Operator Algebras and Dynamics

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对易空间的例子。与I.F.Putnam，C.F.Skau和最近的H.Matui一起，我广泛地研究了Cantor集上与极小同胚有关的非交换空间。我们计划继续研究更大类极小动作的轨道等价性。 Connes和Wood引入了遍历理论的一个新性质，称为近似传递性(AT 刻划了一类von Neumann代数。他们还注意到一些随机游动的泊松边界是AT。最近，D.E.Handelman和我定义了维度空间，这是维度群的一种测度论版本。利用这一新概念，我们将继续研究AT(K)作用，它是近似传递性和矩阵值随机游动的Poisson边界的推广形式。 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。他的出发点是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C*-代数和von Neumann代数的这种对应关系。 由Gromov和Weiss引入的SOFIC群的概念被推广到群的可测动作和可测等价关系。在这项研究中，我将研究与SOFIC等价关系相关的全群的性质

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. With I.F. Putnam, C.F. Skau and more recently H. Matui, I have extensively studied the noncommutative spaces associated to minimal homeomorphisms on a Cantor set. We plan to pursue the study of orbit equivalence for larger class of minimal actions. Connes and Woods introduced a new property of ergodic theory called approximate transitivity (AT) to characterize a class of von Neumann algebras. They noted also that the Poisson boundary of some random walks is AT. Recently D.E. Handelman and I defined dimension spaces, a measure theoretic version of dimension groups. Using this new notion, we will continue the study of AT(k)_actions, a generalized verson of approximate transitivity and of the Poisson boundary of matrix-valued random walks. One of most important recent development in the theory of C*-algebras is Elliott's classification scheme of amenable C*-algebras. His 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. Several properties of von Neumann algebras and C*-algebras translate in properties of their unitary groups. I plan to continue to study and extend this correspondence for both C*-algebras and von Neumann algebras. The notion of sofic groups, introduced by Gromov and Weiss, was extended to measurable actions of groups and to measurable equivalence relations. In this research I will study properties of the full group asociated to sofic equivalence relations