Operator algebras and dynamical systems

20世纪80年代S，A.Connes引入了非交换空间，并开始发展非交换几何。今天，非对易几何是一个迅速发展的数学新领域，对数学和物理的许多领域都有贡献。非对易空间的大量例子与拓扑空间的变换群的自由极小作用有关，直到轨道等价。Bratteli-Vershik模型一直是我们(与I.F.Putnam和C.F.Skau)研究Cantor极小系统拓扑轨道等价性的关键工具。我们计划将Bratteli-Vershik结构推广到自由其他同胚群的极小作用。 Von Neumann和C*-代数的几个性质转化为它们的酉群的性质。我计划与P.W.Ng和A.Sierakowski合作，继续研究和推广C*-代数和von Neumann代数的这种对应关系。 在C*-代数理论中最重要的发展之一是Elliott关于从属C*-代数的分类方案。随着实C*-代数及其K-理论不变量在物理学中发挥着越来越重要的作用，我计划扩大已由其对应的Elliott不变量分类的可服从实C*-代数的类。 Connes和Wood在遍历理论中引入了一个新的性质，称为近似传递性(AT)来刻画一类von Neumann代数。他们还指出，随机游动的渐近边界是AT。我将继续研究AT以及群上随机游动的渐近边界。然后，我将与J.雷诺一起分析群胚上随机游动的渐近边界。利用维空间的概念，这是我与D.E.Handelman引入的维群的一种测度论版本，我们将继续研究AT(K)_Actions，它是近似传递性的推广版本，以及矩阵值随机游动的Poisson边界。

In the 1980's, A. Connes introduced non-commutative spaces, and began to develop non-commutative geometry. Today non-commutative geometry a rapidly growing new area of mathematics that contributes to many fields in mathematics and physics. A large source of examples of non-commutative spaces is associated to free minimal actions, up to orbit equivalence, of group of transformations of a topological space. The Bratteli-Vershik model has been a key tool in our study (with I.F. Putnam and C.F. Skau) of topological orbit equivalence of Cantor minimal systems. We plan to extend the Bratteli-Vershik construction to free minimal actions of other groups of homeomorphisms. Several properties of von Neumann and C*-algebras translate in properties of their unitary groups. I plan to continue to study and extend such correspondences for both C*-algebras and von Neumann algebras in collaboration with P. W. Ng and with A. Sierakowski. One of most important recent development in the theory of C*-algebras is Elliott's classification scheme of amenable C*-algebras. As real C*-algebras and their K-theoretical invariants play an increasingly important role in physics, I plan to enlarge the class of amenable real C*-algebras already classified by their corresponding Elliott's invariant. Connes and Woods introduced a new property in ergodic theory called approximate transitivity (AT) to characterize a class of von Neumann algebras. They also noted that the asymptotic boundary of a random walk is AT. I will continue the study of AT and also of the asymptotic boundary of random walks on groups. Then with J. Renault I will analyse the asymptotic boundary of a random walk on a groupoid. Using the notion of dimension spaces, a measure theoretic version of dimension groups, I introduced with D.E. Handelman, we will continue the study of AT(k)_actions, a generalized version of approximate transitivity and of the Poisson boundary of matrix-valued random walks.