A bridge between complex analysis and operators on Hilbert spaces

本研究的目的是研究复动力系统与Hilbert空间上算子之间的关系。分析几何或动力学对象与与其相关的C^*-代数之间的相互作用是算子代数的基本兴趣。对于分支覆盖，Deaconu和Muhly引入了一个与之相关的C^*-代数，它使用r-离散群胚。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。为了刻画由有理函数R产生的复杂动力系统的分支点信息，我们引入了Julia集J_R上与R相关的C^*-代数O_R的一种略有不同的构造。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。我们的主要研究结果如下：对于R的任何有理函数，如果R的次数至少为2，则C^*-代数O_R是简单的纯无限的。类似地，我们研究了与自相似集上的真压缩系统γ相关的C^*-代数O_γ。我们还得到了如下结果：如果系统γ满足开集条件，则C^*-代数O_γ是单的纯无限的。

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.