Hilbert modules and the structure of C*-algebras

PI将研究C*-代数上的Hilbert模的结构，以及它与代数本身的结构的关系。最近的研究表明，即使对于简单的代数，Hilbert模也比C*-代数上的有限生成投射模承载了更多的信息。这个项目的目的是利用这一事实来研究C*-代数理论的几个方面：C*-动力系统的K-理论的分类，通过相关C*-代数的Hilbert模不变量来协调经典的和动态的覆盖维度，以及寻求核简单可分C*-代数的张量吸收定理，它将提供纯无限简单C*-代数的Kirchberg-S吸收定理的推广。该建议的总体主题是在动力学(具有时间演化的系统)的界面和对数学对象进行分类的非常自然的愿望。我们的目标是使用泛函分析工具来协调和内插经典空间(无时间演化)、周期空间(其中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)的性质。相反，我们将研究动态空间的性质在将这些空间转换为算子族(如果您愿意的话，可以称为无穷维矩阵)后如何表现出来，并将致力于确定何时两个这样的族本质上相同。

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.