2006 Fiscal Year Final Research Report Summary
Analysis and a classification problem on group/quantum group actions on operator algebras
| Project/Area Number |
16540180
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Global analysis
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| Research Institution | Hokkaido University |
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Principal Investigator |
YAMANOUCHI Takehiko Hokkaido Univ., Fac. of Sci., Asso. Prof., 大学院理学研究院, 助教授 (30241293)
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| Co-Investigator(Kenkyū-buntansha) |
KISHIMOTO Akitaka Hokkaido Univ., Fac. of Sci., Prof., 大学院理学研究院, 教授 (00128597)
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| Project Period (FY) |
2004 – 2006
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| Keywords | 1-cocycle / coaction / Cartan subalegbra / discrete equivalence relation / factor / flow of weights / ergodic flow / amenable locally compact group |
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| Research Abstract |
Under this project, I made an intensive study on a certain relationship between 1-cocycles in ergodic theory and actions of cocommutative quantum groups on von Neumann algebras. This research can be divided roughly into two parts as follows.
The first part is concerned with a correspondence between 1-cocyc les on discrete equivalence relations and coactions on von Neumann algebras having Cartan subalgebras. I was able to show that every such cocycle gives rise to a coact ion on the von Neumann algebra A associated with the equivalence relation, whose fixed-point algebra contains the Cartan subalgebra. Conversely, I succeeded to prove that every coaction on A fixing the Cartan subalgebra pointwise arises in the manner described above. As an application, I was able to classify, up to cocycle conjugacy, coactions of amenable locally compact groups on hyperfinite II_{1} factor with full Connes spectra.
The second part is about some relationship between 1-cocycles on ergodic flow spaces and coactions on type III factors. I clarified how every 1-cocycle on the flow of weights on a type III factor N produces a coaction on N whose fixed-point algebra contains the centralizer of a dominant weight on N, and vice versa. I was also able to clarify the structure of the crossed product factors by such coactions by computing explicitly their flows of weights. As an application, I succeeded to give a complete classification on actions of amenable locally compact groups on hyperfinite type III factors with a certain property.
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