Regularity, complexity, and perturbation for C*-algebras

这项研究涉及三个广泛的项目，每个项目都有几个不同难度的子问题。第一个项目旨在证明可分离的简单核C*-代数的三个正则性的等价性：一个拓扑，一个同调和一个代数。这一事实的证明，最近在轨迹的某些限制下取得了快速的进展，将代表Kirchberg对纯无限简单核C*-代数的表征的深刻推广。第二个项目涉及描述集合论和C*-代数之间的相互作用，特别是使用Borel可约性的概念来回答，对于各种类型的功能分析对象，这个问题：同构有多复杂？主要研究者将考虑的对象包括核代数、精确代数和局部自反C*代数，以及算子空间和系统。第三个项目检查C*-代数的均匀扰动和它们保持结构和不变量的程度。这里的重要问题包括z稳定性和稳定性是否被这种扰动所保留。科学探究的许多领域都需要分析无限维系统以及它们可以转换的方式。例子包括量子物理模型、信号分析和天气模式。无限维系统当然是复杂的。理解它们通常是通过用更简单的有限维系统来近似它们。本项目使用这种方法来努力理解称为C*-代数的无限维系统。有限维逼近对象是带有复数项的方阵。我们的目的是确定一个条件，在这个条件下，人们可以用一个固定的有限数量的重叠阵列任意接近无限维系统。我们知道，最后一个属性对原始系统有强大的影响，这些影响揭示了其结构的大量细节。

This research concerns three broad projects, each with several subproblems of varying difficulty. The first project aims to prove the equivalence of three regularity properties for separable simple nuclear C*-algebras: one topological, one homological, and one algebraic. The proof of this fact, which has recently seen rapid progress toward a solution under some restrictions on traces, would represent a deep generalization of Kirchberg's characterization of purely infinite simple nuclear C*-algebras. The second project concerns the interplay between descriptive set theory and C*-algebras, and specifically the use of the notion of Borel reducibility to answer, for various classes of functional analytic objects, the question: How complicated is isomorphism? The objects that the principal investigator will consider include nuclear, exact, and locally reflexive C*-algebras, and operator spaces and systems. The third project examines uniform perturbations of C*-algebras and the degree to which they preserve structure and invariants. Important questions here include whether or not Z-stability and stability are preserved by such perturbations.Many fields of scientific inquiry require analyzing infinite-dimensional systems and the ways in which they can be transformed. Examples include models for quantum physics, signal analysis, and weather patterns. Infinite-dimensional systems are, of course, complicated. Understanding them often proceeds by approximating them with simpler finite-dimensional systems. This project uses this approach in an effort to understand infinite-dimensional systems called C*-algebras. The finite-dimensional approximating objects are square arrays with complex number entries. Our aim is to identify conditions under which one can approximate the infinite-dimensional system arbitrarily closely using only a fixed finite number of overlapping arrays. This last property is known to have powerful consequences for the original system, consequences that reveal a great deal of detail about its structure.