On the interplay between C^*-algebras and topological dynamical systems

以下是我们在1997至1998学年(平成9-10)期间的主要成果。其中，前四项是由Tomiyam.1.证明了最优可能的广义动力系统的限制同构定理；即得到了动力系统之间当其相关C-代数同构时保持此类连续函数子代数同构的条件。2.引入动力系统表示理论作为推论的新立足点，我们加强了这一领域的前人的结果。3.培养了攻击一般同构问题的各种方法，其中一种是泛C-Z-交叉积的概念及其分析，另一种是区分同胚C-代数的理想为好的和坏的(和最坏的)理想以及好的理想的刻画。并从这一点分析了同构的性质。4.得到了动力系统非游荡集的行为的C-代数描述，即Birkhoff中心的深度。在这些动力系统的基本集合中，只有这件事几年来一直没有得到解决。5.Qeda在2维或3维欧氏空间中得到了热对流方程在外部区域的初值问题的整体解。对于4维或5维紧致流形中以环面为一般叶的余维Q Hausdorff叶系，Fujisaki对奇叶的类型进行了分类，并对其局部稳定性进行了分析。

The followings are main points of our results during the academic years through 1997 to 1998(Heisei 9-10). Among them, the first four items are carried out by Tomiyama.1.Proved the restricted isomorphism theorem for the best possible wide classes of dynamical systems ; namely obtained the condition between dynamical systems when their associated C-algebras are isomorphic keeping the subalgebras of continuous functions for such a class.2.Introduced the representation theory of dynamical systems as a new stand pointof the theory As consequences, we have reinforced the previous results in this field.3.Cultivated various kinds of ways to attack the general isomorphism problem, among which one is the idea of universal C-Z-crossed product with its analysis andthe other is to distinguish the ideal of homeomorphism C-algebras as good and bad (and most bad) ones together with the characterizations of good ideals. Also analysed the properties of isomophisms from this point.4.Obtained the C-algebraic descriptions of the behavior of nonwandering sets ofdynamical systems, the depths of the Birkhoff center. Among those elementary sets of dynamical systems, only this thing had been remained unsolved for several years.5.Qeda has obtained the global solutions for the initial value problem of the heat convection equation in an exterior domain in the 2 or 3 dimensional Eucidean spaces. For a codimension q Hausdorff foliation having Torus as general leaves in the 4 or 5 dimensional compact manifold, Fujisaki has classified the types of singular leaves with the analysis of their local stabilities.