Operator Algebras and Dynamics

算子代数和动力学

• 批准号: 105463-2012
• 负责人: Giordano, Thierry
• 金额: $ 2.19万
• 依托单位: University of Ottawa
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=569059
• 关键词: Operator; Algebras; Dynamics

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

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，这是一个快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。非交换空间的例子的大量来源是由等价关系的商给出的。与 I.F.普特南，C.F. Skau 和最近的 H. Matui，我广泛研究了与康托集上的最小同胚相关的非交换空间。我们计划继续研究更大类别的最小作用的轨道等效性。 Connes 和 Woods 引入了遍历理论的一个新属性，称为近似传递性 (AT) 描述一类冯诺依曼代数的特征。他们还指出，一些随机游走的泊松边界是 AT。最近D.E.汉德曼和我定义了维度空间，即维度组的测度理论版本。使用这个新概念，我们将继续研究 AT(k)_actions，这是近似传递性和矩阵值随机游走的泊松边界的广义版本。 C* 代数理论最近最重要的发展之一是 Elliott 的 C* 代数分类方案。他的出发点是 Blackadar 的一个 AF 代数的非近似有限 (AF) 子 C* 代数的例子。我想研究真正的 C* 代数，相当于 Blackadar 的例子和 Elliott 的程序。 冯·诺依曼代数和 C* 代数的几个性质转化为其酉群的性质。我计划继续研究并扩展 C* 代数和冯诺依曼代数的对应关系。 格罗莫夫和韦斯提出的 sofic 群的概念被扩展到群的可测量的行为和可测量的等价关系。在这项研究中，我将研究与 sofic 等价关系相关的整个群的属性