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

符号动力学是一类拓扑动力系统。它包含一类拓扑马尔可夫位移和sofic系统。本研究利用C ~*-代数的泛函分析方法对符号动力学进行了研究。作为本研究的一个具体结果，利用C*-代数方法，证明了由关联C*-代数的Ext-群定义的子移位的Bowen-Franks群在流等价下是不变的（K-Theory 23（2001），67-104）。利用Bowen-Franks关于拓扑马氏位移的论证的推广方法，不用C ~*-代数方法，也证明了这个结果.已经计算了某些子移位如Dyck移位的Bowen-Franks群，从而证明了具有相同拓扑熵的某些子移位彼此不流等价。我们在与子移位相关联的C*-代数内部发现AF-代数的Bratteli图结构中的子移位的表示，并称它们为λ-图系统。在与Wolfgang Krieger教授（Heidelberg University，德国）的合作中，研究了λ-图系统、子移位和Schannon图之间的关系。在联合工作中，引入了一类新的称为原同步的子移位（将出现在J. Math. Soc. Japan）。与λ-图系统相关联的C*-代数的构造已经在题为“与子移位表示相关联的C*-代数”的预印本中引入。这些结果也发表在日本期刊《Suugaku》上，并将在其英文译本《Suugau Exposition》中发表。"

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."