Interactions between C*-algebra and set theory

这个项目涉及集合论和C*-代数之间的相互作用。主要目标是使用集合论工具解决旧的突出的开放问题，这可能涉及证明独立性结果。可以解决的具体问题包括：奈马克关于只有一个不可约表示的非平凡C*代数的存在性问题；对角线上纯态限制纯态的Anderson猜想卡迪逊-辛格问题，它有许多等价的公式，对数学的几个领域都很重要；关于分离纯态的适当子代数存在性的Stone-Weierstrass问题；Brown、Douglas和fillmore关于Calkin代数的自同构的存在性问题，该自同构取单侧移到其伴随矩阵的像；c *代数是一类数学结构，最初主要是为了在物理学中应用而研究的，随着时间的推移，它在数学和物理学的各个领域都有了实质性的应用。目前该领域的许多工作都集中在发展与这些其他领域的进一步联系上，但基本理论的基本问题仍未解决。在过去的十年中，C*代数中一系列长期存在的开放问题已经以一种基本的方式解决了，这些技术来自看似无关的集合论领域。这种令人惊讶的联系丰富了这两个领域。在这个方向上的进一步工作有望在基础水平上更好地理解C*-代数。

This project involves the interaction between set theory and C*-algebras.The primary goal is to settle old outstanding open problems using settheoretic tools, which could involve proving independence results.Specific problems that could be addressed include the following: Naimark's problemabout the existence of nontrivial C*-algebras with only one irreduciblerepresentation; Anderson's conjecture about pure states restricting topure states on diagonal masas; the Kadison-Singer problem that has manyequivalent formulations and is significant to several areas of mathematics;the Stone-Weierstrass problem about the existence of proper subalgebrasthat separate pure states; the problem of Brown, Douglas, and Fillmoreabout the existence of an automorphism of the Calkin algebra taking theimage of the unilateral shift to its adjoint; and Arveson's problem aboutthe existence of sufficiently many boundary representations fornonseparable operator systems.C*-algebras, a class of mathematical structures that were originallystudied primarily for applications in physics, have over time foundsubstantial applications in a broad variety of areas of both mathematicsand physics. Much current work in this field focuses on developingfurther connections to these other areas, but fundamental questions ofbasic theory remain open. In the past decade a series of long-standingopen problems in C*-algebras have been settled using, in an essential way,techniques from the seemingly unrelated area of set theory. This was asurprising connection that has enriched both fields. Further work inthis direction is expected to lead to a better understanding of C*-algebras at a fundamental level.