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

20世纪80年代，A·康奈斯引入了非对易空间，并开始发展非对易几何。非对易几何是数学中一个迅速发展的新领域，对数学和物理的许多学科都有贡献。由等价关系的商给出了大量非交换空间的例子，C.F.Putnam，C.F.Skau和我已经广泛地研究了Cantor集上与极小同胚相关的非交换空间。与H.Matui合作，我们计划继续研究更大类极小动作的轨道等价性。Connes和Wood定义了遍历理论的一个新性质，称为近似传递性(AT)来刻画一类von Neumann代数。他们还指出，一些随机游动的泊松边界是AT。最近，D.E.Handelman和我发展了维度空间，这是维度群的一种测度论版本。我们将继续利用维空间研究随机游动的Poisson边界，并将其应用于von Neumann代数。C*-代数分类中最重要的发展之一是Elliott的顺从C*-代数的分类程序。它的起点是Blackadar关于AF代数的非近似有限(AF子)C*-代数的一个例子。我想研究与Blackadar的例子和Elliott的程序等价的实C*-代数。von Neumann代数和C*-代数的许多性质都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*-代数类和与von Neumann代数的Ping Ng的这种对应关系。

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.