Construction of actions of operator algebraic quantum groups on injective factors and their classification

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(1) Yamanouchi has made an intensive research on actions of compact Kac algebras on von Neumann algebras. He introduced a notion of Connes spectrum for such actions in order to provide a tool for classification of such a class of actions on operator algebras. He proved among others that the crossed product by a compact Kac algebra action is a factor if and only if the action is centrally ergodic and has full Connes spectrum, which clarified that Connes spectrum largely dominates behavior of this kind of actions. He next showed that, when a compact Kac algebra action is minimal, it is at the same time dominant, and studied a Galois correspondence induced by such an action. Meanwhile, in order to capture quantum groups in the framework of von Neumann algebras that are not in the category of Kac algebras, Yamanouchi introduced a notion of a quasi Woronowicz algebra. He showed that this class of operator algebra quatum groups contains, as examples, q-deformations of Lie groups as well as q … More uantum groups that derive from matched piars of locally compact groups by the method of Takeuchi-Majid.(2) Sekine independently succeeded in characterizing factoriality of the crossed product by an action of a compact Kac algebra using a methos different from Yamanouchi's. His result generalizes a classical theorem by Paschke.(3) Kishimoto has made a very unique research on automorphisms on AT-CィイD1*ィエD1-algebras. He has especially studied automorphisms with the Rohlin property. As a result, he was able to give characterizations as to when an automorphism on a simple, real-rank zero AT-CィイD1*ィエD1-algebra has the Rohlin property. He showed also that one can contstruct on such a CィイD1*ィエD1-algebra a one-papameter automorphism group with the Rohlin property, and proved that the crossed product by it is again a simple real-rank zero AT-CィイD1*ィエD1-algebra. Meanwhile, Kishimoto showed that, for an arbitrary pair of simple dimension groups, one can construct a simple real-rank zero AT-CィイD1*ィエD1-algebra and a one-parameter automorphism group on it such that the K-groups of the associated crossed product are exactly the given dimension groups. Less