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

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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.