The relative separable closure of a valued field in its completion

We consider the completion of a topological field whose topology is defined by the valuations of a dependence class of valuation rings. We characterize the separable closure of the field in its completion through henselization of those valuation rings. Actually, we prove that this relative separable closure is the fixed field of some particular subgroup of continuous automorphisms of the total separable closure. We prove also that this relative separable closure has some properties which are analogous to those of the henselization.This paper is part of the author's doctoral dissertation and was finished during his stay in Konstanz though the CNP_q/GMD program.The results 2.11 and 2.12 were firstly announced in C.R. Acad. Sc. Paris, t.283 (1977).     

Some Applications of Rademacher Sequences in Banach Lattices

We give several applications of Rademacher sequences in abstract Banach lattices. We characterise those Banach lattices with an atomic dual in terms of weak* sequential convergence. We give an alternative treatment of results of Rosenthal, generalising a classical result of Pitt, on the compactness of operators from L^p into L^q. Finally we generalise earlier work of ours by showing that, amongst Banach lattices F with an order continuous norm, those having the property that the linear span of the positive compact operators fromE into F is complete under the regular norm for all Banach lattices E are precisely the atomic lattices.     

The smallness problem for -algebras

Akemann showed that any von Neumann algebra with a weak^* separable dual space has a faithful normal representation on a separable Hilbert space. He posed the question: If a C^*-algebra has a weak^* separable state space, must it have a faithful representation on a separable Hilbert space? Wright solved this question negatively and showed that a unital C^*-algebra has the weak^* separable state space if and only if it has a unital completely positive map, into a type I factor on a separable Hilbert space, whose restriction to the self-adjoint part induces an order isomorphism. He called such a C^*-algebra almost separably representable. We say that a unital C^*-algebra is small if it has a unital complete isometry into a type I factor on a separable Hilbert space. In this paper we show that a unital C^*-algebra is small if and only if the state spaces of all n by n matrix algebras over the C^*-algebra are weak^*-separable. It is natural to ask whether almost separably representable algebras are small or not. We settle this question positively for simple C^*-algebras but the general question remains open.     

Completions of Convexly Ordered Valuation Rings

We prove that every convexly ordered valuation ring has a unique completion as a uniform space, which furthermore is a convexly ordered valuation ring. In addition, we give a model theoretic characterisation of complete convexly ordered valuation rings, and give a necessary and sufficient condition for the completion of a convexly ordered valuation ring to be a real closed ring. Mathematics Subject Classification : 03C60, 13L05, 54E15.     

Existence and nonexistence of hypercyclic semigroups

In these notes we provide a new proof of the existence of a hypercyclic uniformly continuous semigroup of operators on any separable infinite-dimensional Banach space that is very different from--and considerably shorter than--the one recently given by Bermúdez, Bonilla and Martinón. We also show the existence of a strongly dense family of topologically mixing operators on every separable infinite-dimensional Fréchet space. This complements recent results due to Bès and Chan. Moreover, we discuss the Hypercyclicity Criterion for semigroups and we give an example of a separable infinite-dimensional locally convex space which supports no supercyclic strongly continuous semigroup of operators.

     

Finite Codimension Subfields of a Field Complete with Respect to a Real Valuation

Let K be a field complete with respect to a real valuation v and not algebraically closed. We will show that every finite codimension subfield of K is closed in the v-adic topology if and only if the degree of imperfection of K is finite. It follows that there are incomplete finite codimension subfields of K when the degree of imperfection of K is infinite. These examples exhibit other interesting pathologies. We are able to give a necessary (and in the case of a discrete real valuation also sufficient) condition for a given finite codimension subfield to be complete. Finally, we give some applications to fields of Laurent series.

Communicated by A. Prestel.     

On the topological structure of the Levi-Civita field

Two topologies on the Levi-Civita field R will be studied: the valuation topology induced by the order on the field, and another weaker topology induced by a family of seminorms, which we will call weak topology. We show that each of the two topologies results from a metric on R, that the valuation topology is not a vector topology while the weak topology is, and that R is complete in the valuation topology while it is not in the weak topology. Then the properties of both topologies will be studied in details; in particular, we give simple characterizations of open, closed, and compact sets in both topologies.     

The ideal property, the projection property, continuous fields and crossed products

Let A be the C^∗-algebra associated to an arbitrary continuous field of C^∗-algebras. We give a necessary and sufficient condition for A to have the ideal property and, if moreover A is separable, we give a necessary and sufficient condition for A to have the projection property. Some applications of these results are given. We also prove that “many” crossed products of commutative C^∗-algebras by discrete, amenable groups have the projection property, generalizing some of our previous results.     

Lattices with Complemented Tolerance Lattice

We characterize lattices with a complemented tolerance lattice. As an application of our results we give a characterization of bounded weakly atomic modular lattices with a Boolean tolerance lattice.     

在再生核空间中方程AuBu+Cu＝f的求解方法

$ 1 引言 本文研究下面一类非线性算子方程求解问题 AμBμ Cμ=f, (1.1)其中f,μ∈W(Ω),μ(O)=1,||f ||=1,A,B,C∈(W(Ω)→W(Ω)),(W(Ω)→W(Ω))是W(Ω)到W(Ω)的连续线性算子空间,W(Ω)是定义在Ω域上的(Ω是实数域R的有界域)再生核空间。 本文是在再生核空间上,通过将一维非线性算子方程(1.1)转化为二维线性算子方     

Uniqueness of Solutions to Hyperbolic Balance Laws in Several Space Dimensions

We introduce the concept of modular family of entropy vectors for general r x r systems of balance laws. We then define the notion of entropy solution to the Cauchy problem compatible with the modular family, assuming that the system admits such a family. We show that this concept reduces to the usual one, introduced by S.N. Kruzkov, in the scalar case and when we restrict ourself to Lipschitz continuous solutions. We also show how the compatibility condition appears in the cases of symmetric systems, 2 x 2 psystems and equations of hyperelasticity and electromagnetism, the last two considered earlier by C.M. Dafermos. We demonstrate that generalized Oleinik's condition implies our compatibility condition in the case of symmetric systems. We prove the uniqueness and stability relatively to the initial data of the entropy solutions compatible with the modular family. This theorem has as corollary uniqueness results due to O.A. Oleinik, S.N. Kruzkov, A.E. Hurd, R.J. DiPerna and C. Bardos. We give also two uniqueness theorems to solutions of equations of hyperelasticity and electromagnetism.     

Compactification of Drinfeld modular varieties and Drinfeld modular forms of arbitrary rank

We give an abstract characterization of the Satake compactification of a general Drinfeld modular variety. We prove that it exists and is unique up to unique isomorphism, though we do not give an explicit stratification by Drinfeld modular varieties of smaller rank which is also expected. We construct a natural ample invertible sheaf on it, such that the global sections of its k-th power form the space of (algebraic) Drinfeld modular forms of weight k. We show how the Satake compactification and modular forms behave under all natural morphisms between Drinfeld modular varieties; in particular we define Hecke operators. We give explicit results in some special cases.     

条件期望作为初等算子在冯-代数上的描述

定义并研究了冯-代数.条件期望基于冯-代数的描述,即作为初等算子的良好性质,会较之一般代数简洁许多.这是因为冯-代数包含一些特殊的子代数.给出了此类代数上置信的初等条件期望的描述及其最小存在的充要条件.并且定义了指标冯-有限条件期望.作为以上结果的推论,得出了条件期望指标有限的充分必要条件和一个重要不等式.     

Ramification of Valuations and Local Rings in Positive Characteristic

In characteristic zero, local monomialization is true along any valuation. However, we have recently shown that local monomialization is not always true in positive characteristic, even in two dimensional algebraic function fields. In this paper we show that local monomialization is true for defectless extensions of two dimensional excellent local rings, extending an earlier result of Piltant and the author for two dimensional algebraic function fields over an algebraically closed field. We also give theorems showing that in many cases there are good stable forms of the extension of associated graded rings in a finite separable field extension.     

Continuity properties of Markov semigroups and their restrictions to invariant <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-spaces

We consider Markov semigroups on the cone of positive finite measures on a complete separable metric space. Such a semigroup extends to a semigroup of linear operators on the vector space of measures that typically fails to be strongly continuous for the total variation norm. First we characterise when the restriction of a Markov semigroup to an invariant L ¹-space is strongly continuous. Aided by this result we provide several characterisations of the subspace of strong continuity for the total variation norm. We prove that this subspace is a projection band in the Banach lattice of finite measures, and consequently obtain a direct sum decomposition.     

Order Continuity of Locally Compact Boolean Algebras

The main purpose of this paper is to exhibit the decisive role that order continuity plays in the structure of locally compact Boolean algebras as well as in that of atomic topological Boolean algebras. We prove that the following three conditions are equivalent for a topological Boolean algebra B: (1) B is compact; (2) B is locally compact, Boolean complete, order continuous; (3) B is Boolean complete, atomic and order continuous. Note that under the discrete topology any Boolean algebra is locally compact.     

Structure des algebres de valuation discrete

We define a topology τ_e, on anyC-algebra of discrete valuation, generalizing the topology of coefficientwise convergence on C[[X]] studied by G. R. Allan. We give a necessary and sufficient condition for τ_e to be complete and prove that the completion provides an algebra of discrete valuation. We also prove that if aC-algebra of discrete valuation is Fréchet andm-convex for τ_e then it is isomorphic to (C[[X]], τ_e) and then τ_e is the uniqueF-algebra topology in A. We prove that a commutative, unital Fréchet l.m.c.a. that is aC-algebra of valuation is in fact aC-algebra of discrete valuation and so is embeddable in (C[[X]], τ_e). Whence a result of H. Bouloussa.     

Nonseparable UHF algebras I: Dixmier's problem

There are three natural ways to define UHF (uniformly hyperfinite) C^∗-algebras, and all three definitions are equivalent for separable algebras. In 1967 Dixmier asked whether the three definitions remain equivalent for not necessarily separable algebras. We give a complete answer to this question. More precisely, we show that in small cardinality two definitions remain equivalent, and give counterexamples in other cases. Our results do not use any additional set-theoretic axioms beyond the usual axioms, namely ZFC.     

Decomposition of Analysis Operators and Frame Ranges for Continuous Frames

In this article, the measure space associated with a continuous frame is supposed to be σ-finite and positive, and a frame range is the range of the analysis operator for a continuous frame. Gabardo and Han in 2003 asked whether two frame ranges can both be contained in another one. To solve this problem, we give two decompositions of analysis operators and frame ranges for continuous frames respectively, which essentially establish a relationship between continuous frames and Hilbert-Schmidt operator valued frames. As applications, it follows that only separable Hilbert space can have a continuous frame, that there exists a continuous frame of Riesz-type if and only if the associated measure space is purely atomic, and that the sum of two frame ranges is still a frame range when the sum is closed. Finally, we construct a counterexample which shows that the Gabardo-Han problem is not necessarily true in general.