Hilbert modules and the structure of C*-algebras

Hilbert 模和 C* 代数的结构

• 批准号: 0969246
• 负责人: Andrew Toms
• 金额: $ 15.28万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-15 至 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0969246&HistoricalAwards=false
• 关键词: Hilbert; modules; structure; algebras

The PI will investigate the structure of Hilbert modules over C*-algebras, and its bearing on the structure of the algebras themselves. Hilbert modules have recently been shown to carry vastly more information than finitely generated projective modules over C*-algebras, even for simple algebras. This project aims to use this fact in studying several aspects of C*-algebra theory: the classification of C*-dynamical systems by their K-theory, the reconciliation of classical and dynamical covering dimension via a Hilbert module invariant of the associated C*-algebras, and the pursuit of a tensorial absorption theorem for nuclear simple separable C*-algebras which will provide a generalization of Kirchberg?s absorption theorem for purely infinite simple C*-algebras.The over-arching theme of the proposal is at the interface of dynamics (systems equipped with a time evolution) and the very natural desire to classify mathematical objects. We aim to use tools from functional analysis to reconcile and interpolate between the properties of classical spaces (no time evolution), periodic spaces (time evolution in which each point follows a finite cycle), and fully dynamic spaces (arbitrary time evolution). Conversely, we will study how the properties of dynamic spaces manifest themselves after translating these spaces into families of operators (infinite-dimensional matrices, if you like), and will work to determine when two such families are essentially the same.

PI将研究C*-代数上的希尔伯特模的结构，以及它对代数本身结构的影响。 希尔伯特模最近被证明比C*-代数上的代数生成投射模携带更多的信息，即使是简单代数。 这个项目的目的是利用这一事实在研究几个方面的C*-代数理论：分类的C*-动力系统的K-理论，调和的经典和动态覆盖维数通过希尔伯特模不变量的相关C*-代数，并追求一个张量吸收定理的核简单可分C*-代数，这将提供一个推广的基希贝格？的吸收定理的纯无限简单的C*-代数。该建议的过度的主题是在接口的动力学（系统配备了时间演化）和非常自然的愿望，以分类数学对象。 我们的目标是使用泛函分析的工具来协调和插值经典空间（无时间演化），周期空间（每个点都遵循有限循环的时间演化）和完全动态空间（任意时间演化）的属性。 相反，我们将研究动态空间的性质如何在将这些空间转化为算子族（无限维矩阵，如果你喜欢的话）后表现出来，并将确定两个这样的族何时本质上相同。