Project/Area Number  09640142 
Research Category 
GrantinAid for Scientific Research (C).

Section  一般 
Research Field 
解析学

Research Institution  HOKKAIDO UNIVERSITY 
Principal Investigator 
YAMANOUCHI Takehiko Hokkaido University, Department of Mathematics, Assistant Professor, 大学院・理学研究科, 助教授 (30241293)

CoInvestigator(Kenkyūbuntansha) 
SEKINE Yoshihiro Hokkaido University, Faculty of Engineering, Associate Professor, 工学部, 助教授 (30243885)
KISHIMOTO Akitaka Hokkaido University, Department of Mathematics, Professor, 大学院・理学研究科, 教授 (00128597)

Project Fiscal Year 
1997 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥3,000,000 (Direct Cost : ¥3,000,000)
Fiscal Year 1999 : ¥900,000 (Direct Cost : ¥900,000)
Fiscal Year 1998 : ¥800,000 (Direct Cost : ¥800,000)
Fiscal Year 1997 : ¥1,300,000 (Direct Cost : ¥1,300,000)

Keywords  Operator algebra / Quantum group / Kac algebra / (Quasi) Woronowicz algebra / (Quantum group) Action / (Injective) Factor / Rohlin property / ATalgebra / 作用素環 / 量子群 / カッツ環 / 擬ヴォロノヴィッツ環 / (量子群)作用 / (単射)因子環 / Rohlin性質 / AT環 / 擬ヴォロヴィッツ環 / 因子環 / 作用 
Research Abstract 
(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, qdeformations of Lie groups as well as q
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uantum groups that derive from matched piars of locally compact groups by the method of TakeuchiMajid. (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 ATCィイD1*ィエD1algebras. 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, realrank zero ATCィイD1*ィエD1algebra has the Rohlin property. He showed also that one can contstruct on such a CィイD1*ィエD1algebra a onepapameter automorphism group with the Rohlin property, and proved that the crossed product by it is again a simple realrank zero ATCィイD1*ィエD1algebra. Meanwhile, Kishimoto showed that, for an arbitrary pair of simple dimension groups, one can construct a simple realrank zero ATCィイD1*ィエD1algebra and a oneparameter automorphism group on it such that the Kgroups of the associated crossed product are exactly the given dimension groups. Less
