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

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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