Tomita's Theory of Modular Hilbert Algebras and its Applications
Tomita's Theory of Modular Hilbert Algebras and its Applications
复制标题
富田的模希尔伯特代数理论及其应用
DOI：
10.1007/bfb0065832
复制
发表时间：
1970
期刊：
影响因子：
3.5
通讯作者：
M. Takesaki
中科院分区：
文献类型：
--
作者：
M. Takesaki
1967年，Tomita在两篇未发表的论文[21]和[22]中阐明了一个von Neumann代数M与它的交换子M <$之间的代数关系，进而证明了von Neumann代数张量积的交换定理（即（M M '= M' ® M½）·为了研究标准表示中M和M '之间的关系，（例如，一个循环表示的M诱导的忠实正常状态），他介绍了两个基本概念，称为广义和模希尔伯特代数，分别，都是相关的，但不同的Dixloves的拟希尔伯特代数[4]。$2的定义证明每个von Neumann代数都同构于广义Hilbert代数的左von Neumann代数并不很困难。然而，通过广义希尔伯特代数，我们看到冯诺依曼代数的对合如何在希尔伯特空间结构中扭曲。为了更清楚地解释他的基本思想，假设P是冯诺依曼代数M的忠实正规态。然后是对合：在希尔伯特空间结构中
In 1967, Tomita clarified the algebraic relation between a von Neumann algebra M and its commutant M¹ in two unpublished papers [21] and [22], and then proved the commutation theorem for tensor products of von Neumann algebras (ie (M M₂)'= M'® M½)· In order to study the relation between M and M'in a standard representation,(for example, a cyclic representation of M induced by a faithful normal state), he introduced two basic notions, called a generalized and modular Hilbert algebras, respectively, both being related to but different from Dixmier's quasi-Hilbert algebra [4]. See $2 for definitions. It is not very difficult to show that every von Neumann algebra is isomorphic to the left von Neumann algebra of a generalized Hilbert algebra. However, by means of generalized Hilbert algebras we see how the involution of a von Neumann algebra is twisted in a Hilbert space structure. To explain his basic idea more clearly, suppose P is a faithful normal state of a von Neumann algebra M. Then the involution: in the Hilbert space structure