Construction of actions of operator algebraic quantum groups on injective factors and their classification

算子代数量子群对内射因子作用的构造及其分类

基本信息

批准号：
09640142
负责人：
YAMANOUCHI Takehiko
金额：
$ 1.92万
依托单位：
HOKKAIDO 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 1999
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640142/
关键词：
Operator algebra Quantum group Kac algebra (Quasi) Woronowicz algebra (Quantum group) Action (Injective) Factor Rohlin property AT-algebra 擬ヴォロヴィッツ環因子環作用

项目摘要

(1) Yamanouchi has made an intensive research on actions of compact Kac algebras on von Neumann algebras. He introduced a notion of Connes spectrum for such actions in order to provide a tool for classification of such a class of actions on operator algebras. He proved among others that the crossed product by a compact Kac algebra action is a factor if and only if the action is centrally ergodic and has full Connes spectrum, which clarified that Connes spectrum largely dominates behavior of this kind of actions. He next showed that, when a compact Kac algebra action is minimal, it is at the same time dominant, and studied a Galois correspondence induced by such an action. Meanwhile, in order to capture quantum groups in the framework of von Neumann algebras that are not in the category of Kac algebras, Yamanouchi introduced a notion of a quasi Woronowicz algebra. He showed that this class of operator algebra quatum groups contains, as examples, q-deformations of Lie groups as well as q … More uantum groups that derive from matched piars of locally compact groups by the method of Takeuchi-Majid.(2) Sekine independently succeeded in characterizing factoriality of the crossed product by an action of a compact Kac algebra using a methos different from Yamanouchi's. His result generalizes a classical theorem by Paschke.(3) Kishimoto has made a very unique research on automorphisms on AT-CィイD1*ィエD1-algebras. He has especially studied automorphisms with the Rohlin property. As a result, he was able to give characterizations as to when an automorphism on a simple, real-rank zero AT-CィイD1*ィエD1-algebra has the Rohlin property. He showed also that one can contstruct on such a CィイD1*ィエD1-algebra a one-papameter automorphism group with the Rohlin property, and proved that the crossed product by it is again a simple real-rank zero AT-CィイD1*ィエD1-algebra. Meanwhile, Kishimoto showed that, for an arbitrary pair of simple dimension groups, one can construct a simple real-rank zero AT-CィイD1*ィエD1-algebra and a one-parameter automorphism group on it such that the K-groups of the associated crossed product are exactly the given dimension groups. Less

（1）Yamanouchi对von Neumann代数的紧凑型KAC代数的作用进行了深入的研究。他引入了针对此类行动的Connes Spectra的概念，以便为操作员代数的类别分类提供一种工具。他被证明，通过紧凑的KAC代数作用交叉的产品是一个因素，并且仅当该动作在中央呈颈并具有完整的Connes Spectra时，这阐明了Conns Spectra在很大程度上主导了这种行为的行为。接下来，他表明，当紧凑的KAC代数作用最少时，它同时占主导地位，并研究了这种动作引起的GALOIS对应关系。同时，为了捕获不在KAC代数类别的von Neumann代数框架中，Yamanouchi引入了Quasi Woronowicz代数的概念。作为示例，组包含谎言组的Q- Q- Q-更多的Q…更多的Uantum群体，这些群体通过Takeuchi-Majid的方法从局部紧凑型组的匹配的匹配中得出。（2）Sekine通过使用与Yamanatouchi的方法不同的方法来独立表征交叉产品的sekoriality of Crositiality conterality of Crossiality fictection。他的结果通常是Paschke的古典定理。（3）Kishimoto进行了一项非常独特的研究，他特别研究了Rohlin物业的自动形态。结果，他能够给出角色，即何时在简单的，现实级的零AT-CII D1*IE D1-Algebra具有Rohlin属性上。他还表明，一个可以与Rohlin属性一起使用这样的CII D1*IE D1代数为单一型自动形态组，并证明其交叉产品再次是一个简单的真实现实级的AT-CII D1*IE D1 D1-Algebra。同时，基希莫托（Kishimoto）表明，对于任意的简单维度组，可以构建一个简单的实级零at-cii d1*ie d1 d1-algebra和一个单参数仪自动形态组，以便相关的交叉产品的k组恰好是给定的尺寸组。较少的