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.

对于出现量子组动作和KMS状态的各种模型，用于Cuntz-Pimsner代数的仪表作用，我确定了（非共同的）泊松边界。特别是，我确定了Connes和Takeaki引入的几种模型引入的权重流。由于权重的定义涉及千古分解，因此一般而言，给出具体描述并不容易。HilbertSpace H的有限运算符B（H）是Von Neumann代数。保存B（h）的单位保存单位的单位分数据说是E_0-序列。 E_0-序列分为三类，I型，II型和III型，除了I型情况外，E_0-semigroups的结构并不理解。使用R. srinivasan，我构建了许多III型的许多相互非循环偶联E_O-序列。在我们建造之前，唯一已知的例子是由Tsirelson构建的。由于Tsirelson的不变性对于我们的例子来说是微不足道的，因此他的方法无法区分我们的例子。对于给定的E_0-序列和单位间隔的开放子集u [0,1]，可以将von Neumann代数A（U）关联，该代数是E_0-序列的共生共轭不变的。 Murray和Von Numann将Von Neumann代数分为三类I型，II型和III型，这与先验E_0-Semigroups的类型分类无关。对于I型和II型E_0-semigroups，von Neumann代数A（U）始终是I型。我们证明，对于我们的示例，von Neumann代数A（U）可以根据集合U的形状为III类型。