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

C*-代数的 K 理论及其在符号动力学中的应用

基本信息

批准号：
12640204
负责人：
MATSUMOTO Kengo
金额：
$ 1.66万
依托单位：
Yokohama City University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2001
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640204/
关键词：
C^*-algebras symbolic dynamics subshifts K-Theory Ext-groups

项目摘要

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."

符号动力学是一类拓扑动力系统。它包含拓扑马尔可夫位移和SOFIC系统的CLSS。本研究利用C~*-代数的泛函分析方法研究了符号动力学问题。作为这项研究的一个具体结果，用C*-代数方法证明了相关C*-代数的Ext-群所定义的子移位的Bowen-Franks群在流等价下是不变的(K-Theort23(2001)，67-104)。在不使用C*-代数方法的情况下，用Bowen-Franks关于拓扑马尔可夫位移的一个推广方法证明了这一结果。计算了Dyck移位等某些子移位的Bowen-Franks群，从而证明了某些具有相同拓扑熵的子移位不是流等价的。在与子移位相关的C*-代数中，我们发现了AF-代数的Bratteli图结构中的子移位的表示，并称之为λ-图系统。与德国海德堡大学的Wolfgang Krieger教授合作研究了λ-图系统、子移位和流道图之间的关系。在联合工作中，引入了一类新的子移位，称为原同步(出现在J.Math中。SoC。日本)。与λ-图系统相关的C*-代数的构造已在预印本《C*-Algebras Associates with Presentation of SubShift》中被介绍，这些结果也已发表在日文期刊《Suugaku》上，并将发表在其英译本《Suugau Exposal》上。