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.康纳斯介绍了非交换空间，并开始发展非交换几何，一个迅速发展的新领域的数学，有助于许多学科的数学和物理。非交换空间的大量例子是由等价关系的商给出的。与IF Putnam，C.F. Skau和最近的H. Matui，我已经广泛地研究了与Cantor集上的极小同胚相关的非交换空间。我们计划继续研究更大类的最小作用量的轨道等价性。 Connes和Woods引入了遍历理论的一个新性质，称为近似传递性（AT）， 刻画了一类von Neumann代数。他们还指出，一些随机游动的泊松边界是AT。最近DE Handelman和我定义了维数空间，这是维数群的测度论版本。利用这个新概念，我们将继续研究AT（k）作用，一个广义的近似传递性和Poisson边界的矩阵值随机游动。 C ~*-代数理论的一个重要发展是Elliott提出的顺从C ~*-代数的分类方案。他的出发点是一个例子Blackadar的非近似有限（AF）子C*-代数的AF代数。我想研究与Blackadar的例子和Elliott的程序等价的真实的C*-代数。 冯诺依曼代数和C*-代数的一些性质转化为它们的酉群的性质。我计划继续研究和扩展这两个C*-代数和冯诺依曼代数的对应关系。 Gromov和韦斯引入的sofic群的概念被推广到群的可测作用和可测等价关系。在本研究中，我将研究与sofic等价关系相关的全群的性质