Hilbert modules and the structure of C*-algebras

PI将研究C*-代数上Hilbert模的结构，以及它对代数本身结构的影响。Hilbert模最近被证明比C*-代数上有限生成的射影模携带更多的信息，即使对于简单代数也是如此。本项目旨在利用这一事实来研究C*-代数理论的几个方面：通过k理论对C*-动力系统进行分类，通过相关C*-代数的Hilbert模不变量调和经典和动态覆盖维，以及追求核简单可分C*-代数的张量吸收定理，该定理将提供Kirchberg？纯粹无限简单C*代数的s吸收定理。该提案的首要主题是动力学（配备时间演化的系统）和对数学对象进行分类的非常自然的愿望的接口。我们的目标是使用功能分析的工具来调和和插值经典空间（无时间演化），周期空间（每个点遵循有限周期的时间演化）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）之后表现出来，并将确定两个这样的族何时本质上相同。

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.