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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 群，允许具有 Rohlin 性质的有限群作用。更准确地说，这样的 K 群被描述为完全上同调平凡的 G 模。作为一个应用，我证明了在核 C^* 代数的两个“可分类”类别中，具有 Rohlin 性质的有限群作用根据它们对 K 群的作用被完全分类。证明每个完全上同调的平凡 G 模都是诱导 G 模的归纳极限，我为给定的 K 理论不变量构造了具有 Rohlin 性质的模型动作。这些结果表明，人们总是可以处理模型来研究此类行为。预计将来会出现该事实的多种应用。Rohlin 属性的双重概念是近似可表示性。作为上述结果的应用，我完整地表征了素数幂阶循环群的准自由作用在 Cuntz 代数上何时可近似表示。这个结果没有直观的解释，它是鲁普上同调论证的一个有趣的结果。在与S. Neshveyev和L. Tuset的合作中，我们推测量子群SUq(n)的对偶的泊松边界是量子旗流形SUq(n)/T^<n-1>，并给出了n=3的证明。我们注意到我之前介绍过的非交换泊松积分图和 Berezin 量化之间有很强的相似性。利用这一观察结果，我们的证明最终分析了作用于量子带状球函数空间的某个马尔可夫算子。我们的方法可能适用于紧致半单李群的一般 q 变形，我们现在正在研究它。