K-theory for C^* algebras aud its applications to symbolic dynamics

在本研究项目中，我们利用C^*-代数的结构理论及其k理论，发展了符号动力学理论。我们特别与Wolfgang Krieger教授（德国海德堡大学）合作发表了两篇联合研究论文。一篇是论文A lambda-graph system for the Dyck shift and its K-groups (with W. Krieger), Documenta Math. 8(2003), 79-96，其中我们引入了Cantor horizon lambda-graph system for Dyck shifts的概念，并计算了它的k理论群。另一篇是论文《Dyck移位的λ -图系统及其k群》（与W. Krieger合著），《文献数学》8(2003),79-96，其中我们引入了λ -图系统的λ -熵概念，作为拓扑熵的推广。我们还研究了符号矩阵系统的强位移等价，《遍历理论与动力系统》23(2003),1551-1774；符号动力系统的强位移等价和C^*代数的Morita等价，《遍历理论与动力系统》24(2004),199-215。在Motzkin位移-图系统的简单纯无限C^*-代数，数学中，我们也研究了与Motzkin位移-图系统相关的C^*-代数。Z. 248(2004), 369-39。符号动力学中的因子映射在C^*-代数中的蕴涵，数学学报，15(2004),313-339。C^*-代数上的非交换拓扑熵（D. Voiculescu），载于论文《C^*-代数上的拓扑熵》，《遍历理论与动力系统》（已出版）。

In this research project, we have developed a theory of symbolic dynamics by using structure theory of C^*-algebras and its K-theory. We in particular have produced two joint research papers with professor Wolfgang Krieger (Germany, Heidelberg university). One is the paper A lambda-graph system for the Dyck shift and its K-groups (with W. Krieger), Documenta Math. 8(2003), 79-96 in which we have introduced a notion of Cantor horizon lambda-graph system for Dyck shifts and computed its K-theory group. The other one is the paper A lambda-graph system for the Dyck shift and its K-groups (with W. Krieger), Documenta Math. 8(2003), 79-96 in which we have introduced a notion of lambda-entropy for lambda-graph systems as a generalization of topological entropy. We also studied strong shift equivalence in symbolic dyanical systems in the two papers On strong shift equivalence of symbolic matrix systems, Ergodic Theory and Dynamical System 23(2003), 1551-1774 and Strong shift equivalence of symbolic dynamical systems and Morita equivalence of C^*-algebras, Ergodic Theory and Dynamical System 24(2004), 199-215. We also studied the C^*-algebra associated with a lambda-garph system of Motzkin shift in A simple purely infnite C^*-algebra associated with a lambda-graph system of the Motzkin shift, Math. Z. 248(2004), 369-39. Factor maps in symbolic dynamics have been studied in Factor maps of symbolic dynamics and inclusions of C^*-algebras, International J. Math. 15(2004), 313-339. Noncommutative topological entropy or D. Voiculescu on the C:-algebras associated with lambda-graph systems have been computed in the paper Topological entropy in C^*-algebras associated with lambda-graph systems, prerint, Ergodic Theory and Dynamical System (in press)