Research for Hilbert C*bimodules and its application to analysis of discrete dynamical systems
Project/Area Number 
15540207

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Global analysis

Research Institution  OKAYAMA UNIVERSITY 
Principal Investigator 
KAJIWARA Tsuyoshi Okayama University, Graduate School of Environmental Sciences, Professor, 大学院環境学研究科, 教授 (50169447)

CoInvestigator(Kenkyūbuntansha) 
WATATANI Yasuo Kyushu University, Graduate School of Mathematical Science., Professor, 大学院数理学研究院, 教授 (00175077)
SASAKI Toru Okayama University, Graduate School of Environmental Sciences., Lecturer, 大学院環境学研究科, 講師 (20260664)

Project Period (FY) 
2003 – 2006

Project Status 
Completed (Fiscal Year 2006)

Budget Amount *help 
¥2,500,000 (Direct Cost: ¥2,500,000)
Fiscal Year 2006: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2005: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2004: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 2003: ¥700,000 (Direct Cost: ¥700,000)

Keywords  Hilbert C*bimodule / Complex dynamical systems / Selfsimilar sets / Branch points / KMS states / ヒルベルトC*双加群 / 離散力学系 / 自己相似写像 / KMS状態 / ヒルベルトC^*双加群 
Research Abstract 
1. We proved that the fact a countably generated Hilbert C^*bimodule if of finite index and the fact that it has a conjugation is equivalent. Moreover, we constructed many examples of countably generated Hilbert C^*bimodules of finite index. We clarified some properties of bases of Hilbert C^*modules. These results has been published in "Jones index theory for Hilbert C^*bimodules and its equivalence with conjugation theory". 2. We constructed Hilbert C^*bimodules from the dynamical systems on the Riemannian sphere given by rational functions, and constructed C^*algebras using Pimsner construction. We proved simplicity and pure infiniteness of these C^*algebras. We calculated Kgroups for some examples. These results has been published in "C^*algebras associated with complex dynamical systems". 3. We constructed Hilbert C^*bimodules from selfsimilar sets given by families of proper contractions, and constructed C^*algebras using Pimsner construction. Under appropriate conditio
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n, we proved simplicity and pure infiniteness of theseC^*algebras. We showed that two C^*algebras differently constructed for SierPinski Gasket, which is a typical fractal, are different using Kgroup. We also calculated Kgroup of the C^*algebra constructed from Koch curve, which is also a typical example. These results has been published in "C^*algebras associated with selfsimilar sets". 4. We constructed countable basis explicitly for the Hilbert C^*modules constructed from complex dynamical system and selfsimilar sets. This construction is a generalization of that given for the Hilbert C^*module constructed from tent map using the idea of wavelet basis. This seems the first explicit example of countable bases. Although this construction is not contained in the papers which is already published, it gives some help for research of KMS states of C^*algebras constructed from rational functions and selfsimilar sets. This research continues in the next period. 5. We constructed C^*algebras for transcendental functions and studied them. We showed simplicity for exponential map case. But there exist a difficulty arising from the existence of pure singularity, and this research also continues. Less

Report
(5 results)
Research Products
(12 results)