Amenabilty, soficity, dynamics, and operator algebras

适应性、社交性、动力学和算子代数

• 批准号: 1500593
• 负责人: David Kerr
• 金额: $ 18万
• 依托单位: Texas A&M University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1500593&HistoricalAwards=false
• 关键词: Amenabilty; soficity; dynamics; operator; algebras

The subject of functional analysis provides a powerful mathematical language for the investigation of infinite-dimensional structures. The abstract nature of this language allows for a wide spectrum of applications, whose various settings include quantum mechanics, the theory of probability, and the study of systems, physical or otherwise, as they evolve in time. The project will concentrate on several aspects and generalized forms of the last of these three settings, and it will largely revolve around phenomena related to entropy.Finite or finite-dimensional approximation has long played a fundamental role in both dynamics and operator algebras and is manifest in its most basic and influential forms through the notions of amenability and soficity and their algebraic analogues. In the domain of C*-algebras, amenability has been subject to various refinements like finite decomposition rank and finite nuclear dimension that express the kind of topological regularity necessary for K-theoretic classification theorems within the Elliott program. This project aims to broaden our understanding of the relationships between these regularity properties and phenomena in topological dynamics, in particular mean dimension and Rokhlin-type dimensional invariants, and to test this understanding on a variety of examples that have so far remained beyond the scope of classification, such as actions of branch groups. Combinatorial independence will be brought into this picture as a tool for analyzing mixing and paradoxicality in dynamics and their relation to quasidiagonality and dimensional behavior, and it will be also applied to actions of sofic groups in the study of entropy and orbit equivalence.

泛函分析这门学科为研究无限维结构提供了一门强大的数学语言。这种语言的抽象性质允许广泛的应用，其各种设置包括量子力学、概率论，以及研究系统，无论是物理的还是其他的，因为它们随着时间的演变。这个项目将集中在这三个背景中最后一个的几个方面和推广形式，它将主要围绕与熵有关的现象。有限维近似长期以来在动力学和算子代数中都扮演着基本的角色，并通过亲和性和柔软性的概念及其代数类似物体现在其最基本和最有影响力的形式中。在C*-代数域中，顺从性受到各种细化的影响，如有限分解秩和有限核维度，它们表示了Elliott程序中K-理论分类定理所必需的那种拓扑正则性。这个项目旨在扩大我们对这些正则性与拓扑动力学现象之间的关系的理解，特别是平均维度和Rokhlin型维度不变量，并在迄今仍超出分类范围的各种例子上检验这一理解，例如分支群的作用。组合独立性将作为一种分析动力学中的混合和悖论及其与拟对角线和维度行为的关系的工具被引入这一图景，它也将被应用于研究熵和轨道等价中SOFIC群的行为。