Group actions on operator algebras

利用Rohlin性质完整地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群被表征为完全上同调平凡g模。作为一个应用，我证明了在两个“可分类”的核C^*-代数中，具有Rohlin性质的有限群作用根据它们在k群上的作用是完全分类的。为了证明每一个完全上同调平凡g模都是归纳g模的归纳极限，我对给定的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，为了研究这类行为，人们总是可以处理模型。这一事实有望在未来得到若干应用。罗林性质的对偶概念是近似可表征性。作为上述结果的一个应用，我完全刻画了一个素数幂次循环群在Cuntz代数上的拟自由作用是近似可表示的。对于这个结果没有直观的解释，这是群上同调论证的一个有趣的结果。在与S. Neshveyev和L. Tuset的联合工作中，我们推测量子群SUq(n)对偶的泊松边界是量子标志流形SUq(n)/T^<n-1>，并给出了n=3的证明。我们注意到我之前介绍的非交换泊松积分映射与Berezin量化之间有很强的相似性。利用这一观察结果，我们的证明以分析作用于量子带球函数空间的某个马尔可夫算子结束。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。

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.