K-Theory for C*- algebras and its applications to Symbolic dynamics

符号动力学是一类拓扑动力学系统。它包含了一类拓扑马尔可夫移位和数学系统。本课题利用C*-代数的泛函解析方法研究符号动力学。作为本研究的具体结果，利用C*-代数方法证明了相关C*-代数的ext群所定义的子位移的Bowen-Franks群在流动等价下是不变性的（K-Theory 23(2001), 67-104）。在不使用C*-代数方法的情况下，利用Bowen-Franks的拓扑马尔可夫位移论证的推广方法证明了这一结果。对某些子位移（如Dyck位移）的Bowen-Franks群进行了计算，从而表明具有相同拓扑熵的一些子位移彼此之间并不相等。在与子移相关的C*-代数中，我们发现了af -代数的Bratteli图结构中子移的表示，并将其称为λ-图系统。与德国海德堡大学Wolfgang Krieger教授共同研究了λ图系统、子位移和Schannon图之间的关系。在联合工作中，引入了一类新的子迁移，称为原同步（protosynchronizations）。Soc。日本)。在一篇题为“与子位移表示相关的C*-代数”的预印本中，介绍了与λ-图系统相关的C*-代数的构造。这些研究结果也发表在日本期刊《Suugaku》上，并将以英文译本《Suugau Exposition》发表。

Symbolic dynamics is a class of topological dynamical systems. It contains a clss of topological Markov shifts and sofic systems. In this research project, symbolic dynamics has been studied by using functional analytic method of C*-algebras. As a concrete result of this research, the Bowen-Franks groups for subshifts defined by the Ext-groups for the associated C*-algebras have been proved to be invariant under flow equivalence by using C*-algebra method (K-Theory 23(2001), 67-104). The result has been also proved by using a generalizing method of Bowen-Franks's argument for topological Markov shifts without using C*-algebra method. The Bowen-Franks groups for certain subshifts such as Dyck shifts have been computed so that it has been shown that some subshifts with the same topological entropy are not flow equivalent each other. We find presentations of subshifts in the structure of the Bratteli diagram of the AF-algebras inside the C*-algebras associated with subshifts and call them λ-graph systems. A relationship among λ-graph systems, subshifts and Schannon graphs has been studied in a joint work with Professor Wolfgang Krieger (Heidelberg University, Germany). In the joint work, a new class of subshifts called protosynchronizations has been introduced (to appear in J. Math. Soc. Japan). A construction of C*-algebras associated with λ-graph systems has been introduced in a preprint entitled "C*-algebras associated with presentations of subshifts. "These results hav been also published in the Japanese journal "Suugaku" and will be published in its english translation "Suugau Exposition."