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A bridge between complex analysis and operators on Hilbert spaces

希尔伯特空间上复分析和算子之间的桥梁

基本信息

批准号：
14340050
负责人：
WATATANI Yasuo
金额：
$ 7.36万
依托单位：
KYUSHU UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2005
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14340050/
关键词：
complex dynamical system C^*-algebra Julia set Fractal set Cuntz algebra rational function contraction operator C^*環ヒルベルト双加群トエプリッツ作用素

项目摘要

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.

研究的目的是研究Hilbert空间上的复杂动力系统和算子之间的关系。分析一个几何或动力学对象和与之相关联的C^*-代数之间的相互作用是算子代数中的一个基本兴趣。对于分支覆盖，Deaconu和Muhly利用r-离散广群引入了与之相关联的C^*-代数。分支覆盖的一个典型例子是被视为黎曼球面的自映射的有理函数。为了捕捉由有理函数R产生的复杂动力系统的分支点信息，我们在Julia集J_R上引入了一个与R相关联的C^*-代数O_R的略有不同的构造。C^*-代数O_R是J_R上连续函数集的C^*-代数C（J_R）上的Hilbert双模的Cuntz-Pimsner代数.本文的主要结果是：对R的任意有理函数，若R的次数至少为2，则C^*-代数O_R是单的纯无限的.类似地，我们研究了与自相似集上的真压缩系统γ相关联的C^*-代数O_γ。我们还得到如下结果：若系统γ满足开集条件，则C^*-代数O_γ是单的纯无限的.