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

我们考虑了一个算子代数的代数结构如何决定或被其基础希尔伯特空间的结构所决定的问题。为了给出这个问题的答案，我们在具有自对偶锥族的矩阵有序Hilbert空间上引入了完全序同构和完全正交分解同构的概念。矩阵有序希尔伯特空间是适用于完全正映射的合适对象，它使我们能够处理非交换序。在本研究中，我们证明了Hilbert空间上的完全正投影在von Neumann代数上引起条件期望，以及完全序同构引起von Neumann代数的自同构。研究了von Neumann代数的类型与完全正映射之间的关系，并刻画了有限von Neumann代数的矩阵有序标准形式。我们还引入了Hilbert空间上基于对自对偶锥的正性的算子的更多阶的概念。这种顺序与通常操作顺序的不同之处在于与产品的兼容性。我们研究了这一阶算子不等式的基本性质，并在有限维可交换冯·诺伊曼代数的有序希尔伯特空间范畴中强化了算术和几何平均不等式。此外，我们还在几何学和数论方面取得了一些成果。其中之一是研究球定理和Dehn引理之间的关系。然后计算了代数数域有限伽罗瓦扩展的相对单元群的秩。有必要确定其阿贝尔子群都是循环的有限群。我们已经用分环多项式很简单地描述了一个由无限多个具有两种不同交替质量的粒子组成的一维硬核链的力学系统，最后，我们推广并给出了格兰特公式的非常普遍适用的证明，这是分环场中指数函数或每一个具有复乘法的椭圆函数的周期分值的乘积公式的推广，对每一个切圆型的超椭圆函数。少

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