Dynamics and C*-algebras

动力学和 C* 代数

• 批准号: RGPIN-2016-04104
• 负责人: Putnam, Ian
• 金额: $ 2.4万
• 依托单位: University of Victoria
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=684667
• 关键词: Dynamics; algebras

Operator algebras were initially studied as models for quantum mechanical systems, because the noncommutativity of multiplication of operators can be used to encode the Heisenberg uncertaintly principle. Since then, this same noncommutative nature of operator algebras has been applied in many other areas of mathematics. This has been particularly successful for dynamical systems, the mathematical models for the spatial and temporal evolution of physical systems. This interaction between operator algebras and dynamical systems is the subject of my research.******There are quite general constructions of operator algebras from dynamical systems. In the view of Alain Connes' program of noncommutative geometry, these act as a replacement for the space of orbits of the system. This first provides a range of tools for understanding the dynamics through their associated operator algebras. This has been particularly effective over the past thirty years, as Connes' program has provided powerful new ideas. As an example, Krieger gave new invariants for symbolic systems through K-theory of the C*-algebras. Results with my co-authors show how analogous results can provide complete invariants for the orbit structure of other systems. ***The construction from dynamical systems also provides a rich source of examples of operator algebras and understanding their structure has been a major goal for the field. George Elliott's classification program for amenable C*-algebras has been one of the largest areas of operator algebras over the past twenty-five years and many of the most impressive results, such as those of Toms and Winter, have been for examples arising from dynamical systems. My own work in this area in the early days produced ideas and technical tools which are still in use today.******My proposal is to continue my investigations into these interactions. There is special emphasis on chaotic systems for which I have recently extended Krieger's invariant for to a much broader class of chaotic dynamical systems. This gives innovative tools for the study of the geometry of fractals. The goal is a better understanding of the invariant but also the development of tools within C*-algebra theory which elucidate the dynamical structure.******In another direction, my goal is to develop tools that provide quantitative measures for a pair of C*-algebras which, although each is constructed from complicated dynamics, are related to each other in a fairly simple fashion.***********

最初将操作员代数作为量子机械系统的模型进行了研究，因为可以使用操作员乘法的非交换性来编码Heisenberg不确定性的原理。从那时起，在许多其他数学领域都应用了操作员代数的这种非交通性质。对于动态系统，这是物理系统的空间和时间演变的数学模型，这是特别成功的。操作器代数与动态系统之间的这种相互作用是我的研究的主题。******是动态系统的运算符代数的一般结构。在Alain Connes的非交通几何形状的图表中，这些程序可用于替代系统轨道的空间。这首先提供了一系列工具，可通过其相关的操作员代数来理解动态。在过去的三十年中，这特别有效，因为Connes的计划提供了有力的新想法。例如，克里格（Krieger）通过c*-ergebras的K理论为符号系统提供了新的不变性。我的合着者的结果表明，类似结果如何为其他系统的轨道结构提供完整的不变性。 ***动态系统的结构还提供了广泛的操作员代数示例，了解其结构已成为该领域的主要目标。在过去的二十五年中，乔治·埃利奥特（George Elliott）针对的C* - 代数分类计划一直是运营商代数的最大领域之一，许多最令人印象深刻的结果，例如Toms和Winter的结果，都是由动态系统引起的示例。在早期，我自己在这一领域的工作产生了当今仍在使用的想法和技术工具。******我的建议是继续对这些互动进行调查。我最近特别强调了混乱的系统，我最近将克里格的不变性扩展到了更广泛的混乱动力学系统。这为研究分形的几何形状提供了创新的工具。目的是更好地理解不变性的c*-Algebra理论中工具的开发，该工具阐明了动态结构。******在另一个方向上，我的目标是开发工具，为一对C*-Algebras提供定量措施，尽管每种都来自复杂的动力学，但在一个相当简单的时尚中都相互关联。