Group actions on operator algebras

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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.