Topological orbit equivalence, amenability, approximate transitivity, antiautomorphisms, unitary groups and operator algebras

在20世纪80年代，a . Connes引入了非交换空间，并开始发展非交换几何，这是一个迅速发展的数学新领域，对数学和物理的许多学科做出了贡献。等价关系的商给出了非交换空间的大量例子。I.F. Putnam， C.F. Skau和我对Cantor集合上与极小同胚相关的非交换空间进行了广泛的研究。在与H. Matui的合作下，我们计划继续研究更大类最小作用的轨道等效性。cones和Woods定义了遍历理论的一个新性质，称为近似传递性（AT）来表征一类von Neumann代数。他们还注意到一些随机游走的泊松边界是AT。最近，我和D.E. Handelman发展了维空间，这是维群的一种度量理论。我们将继续使用维空间研究随机游走的泊松边界，并将其应用于冯·诺伊曼代数。最近在C*-代数分类方面最重要的发展之一是Elliott的可调C*-代数分类程序。它的出发点是由Blackadar给出的AF代数的非近似有限（AF）子C*-代数的一个例子。我想研究等价于Blackadar的例子和Elliott的程序的真正的C*代数。冯诺依曼代数和C*-代数的许多性质转化为它们的酉群的性质，在某些情况下被认为是离散的或拓扑群。我计划继续研究C*-代数类的这种对应关系，以及与冯诺依曼代数的Ng的对应关系。

In the 1980's, A. Connes introduced noncommutative spaces, and began to develop noncommutative geometry, a rapidly growing new area of mathematics which contributes to many disciplines in mathematics and physics. A large source of examples of noncommutative spaces is given by the quotient of equivalence relations.  I.F. Putnam, C.F. Skau and I have extensively studied the noncommutative spaces associated with minimal homeomorphisms on a Cantor set. In collaboration with H. Matui, we plan to pursue the study of orbit equivalence for larger classes of minimal actions.Connes and Woods defined a new property of ergodic theory called approximate transitivity (AT) to characterize a class of von Neumann algebras. They also noted that the Poisson boundary of some random walks is AT. Recently D.E. Handelman and I developed dimension spaces, a measure theoretic version of dimension groups. We will continue the study of the Poisson boundary of random walks using dimension spaces and will apply it to von Neumann algebras.One of most important recent developments in the classification of C*-algebras is Elliott's classification program of amenable C*-algebras. Its starting point was an example by Blackadar of a non approximately finite (AF) sub C*-algebra of an AF algebra. I want to study the real C*-algebra equivalent to both Blackadar's example and Elliott's program.Many properties of von Neumann algebras and C*-algebras translate in properties of their unitary groups, in some cases considered as discrete or as topological groups. I plan to continue the study of such correspondences for classes of both C*-algebras and with Ping Ng of von Neumann algebras.