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Harmonic Analysis on Operator Algebras

算子代数的调和分析

基本信息

批准号：
16540190
负责人：
IZUMI Masaki
金额：
$ 2.37万
依托单位：
Kyoto University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2006
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540190/
关键词：
Operator Algebras Poisson Boundary C^*-Algebras von Neumann Algebras 双曲群 KMS状態 von Neumann環 Poisson boundary quantum groups von Neumann algebras C^*-algebras quantum probability

项目摘要

For various models appearing quantum group actions and KMS states for the gauge action of the Cuntz-Pimsner algebra, I determined the (non-commutative) Poisson boundaries. In particular, I determined the flow of weights introduced by Connes and Takesaki for several models. Since the definition of the flow of weights involves ergodic decomposition, it is not so easy to give a concrete description in general.The set of bounded operators B(H) of a Hilbert space H is a von Neumann algebra. A one-parameter semigroup of unit preserving endomorphism of B(H) is said to be an E_0-semigroup. E_0-semigroups are classified into three categories, type I, type II, and type III, and except for the type I case, the structure of E_0-semigroups is not well-understood. With R. Srinivasan, I constructed uncountably many mutually non-cocycle conjugate E_O-semigroups of type III. Before our construction, the only known such examples were constructed by Tsirelson. Since Tsirelson's invariant is trivial for our examples, his method can not distinguish our examples. For a given E_0-semigroup and for an open subset U of the unit interval [0,1], one can associate a von Neumann algebra A(U), which is a cocycle conjugacy invariant of the E_0-semigroup. Murray and von Numann classified von Neumann algebras into three categories, type I, type II, and type III, which is nothing to do with the type classification of E_0-semigroups a priori. For type I and type II E_0-semigroups, the von Neumann algebra A(U) is always of type I. We show that for our examples, the von Neumann algebra A(U) may be of type III according to the shape of the set U.

对于出现量子群作用和昆兹-皮姆斯纳代数规范作用的KMS状态的各种模型，我确定了（非交换的）泊松边界。特别是，我确定了Connes和Takesaki为几个模型引入的权重流。由于权重流动的定义涉及到遍历分解，所以不容易给出一般的具体描述。希尔伯特空间H的有界算子集B(H)是一个冯·诺伊曼代数。我们称B(H)的保持单位自同态的单参数半群为e_0半群。e_0 -半群分为I型、II型和III型三种类型，除I型外，e_0 -半群的结构尚不清楚。与R. Srinivasan一起构造了不可数的互非环共轭e_o半群。在我们建造之前，唯一已知的这样的例子是由Tsirelson建造的。由于Tsirelson的不变量对我们的例子来说是微不足道的，所以他的方法不能区分我们的例子。对于给定的e_0 -半群和单位区间[0,1]上的开子集U，可以关联一个von Neumann代数a (U)，它是e_0 -半群的一个环共轭不变量。Murray和von Numann将von Neumann代数分为I型、II型和III型三种类型，这与e_0 -半群的先验类型分类无关。对于I型和II型e_0半群，von Neumann代数A(U)总是I型的。我们证明，对于我们的例子，根据集合U的形状，von Neumann代数A(U)可以是III型的。