2007 Fiscal Year Final Research Report Summary
Representation and measure theory of infinite dimensional moues and its applications
| Project/Area Number |
18540184
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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 |
Basic analysis
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| Research Institution | Kochi University |
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Principal Investigator |
SHIMOMURA Hiroaki Kochi University, Department of Mathematics Faculty of Education, Professor (20092827)
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| Project Period (FY) |
2006 – 2007
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| Keywords | infinite dimensional group / representation theory / measure theory / diffeomorphism / irreducible representation / positive-deftnite-function / extreme decomposition |
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| Research Abstract |
The main subjects of this period concerns with positive-definite functions on topological groups :
The first one is an extension of the G-N-S construction method for the positive-definite functions on the usual locally compact groups to that for infinite dimensional groups. We showed that the extension is also possible, if we have a quasi-invariant probability measure whose admissible shifts form a denge subgroup. Such a measure plays a key role in place of the Haar measures on the locally compact groups. An interesting and important example of the infinite dimensional groups which admit such measures is the group of diffeomorphisms on smooth manifolds. The detailed arguments are already published in Math Z.
The second one is the following problem :
When and only when do extreme decompositions of positive-definite functions coincide with irreducible decompositions of unitary representations through the G-N-S construction on the usual locally compact groups ?
Up to the present time, we already have obtained a necessary and sufficient condition for the above problem. However its description is not so clear that we are looking for another more concrete condition than that one.
Of course, the problem is not always affirmative. We give the negative interesting examples whose origin is due to Thoma :
In 1964, Thoma had a complete classification of the characters which is by definition normal traces on factors of type H. Then after about 30 years of this work, Obata had a disintegration of these characters.
Hence, we have a problem whether the disintegration corresponds to irreducible decomposition or not.
We will write here a partial answer for this problem briefly : It depends on the Thoma parameter which characterize his description ; It is negative if the parameter takes some values. Whether it is always negative or not is left for us, as well as the study of more specific equivalent conditions for the main problem.
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