Research on Inclusion Relations of Subalgebras
| Project/Area Number |
03640156
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| Research Category |
Grant-in-Aid for General Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Research Field |
解析学
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| Research Institution | Nara University of Education |
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Principal Investigator |
KAWAKAMI Satoshi Nara univ.Educ, Educ., Ass.Prof., 教育学部, 助教授 (20161284)
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| Co-Investigator(Kenkyū-buntansha) |
ASAI Teruaki Nara univ.Educ, Educ., Ass.Prof., 教育学部, 助教授 (60094497)
KIKUCHI Teppei Nara univ.Educ, Educ., Professor, 教育学部, 教授 (50031589)
JIMBO Toshiya Nara univ.Educ, Educ., Professor, 教育学部, 教授 (80015560)
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| Project Period (FY) |
1991 – 1992
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| Project Status |
Completed (Fiscal Year 1992)
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| Budget Amount *help |
¥1,700,000 (Direct Cost: ¥1,700,000)
Fiscal Year 1992: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 1991: ¥900,000 (Direct Cost: ¥900,000)
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| Keywords | von Neumann algebra / operator valued weight / index / conditional expectation / entropy / analytic function algebra / peak set / 条件付期待値 / エントロピ- / 多変数関数論 / 正則環 |
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| Research Abstract |
In a field of operator algebras, we studied global analysis of inclusion relations of von Neumann subalgebras. We have made it clear what played principal roles on the analysis and have established some reduction theory to irreducible inclusion relations.In order to do so, first of all, we defined some types of inclusion relations, so called, semifinite type and finite type. Secondly, we developed some new notions and their properties on Radon-Nikodym derivative and disintegration of operator valued weights. For an operator valued weight E from a von Neumann algebra onto a subalgebra, we define the standard correspondent (or commutant) E' of E, which satisfies a chain rule property. Applying these results, we can get the following. (1) A chain rule property is also true to hold for indicial derivative of an operator valued weight, whose notion had been developed by us. (2) For a general pair of von Neumann algebras, if there is a conditional expectation whose index is a bounded operator, there exists uniquely the conditional expectation of minimal index type. This is an extension of Hiai's result. (3) It is easy to show that a convolution of conditional expectations of minimal type is also of minimal type in a general setting. (4) For a problem on additivity of Pimsner-Popa's entropy, we have succeeded to give a complete answer.
In a field of function algebra, we have made clear some relation among a zero set, a peak interpolation set, and so on for an analytic function algebra.
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Report
(3 results)
Research Products
(3 results)
