Group actions on operator algebras

算子代数的群作用

基本信息

批准号：
13640210
负责人：
IZUMI Masaki
金额：
$ 2.5万
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：
2001
资助国家：
日本
起止时间：
2001 至 2003
项目状态：
已结题

项目摘要

I completely characterized the K-groups of C^*-algebras allowing finite group actions with the Rohlin property. More precisely, such K-groups are characterized as completely cohomologically trivial G-modules. As an application, I showed that in two "classifiable" classes of nuclear C^*-algebras, finite group actions with the Rohlin property are completely classified in terms of their actions on the K-groups. Showing that every completely cohomological trivial G-module is inductive limit of induced G-modules, I construct model actions with the Rohlin property for a given K-theoretical invariant. These results show that one can always deal with models in order to investigate this class of actions. Several applications of this fact are expected in the future.The dual notion of the Rohlin property is approximate representability. As an application of the above-mentioned result, I completely characterized when a quasi-free action of a prime power order cyclic group on the Cuntz algebra is approximately representable. There is no intuitive explanation for this result and it is an interesting consequence of a croup cohomology argument.In a joint work with S. Neshveyev and L. Tuset, we conjectured that the Poisson boundary of the dual of the quantum group SUq(n) is the quantum flag manifold SUq(n)/T^<n-1>, and we gave a proof for n=3. We noticed strong similarity between the non-commutative Poisson integral map, which I introduced before, and Berezin quantization. Using this observation, our proof ends up with analysis of a certain Markov operator acting on the space of quantum zonal spherical functions. Our approach probably works for general q-deformation of compact semi-simple Lie groups and we are pursuing it now.

我完全表征了C^* - 代数的K组，允许使用Rohlin属性有限的组动作。更确切地说，此类K组被特征在于完全既定的G模块。作为一种应用，我表明，在两种“可分类”类别的核C^* - 代数中，与Rohlin属性的有限群体行动完全根据其对K组的行为进行了完全分类。表明每一个完全共同的琐事G模块都是诱导的G模块的电感限制，我使用Rohlin属性构建模型动作，以给定的K理论不变。这些结果表明，人们总是可以处理模型以研究此类动作。预计将来，这一事实的几个应用。Rohlin属性的双重概念近似表示。作为上述结果的应用，当我在Cuntz代数上的Prime Power Order Cyclot cyclot cyclage基团的准无作用近似代表时，我完全表征了。对此结果没有直观的解释，这是croup copomology参数的有趣结果。在与S. Neshveyev和L. tuset的联合合作中，我们猜想量子组SUQ（N）的双重泊松边界是量子flag（N）量子flagord suq（n）/t^<n-1>，我们给了N = 3 = 3 = 3 = 3 = 3。我们注意到我之前介绍的非交通泊松积分图与伯雷唱的量化之间的强烈相似性。利用此观察结果，我们的证明最终以对量子区域球形函数空间作用的特定马尔可夫操作员的分析。我们的方法可能对紧凑的半简单谎言组的一般Q规定作用，我们现在正在追求它。