2005 Fiscal Year Final Research Report Summary
A bridge between complex analysis and operators on Hilbert spaces
| Project/Area Number |
14340050
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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 |
Basic analysis
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| Research Institution | KYUSHU UNIVERSITY |
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Principal Investigator |
WATATANI Yasuo Kyushu University, Faculty of Mathematics, Professor, 大学院・数理学研究院, 教授 (00175077)
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| Co-Investigator(Kenkyū-buntansha) |
KOSAKI Hideki Kyushu University, Faculty of Mathematics, Professor, 大学院・数理学研究院, 教授 (20186612)
HAMACHI Toshihiro Kyushu University, Faculty of Mathematics, Professor, 大学院・数理学研究院, 教授 (20037253)
MATSUI Taku Kyushu University, Faculty of Mathematics, Professor, 大学院・数理学研究院, 教授 (50199733)
KAJIWARA Tsuyoshi Okayama University, Department of Environmental and Mathematical Sciences, Professor, 大学院・環境学研究科, 教授 (50169447)
NAKAZI Takahiko Hokkaido University, Department of Sciences, Professor, 大学院・理学研究科, 教授 (30002174)
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| Project Period (FY) |
2002 – 2005
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| Keywords | complex dynamical system / C^*-algebra / Julia set / Fractal set / Cuntz algebra / rational function / contraction / operator |
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| Research Abstract |
The aim of the research is to study a relation between complex dynamical systems and operators on Hilbert spaces. It is of fundamental interest in operator algebras to analyze interplay between a geometric or dynamical object and a C^*-algebra associated with it. For a branched covering, Deaconu and Muhly introduced a C^*-algebra associated with it using a r-discrete groupoid. A typical example of a branched covering is a rational function regarded as a self-map of the Riemann sphere. In order to capture information of the branched points for the complex dynamical system arising from a rational function R, we introduced a slightly different construction of a C^*-algebra O_R associated with R on the Julia set J_R. The C^*-algebra O_R is the Cuntz-Pimsner algebra of a Hilbert bimodule over the C^*-algebra C(J_R) of the set of continuous functions on J_R.
The main result of our research is the following : For any rational function of R, If the degree of R is at least two, then C^*-algebra O_R is simple and purely infinite. Similarly we study a C^*-algebra O_γ associated with a system γ of proper contractions on a self-similar set. We also obtain the following result : If the system γ satisfies the open set condition, then C^*-algebra O_γ is simple and purely infinite.
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