Research for Hilbert C*-bimodules and its application to analysis of discrete dynamical systems

1.我们证明了可数生成的Hilbert C^*-双模若为有限指标模与它有共轭是等价的。此外，我们构造了许多有限指标的可数生成Hilbert C^*-双模的例子。阐明了Hilbert C^*-模基的一些性质。这些结果已发表在“Jones index theory for Hilbert C^*-bimodules and its equivalence with conjugation theory”. 2.我们从黎曼球面上由有理函数给出的动力系统构造了Hilbert C^*-双模，并利用Pimsner构造法构造了C^*-代数。我们证明了这些C^*-代数的简单性和纯无限性。我们计算了一些例子的K-群。这些结果已发表在“C^*-algebras associated with complex dynamic systems”上。我们利用真压缩族给出的自相似集构造了Hilbert C^*-双模，并利用Pimsner构造法构造了C^*-代数。在适当的条件下 ...更多信息 n，证明了这类C ^*-代数的单性和纯无限性。利用K-群证明了对SierPinski Gasket这一典型分形构造的两个不同的C^*-代数是不同的。我们还计算了由Koch曲线构造的C^*-代数的K-群，这也是一个典型的例子。这些结果已发表在“C^*-algebras associated with self-similar sets”上。对于由复动力系统和自相似集构造的Hilbert C^*-模，我们构造了可数基。这种构造是利用小波基的思想从帐篷映射构造Hilbert C^*-模的推广。这似乎是可数基的第一个明确的例子。虽然这种构造在已发表的文献中还没有出现，但它对研究由有理函数和自相似集构造的C^*-代数的KMS态有一定的帮助。这项研究将在下一阶段继续进行。我们构造了超越函数的C^*-代数并对其进行了研究。我们证明了指数映射的简单性。但由于纯奇点的存在，这方面的研究仍在继续。少

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 K-groups for some examples. These results has been published in "C^*-algebras associated with complex dynamical systems".3. We constructed Hilbert C^*-bimodules from self-similar sets given by families of proper contractions, and constructed C^*-algebras using Pimsner construction. Under appropriate conditio … More 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 K-group. We also calculated K-group of the C^*-algebra constructed from Koch curve, which is also a typical example. These results has been published in "C^*-algebras associated with self-similar sets".4. We constructed countable basis explicitly for the Hilbert C^*-modules constructed from complex dynamical system and self-similar 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 self-similar 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