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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对紧Kac代数在von Neumann代数上的作用作了深入的研究。他引入了这类作用的Connes谱的概念，以便为算子代数上这类作用的分类提供一种工具。他证明了紧Kac代数作用的交叉积是一个因子当且仅当该作用是中心遍历的且具有全Connes谱，这阐明了Connes谱在很大程度上支配着这类作用的行为。其次，他证明了当一个紧Kac代数作用是极小的时，它同时是占优的，并研究了由这种作用诱导的Galois对应。同时，为了在von Neumann代数的框架下捕捉不属于Kac代数范畴的量子群，Yamanouchi引入了拟Woronowicz代数的概念。他证明了这类算子代数拟群包含，作为例子，李群的q-变形以及q…(2)Sekine利用与Yamanouchi不同的方法，独立地利用紧Kac代数的作用刻画了交叉积的阶乘性。他的结果推广了Paschke的一个经典定理。(3)岸本对AT-CィイD_1*ィエD_1-代数的自同构做了非常独特的研究。他特别研究了具有Rohlin性质的自同构。因此，他能够刻画简单的实秩零AT-CィイD_1*ィエD_1-代数上的自同构何时具有Rohlin性质。他还证明了在这样的C-ィイD_1*ィエD_1-代数上可以构造一个具有Rohlin性质的单参数自同构群，并证明了它的交叉积又是一个简单的实秩零的AT-CィイD_1*ィエ_1-代数。同时，Kishimoto证明了，对于任意一对简单维群，我们可以构造一个简单的实数秩零的AT-CィイD_1*ィエD_1-代数和它上的一个单参数自同构群，使得相关交叉积的K-群正好是给定的维群。较少