type of valuation of a ring that combines the notion of Krull valuation with that of multiplicative seminorm. This definition partially restores the broken symmetry between archimedean and non-archimedean valuations. This also allows us to define a notion of global analytic space that reconciles Berkovich's notion of analytic space of a (Banach) ring with Huber's notion of non-archimedean analytic space. After defining natural generalized valuation spectra and computing the spectrum of Z and Z[X], we define analytic spectra and sheaves of analytic functions on them.

关于Banach环谱的层性性质

On the Sheafyness Property of Spectra of Banach Rings

本文通过引入导出有理局部化，为非阿基米德Banach环构建了同伦Huber谱Spa^h(R)及严格正合的导出结构层，从而在传统结构层失效时利用导出几何工具研究其几何性质。

Berkovich spectra of elements in Banach Rings

Adapting the notion of the spectrum $Σ_a$ for an element $a$ in an ultrametric Banach algebra (as defined by Berkovich), we introduce and briefly study the Berkovich spectrum $σ^{Ber}_R(u)$ of an element $u$ in a Banach ring $R$. This spectrum is a compact subset of the affine analytic space $A_Z^1$ over $Z$, and the later can be identified with the "equivalence classes" of all elements in all complete valuation fields. If $R$ is generated by $u$ as a unital Banach ring, then $σ^{Ber}_R(u)$ coincides with the spectrum of $R$ (as defined by Berkovich). If $R$ is a unital complex Banach algebra,…

Pentes des fibrés vectoriels adéliques sur un corps global

At the end of the twentieth century, J.-B. Bost developped a slope theory of hermitian vector bundles over number fields. A new method of diophantine approximation, the so-called slope method, has emerged from his research. Our article proposes a generalisation to adelic vector bundles over global fields. The norms at the archimedean places are no longer supposed to be hermitian. The link with adelic successive minima is also mentioned. To get these results, we use several concepts from the geometry of finite dimensional Banach spaces.

On the restriction of representations of $\GL_2(F)$ to a Borel subgroup

Let $F$ be a non-Archimedean local field and let $p$ be the residual characteristic of $F$. Let $G=GL_2(F)$ and let $P$ be a Borel subgroup of $G$. In this paper we study the restriction of irreducible representations of $G$ on $E$-vector spaces to $P$, where $E$ is an algebraically closed field of characteristic $p$. We show that in a certain sense $P$ controls the representation theory of $G$. We then extend our results to smooth $\oK[G]$- modules of finite length and unitary $K$-Banach space representations of $G$, where $\oK$ is the ring of integers of a complete discretely valued field $K…