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Researches on the classification of von Neumann algebras by selfdual cones and operator inequalities

冯诺依曼代数自对偶锥和算子不等式分类的研究

基本信息

批准号：
09640144
负责人：
MIURA Yasuhide
金额：
$ 1.6万
依托单位：
Iwate University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1998
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640144/
关键词：
matrix ordered Hilbert space selfdual cone completely positive map type of von Neumann algebra operator inequality ノイマン環不等式

项目摘要

We considered the question how the algebraic structure of a operator algebra determines or is determined by the structure of its underlying Hilbert space. In order to give the answer to this problem, we have introduced the notion of the complete order isomorphisms and complete orthogonal decomposition isomorphisms on the matrix ordered Hilbert space with a family of selfdual cones. Matrix ordered Hilbert spaces are the appropriate objects to which completely positive maps apply and enabled us to handle non-commutative order. In this research, we have proven that a completely positive projection on the Hilbert space induces a conditional expectation on a von Neumann algebra, and a complete order isomorphism induces an automorphism of the von Neumann algebra. We have then investigated the relationship between the type of von Neumann algebras and completely positive maps, and characterized the matrix ordered standard forms of finite von Neumann algebras.We have also introduced the notion … More of the order of operators on a Hilbert space based on the positivity with respect to the selfdual cone. The different point of this order from the usual order of operators is the compatibility with the products. We have investigated the fundamental properties on the operator inequalities with respect to this order, and sharpened the arithmetic and geometric mean inequality in the category of the ordered Hilbert space associated with a finite dimensional commutative von Neumann algebra.Furthermore, we have obtained some results in the geometry and the number theory. One of them is the investigation of the relationship between the sphere theorem and the Dehn lemma. The rank of the group of relative units of a finite Galois extension of the algebraic number field has then been calculated. It is necessary to determine the finite groups all whose abelian subgroups are cyclic. We have described by cyclotomic polynomials quite simply a mechanical system of one-dimensional hard core chain which consists of infinite many particles with two different alternating mass Finally, we have generalized and given very generally applicable proof of the formula of Grant, which is a generalization of product formulae for division values of periods of the exponential function or each elliptic function with complex multiplication in a cyclotomic field, to every hyperelliptic function of cyclotomic type. Less

我们考虑了算子代数的代数结构如何决定或由其基础希尔伯特空间的结构决定的问题。为了解决这个问题，我们在具有自对偶锥族的矩阵序Hilbert空间上引入了完全序同构和完全正交分解同构的概念。矩阵序希尔伯特空间是完全正映射适用的适当对象，使我们能够处理非交换序。本文证明了Hilbert空间上的完全正投影导出von Neumann代数上的条件期望，完全序同构导出von Neumann代数的自同构。研究了vonNeumann代数的类型与完全正映射之间的关系，刻画了有限vonNeumann代数的矩阵序标准形，并引入了vonNeumann代数的矩阵序标准形的概念 ...更多信息基于自对偶锥的正性，讨论了Hilbert空间上算子的阶。这种顺序与通常的算子顺序的不同之处在于与乘积的相容性。本文研究了这类算子不等式的基本性质，并在有限维交换vonNeumann代数所对应的序Hilbert空间范畴内，加强了算术平均和几何平均不等式，得到了一些几何和数论上的结果.其中之一是对球面定理和Dehn引理之间关系的研究。然后计算了代数数域的有限伽罗瓦扩张的相对单位群的秩。确定交换子群是循环的有限群是必要的。本文用分圆多项式十分简单地描述了由无限多个不同质量的粒子组成的一维硬核链力学系统。最后，推广了Grant公式，并给出了其非常普遍适用的证明。Grant公式是分圆域上指数函数或每个椭圆函数的周期与复数乘法的乘积公式的推广。每一个分圆型超椭圆函数少