Closed graph theorems for bornological spaces

The aim of this paper is that of discussing Closed Graph Theorems for bornological vector spaces in a way which is accessible to non-experts. We will see how to easily adapt classical arguments of functional analysis over $\mathbb{R}$ and $\mathbb{C}$ to deduce Closed Graph Theorems for bornological vector spaces over any complete, non-trivially valued field, hence encompassing the non-Archimedean case too. We will end this survey by discussing some applications. In particular, we will prove de Wilde's Theorem for non-Archimedean locally convex spaces and then deduce some results about the automatic boundedness of algebra morphisms for a class of bornological algebras of interest in analytic geometry, both Archimedean (complex analytic geometry) and non-Archimedean

Dagger Geometry As Banach Algebraic Geometry

In this article, we apply the approach of relative algebraic geometry towards analytic geometry to the category of bornological and Ind-Banach spaces (non-Archimedean or not). We are able to recast the theory of Grosse-Kl\"onne dagger affinoid domains with their weak G-topology in this new language. We prove an abstract recognition principle for the generators of their standard topology (the morphisms appearing in the covers). We end with a sketch of an emerging theory of dagger affinoid spaces over the integers, or any Banach ring, where we can see the Archimedean and non-Archimedean worlds coming together

Analytic geometry over F_1 and the Fargues-Fontaine curve

This paper develops a theory of analytic geometry over the field with one element. The approach used is the analytic counter-part of the Toen-Vaquie theory of schemes over F_1, i.e. the base category relative to which we work out our theory is the category of sets endowed with norms (or families of norms). Base change functors to analytic spaces over Banach rings are studied and the basic spaces of analytic geometry (like polydisks) are recovered as a base change of analytic spaces over F_1. We end by discussing some applications of our theory to the theory of the Fargues-Fontaine curve and to the ring Witt vectors.Comment: Small corrections have been made in the last section of the paper and some typos have been correcte

On the uniqueness of invariant states

Given an abelian group G endowed with a T-pre-symplectic form, we assign to it a symplectic twisted group *-algebra W_G and then we provide criteria for the uniqueness of states invariant under the ergodic action of the symplectic group of automorphism. As an application, we discuss the notion of natural states in quantum abelian Chern-Simons theory.Comment: 29 pages -- accepted in Advances in Mathematic

On a generalization of affinoid varieties

In this thesis we develop the foundations for a theory of analytic geometry over a valued field, uniformly encompassing the case when the base field is equipped with a non-archimedean valuation and the case when it has an archimedean one. We will use the efficient language of bornological algebras to reach this goal. Since, at our knowledge, there is not a strong literature on bornological algebras on which we can base our results, we need to start from scratch with theory of bornological algebras and we develop it as far as we need for our scope. In this way we can construct our theory of analytic spaces taking as building blocks affinoid dagger algebras, i.e. bornological algebras isomorphic to quotients of the algebras of germs of analytic functions on polycylinders. It turns out that we can show for the category of dagger affinoid algebras the analogous of all the main results of the category of affinoid algebras and hence we obtain a good theory of dagger affinoid spaces, which behave very similarly to affinoid spaces used to construct rigid analytic spaces. We underline that the archimedean case has special features that we study in the first section of the fifth chapter, and that these special properties enable us to prove the generalization, in archimedean context, of the main result of affinoid algebras theory: the Gerritzen-Grauert theorem (we remark that our proof is inspired by the new proof of Gerritzen-Grauert theorem in Berkovich geometry given by Temkin). With a good affine theory at our disposal we end this thesis by constructing the global theory of dagger analytic spaces. We use the theory of Berkovich nets and hence a Berkovich-like approach to the globalization. In this way we obtain the category of dagger analytic spaces and we study the relations of the spaces we found with the ones already present in literature. In particular we see that our spaces are very strongly related to dagger spaces of Grosse-Klonne, and that, in the archimedean case, the category of classical complex analytic spaces embeds fully faithfully in the category of complex dagger analytic spaces. In conclusion, in this work we obtain a complete affinoid theory of dagger spaces over any valued field and we show that on this theory can be developed a good theory of dagger analytic spaces, which deserves to be strengthened in the future