1,720,967 research outputs found
Closed graph theorems for bornological spaces
Full text linkThe aim of this paper is that of discussing closed graph theorems for bornological vector spaces in a self-contained way, hoping to make the subject more accessible to non-experts. We will see how to easily adapt classical arguments of functional analysis over R and 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
Theorems A and B for dagger quasi-Stein spaces
No full textIn this article, we use the homological methods of the theory of quasi-abelian categories and results from functional analysis to prove Theorems A and B for (a broad sub-class of) dagger quasi-Stein spaces. In particular, we show how to deduce these theorems from the vanishing, under certain hypothesis, of the higher derived functors of the projective limit functor. Our strategy of the proof generalizes and puts in a more formal framework Kiehl's proof for rigid quasi-Stein spaces
On the sheafyness property of spectra of Banach rings
Full text linkLet (Formula presented.) be a non-Archimedean Banach ring, satisfying some mild technical hypothesis that we will specify later on. We prove that it is possible to associate to (Formula presented.) a homotopical Huber spectrum (Formula presented.) via the introduction of the notion of derived rational localization. The spectrum so obtained is endowed with a derived structural sheaf (Formula presented.) of simplicial Banach algebras for which the derived C̆ech–Tate complex is strictly exact. Under some hypothesis, we can prove that there is a canonical morphism of underlying topological spaces (Formula presented.) that is a homeomorphism in some well-known examples of non-sheafy Banach rings, where (Formula presented.) is the usual Huber spectrum of (Formula presented.). This permits the use of the tools from derived geometry to understand the geometry of (Formula presented.) in cases when the classical structure sheaf (Formula presented.) is not a sheaf
Derived Analytic Geometry for Z-Valued Functions Part I: Topological Properties
No full textWe study the Banach algebras C(X,R) of continuous functions from a compact Hausdorff topological space X to a Banach ring R whose topology is discrete. We prove that the Berkovich spectrum of C(X,R) is homeomorphic to ζ(X)×M(R), where ζ(X) is the Banaschewski compactification of X and M(R) is the Berkovich spectrum of R. We study how the topology of the spectrum of C(X,R) is related to the notion of homotopy Zariski open embedding used in derived geometry. We find that the topology of ζ(X) can be easily reconstructed from the homotopy Zariski topology associated with C(X,R). We also prove some results about the existence of Schauder bases on C(X,R) and a generalization of the Stone–Weierstrass Theorem, under suitable hypotheses on X and R
Rigidity for rigid analytic motives
Full text linkIn this paper we prove the Rigidity Theorem for motives of rigid analytic varieties over a non-Archimedean valued field. We prove this theorem both for motives with transfers and without transfers in a relative setting. Applications include the construction of étale realization functors, an upgrade of the known comparison between motives with and without transfers and an upgrade of the rigid analytic motivic tilting equivalence, extending them to-coefficients
On the uniqueness of invariant states
No full textGiven an abelian group G endowed with a T=R/Z-pre-symplectic form, we assign to it a symplectic twisted group ⁎-algebra WG 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
Analytic geometry over F1 and the Fargues-Fontaine curve
No full textThis paper develops a theory of analytic geometry over the field with one element. The approach used is the analytic counter-part of the Toën-Vaquié theory of schemes over F1, 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 (e.g. polydisks) are recovered as a base change of analytic spaces over F1. We conclude by discussing some applications of our theory to the theory of the Fargues-Fontaine curve and to the ring of Witt vectors
On a family of circulant matrices for quasi-cyclic low-density generator matrix codes
No full textWe present a new class of sparse and easily invertible circulant matrices that can have a sparse inverse though not being permutation matrices. Their study is useful in the design of quasi-cyclic low-density generator matrix codes that are able to join the inner structure of quasi-cyclic codes with sparse generator matrices, so limiting the number of elementary operations needed for encoding. Circulant matrices of the proposed class permit to hit both targets without resorting to identity or permutation matrices that may penalize the code minimum distance and often cause significant error floors. © 2011 IEEE
Stein domains in Banach algebraic geometry
No full textIn 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 derived perspective on analytic geometry
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