Research on Inclusion Relations of Subalgebras
Project/Area Number  03640156 
Research Category 
GrantinAid for General Scientific Research (C)

Allocation Type  Singleyear Grants 
Research Field 
解析学

Research Institution  Nara University of Education 
Principal Investigator 
KAWAKAMI Satoshi Nara univ.Educ, Educ., Ass.Prof., 教育学部, 助教授 (20161284)

CoInvestigator(Kenkyūbuntansha) 
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)

Project Period (FY) 
1991 – 1992

Project Status 
Completed(Fiscal Year 1992)

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)

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 
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 RadonNikodym 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 PimsnerPopa'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.

Report
(3results)
Research Output
(3results)