1996 Fiscal Year Final Research Report Summary
Development of index theory for C-algebras
| Project/Area Number |
08454032
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
解析学
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| Research Institution | KYUSHU UNIVERSITY |
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Principal Investigator |
WATATANI Yasuo Kyushu U.Mathematics, Prof., 大学院・数理学研究科, 教授 (00175077)
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| Co-Investigator(Kenkyū-buntansha) |
MAESONO Yoshihiko Kyushu U.Mathematics, Assoc.Prof., 大学院・数理学研究科, 助教授 (30173701)
ISHIKAWA Nobuhiro Kyushu U.Mathematics, Prof., 大学院・数理学研究科, 教授 (10037806)
KAZAMA Hideaki Kyushu U.Mathematics, Prof., 大学院・数理学研究科, 教授 (10037252)
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| Project Period (FY) |
1996
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| Keywords | subfactor / Jones index / Hilbert C^<**>-module / bimodule / crossed product / duality / free / ergodic |
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| Research Abstract |
Let M be a type II_1-factor and N a subfactor of M.V.Jones introduced Jones index [M : N] as a certain rank of M as N-module using the coupling constant. Kosaki and Longo extended Jones index theory for type III subfactors. The head investigator had extended Jones index theory for simple C^<**>-lgebras.
In this research we introduced the notion of Hilbert C^<**>-bimodule by replacing the associativity condition in Rieffel's imprimitivity bimodules by the Pimsner-Popa type ineqality. The index of Hilbert C^<**>-bimodules measures the failure from imprimitivity bimodules. We also studied actions of countable discrete groups on Hilbert C^<**>-bimodules from the motivation of equivariant Morita equivalence for C^<**>-algebras and orbifold construction for subfactors. We considered crossed products of Hilbert C^<**>-bimodules and showed that a certain kind of free and ergodic action guarantees the irreducibility of the crossed product. We also have Takesaki-Takai duality for Hilbert C^<**>-bimodules. We gave a concrete example of the duality of Dynkin diagrams of type A and type D in the level of bimoules. We investigated a fusion rule of irreducible decomposition of iterative tensor products of a Hilbert C^<**>-bimodule and its conjugate.
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