Research for Hilbert C^* -bimodules and its application to dicrite dynamical systems

1. 在之前有限生成Hilbert C^*-双模研究的基础上，我们发展了可数生成Hilbert C^*-双模的基本理论。特别地，我们利用共轭的概念给出了有限指数性质的一个表征。并给出了可数生成Cuntz - Krieger双模是有限索引的一个条件。我们计划给出许多可数生成双模的例子，并将我们的理论应用到具体的例子中。我们定义了连续群的交叉积双模作为可数生成双模的例子，并证明了它们是有限索引的。研究了由可数生成的Cuntz-Krieger双模生成的C^*-代数的结构，得到了它们的简单性和理想分类可能性的条件。研究了顶点集为一维环面元素的连续图所生成的C^*-代数的结构。我们得到的结果与3.5相似。以非有限指数的可数生成Hilbert C^*-双模为例，研究了由“帐篷映射”动力系统生成的C^*-代数。我们构造了一个完备的可数基，证明了它的简单性和纯无穷性，并计算了它的k群。对于黎曼球上有理映射迭代给出的复杂动力系统，给出了它们在Julia集合上有分支点时的Hilbert C^*-双模的定义。我们研究了相关C^*代数的结构，并以二维多项式和切比切夫多项式为具体的例子，计算了它们的k群。我们计划发展这一理论，同时研究复杂动力系统的遍历性质。

1. Following the preceding our research on finitely generated Hilbert C^*-bimodules, we develop fundamental theory for countably generated Hilbert C^*-bimodules. In particular, we give a characterization of finite index property using the concept of conjugate. Moreover, we give a condition for countably generated Cuntz Krieger bimodules to be of finite index. We have a plan to give many examples of countably generated bimodules, and will develop application of our theory to concrete examples.2. We define the crossed product bimodules by continuous groups as examples of countably generated bimodules, and show that they are of finite index.3. We study the structure of the C^*-algebras generated by countably generated Cuntz-Krieger bimodule, and get conditions for their simplicity and the possibility of the classification of ideals.4. We study the structure of C^*-algebras generated by continuous graphs whose vertex sets are elements in 1-dimensional torus. We get results similar as in 3.5. We study the C^*-algebra generated by "tent map" dynamical system as an example of countably generated Hilbert C^*-bimodule which is not of finite index. We construct a well behaved countable basis for this, and prove the simplicity and pure infiniteness, and calculate its K-groups.6. For complex dynamical systems given by iteration of rational maps on Riemannian sphere, we give a definition of Hilbert C^*-bimodules for them even when they have branched points on their Julia sets. We study the structures of the associated C^*-algebras for concrete examples, for example 2-dimensional polynomial and Tchebychev polynomials, calculate their K-groups. We have a plan to develop this theory and also study ergodic properties of complex dynamical systems.