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Vol 8, No 2 (2016)
- Year: 2016
- Articles: 5
- URL: https://journals.rcsi.science/2070-0466/issue/view/12497
Research Articles
Invariant ultrametrics and Markov processes on the finite adèle ring of ℚ
Abstract
This article introduces several rotation and additive invariant ultrametrics on the finite adèle ring Af of the rational numbers ℚ. Symmetry, regularity and uniqueness properties of these ultrametrics are provided. With these non-Archimedean metrics at hand it is possible to define a wide class of rotation and additive invariant Markov processes on Af.
89-114
The Corona problem on a complete ultrametric algebraically closed field
Abstract
Let IK be a complete ultrametric algebraically closed field and let A be the Banach IK-algebra of bounded analytic functions in the ”open” unit disk D of IK provided with the Gauss norm. Let Mult(A, ‖. ‖) be the set of continuous multiplicative semi-norms of A provided with the topology of simple convergence, let Multm(A, ‖. ‖) be the subset of the ϕ ∈ Mult(A, ‖. ‖) whose kernel is amaximal ideal and let Mult1(A, ‖. ‖) be the subset of the ϕ ∈ Mult(A, ‖. ‖) whose kernel is a maximal ideal of the form (x − a)A with a ∈ D. By analogy with the Archimedean context, one usually calls ultrametric Corona problem the question whether Mult1(A, ‖. ‖) is dense in Multm(A, ‖. ‖). In a previous paper, it was proved that when IK is spherically complete, the answer is yes. Here we generalize this result to any algebraically closed complete ultrametric field, which particularly applies to ℂp. On the other hand, we also show that the continuous multiplicative seminorms whose kernel are neither a maximal ideal nor the zero ideal, found by Jesus Araujo, also lie in the closure of Mult1(A, ‖. ‖), which suggest that Mult1(A, ‖. ‖)might be dense in Mult(A, ‖. ‖).
115-124
Non-archimedean transportation problems and Kantorovich ultra-norms
Abstract
We study a non-archimedean (NA) version of transportation problems and introduce naturally arising ultra-norms which we call Kantorovich ultra-norms. For every ultra-metric space and every NA valued field (e.g., the field Qp of p-adic numbers) the naturally defined inf-max cost formula achieves its infimum. We also present NA versions of the Arens-Eells construction and of the integer value property. We introduce and study free NA locally convex spaces. In particular, we provide conditions under which these spaces are normable by Kantorovich ultra-norms and also conditions which yield NA versions of Tkachenko-Uspenskij theorem about free abelian topological groups.
125-148
149-159
A local mean value theorem for functions on non-archimedean field extensions of the real numbers
Abstract
In this paper, we review the definition and properties of locally uniformly differentiable functions on N, a non-Archimedean field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order. Then we define and study n-times locally uniform differentiable functions at a point or on a subset of N. In particular, we study the properties of twice locally uniformly differentiable functions and we formulate and prove a local mean value theorem for such functions.
160-175