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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.

我用Rohlin性质完全刻画了允许有限群作用的C^*-代数的K-群。更确切地说，这样的K-群被刻画为完全上同调平凡G-模。作为应用，我证明了在两类“可分类”的核C^*-代数中，具有Rohlin性质的有限群作用可以根据它们在K-群上的作用而完全分类。证明了每一个完全上同调平凡G-模都是诱导G-模的诱导极限，并对给定的K-理论不变量构造了具有Rohlin性质的模型作用。这些结果表明，人们总是可以处理模型，以研究这类行动。预计未来会出现这一事实的几个应用。洛林性质的对偶概念是近似可表示性。作为上述结果的一个应用，完全刻划了素数幂阶循环群在Cuntz代数上的拟自由作用何时是近似可表示的。这个结果没有直观的解释，它是一个有趣的后果，一个croup上同调的论点。Neshveyev和L. Tuset，我们证明了量子群SUq（n）的对偶的Poisson边界是量子旗流形SUq（n）/T^<n-1>，并对n=3给出了证明.我们注意到我之前介绍过的非交换泊松积分映射与Berezin量子化之间有很强的相似性。利用这一观察，我们的证明最终分析了一个特定的马尔可夫算子作用于量子带状球函数空间。我们的方法可能适用于紧致半单李群的一般q-变形，我们现在正在追求它。