Operator algebras and dynamical systems

20世纪80年代，A.康纳斯介绍了非交换空间，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新的数学领域，对数学和物理学的许多领域都有贡献。非交换空间的大量例子与拓扑空间的变换群的自由极小作用（直到轨道等价）有关。Bratteli-Vershik模型一直是我们研究的关键工具（与I.F. Putnam和C.F. Skau）关于Cantor极小系统的拓扑轨道等价性。我们计划将Bratteli-Vershik构造扩展到其他同胚群的自由极小作用。 * * von Neumann和C *-代数的几个性质转化为它们的酉群的性质。我计划继续研究和推广这种对应的C *-代数和冯诺依曼代数在合作P。Ng和A.谢拉科夫斯基* * C *-代数理论最重要的最新发展之一是Elliott的顺从C *-代数分类方案。由于真实的C *-代数和它们的K-理论不变量在物理学中起着越来越重要的作用，我计划扩大已经由它们相应的Elliott不变量分类的顺从的真实的C *-代数的类。* * Connes和Woods在遍历理论中引入了一个新的性质，称为近似传递性（AT），以刻画一类冯诺依曼代数。他们还指出，随机游走的渐近边界是AT。我将继续研究的AT和渐近边界的随机游动的群体。然后与雷诺我将分析的渐近边界的随机游动的广群。使用维数空间的概念，维数群的测度论版本，我介绍了与D. E。Handelman，我们将继续研究AT（k）作用，近似传递性和矩阵值随机游动的Poisson边界的推广版本。

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.