Regularity, complexity, and perturbation for C*-algebras

C* 代数的正则性、复杂性和扰动

• 批准号: 1301673
• 负责人: Andrew Toms
• 金额: $ 19.44万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-15 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1301673&HistoricalAwards=false
• 关键词: Regularity; complexity; perturbation; algebras

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.

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。第一个项目的目的是证明可分简单核C*-代数的三个正则性的等价性：一个拓扑的，一个同调的，一个代数的。这个事实的证明，最近已经看到了快速进展的解决方案下的一些限制的痕迹，将代表一个深刻的推广基希贝格的特点，纯粹无限简单核C*-代数。 第二个项目涉及的相互作用之间的描述集理论和C*-代数，特别是使用的概念，波莱尔reductible回答，为各类功能分析对象，问题：如何复杂的是同构？主要研究者将考虑的对象包括核，精确和局部自反C*-代数，算子空间和系统。 第三个项目研究C*-代数的一致扰动以及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样的扰动所保持。许多科学研究领域需要分析无限维系统以及它们可以转换的方式。 例子包括量子物理学、信号分析和天气模式的模型。 无限维系统当然是复杂的。 理解它们通常通过用更简单的有限维系统近似它们来进行。 该项目使用这种方法来理解称为C*-代数的无限维系统。 有限维近似对象是具有复数项的方阵。 我们的目的是确定条件下，可以近似的无限维系统任意密切使用只有一个固定的有限数量的重叠阵列。 这最后一个属性对原始系统有着强有力的影响，这些影响揭示了关于其结构的大量细节。