C*-algebras and set theory

这是一个跨学科的建议之间的接口C *-代数和集合论。 这两个主题曾经被认为只是表面上的相关，但在过去的几年里，已经发现了令人兴奋的联系和深层定理。一些著名的和长期存在的C *-代数的开放问题最近得到解决，使用集合论。此外，还证明了这些问题有一个内在的基础方面，在解决这些问题时使用集合论是必要的。 我将描述我计划解决的C *-代数上的几个问题中的两个，它们的解决方案似乎需要集合论。 第一个是Naimark的问题。它询问是否每个具有唯一（直到共轭）不可约表示的C *-代数同构于某些希尔伯特空间上的紧算子代数。这个问题的一个否定的答案被证明与集合论的标准公理是相对一致的。Akemann和N. 2002年的韦弗。我猜想，积极的答案Naimark的问题也是相对一致的标准公理集理论，这将导致一个有趣的二分法表示理论的任意C *-代数。 第二个问题，或者更确切地说是一系列问题，具有非常不同的性质，因为它不涉及独立性结果。Elliott利用K-理论不变量对核C *-代数进行分类的计划是现代算子代数理论中的一个重要主题。近年来，随着M. Rordam和A.汤姆斯和引进新的不变量，如Cuntz半群。我建议使用Hjorth，Kechris等人开发的抽象分类理论来分析可分C *-代数的分类问题。该理论提供了比较分类问题复杂性的方法，它可以通过简单的不变量（如可数群）来隔离分类的具体障碍。

This is an interdisciplinary proposal on the interface between C*-algebras and set theory. These two subjects were once considered to be only superficially related but exciting connections and deep theorems have been discovered during the last few years. Some well-known and long-standing open problems about C*-algebras were recently resolved using set theory. Moreover, it was proved that these problems have an inherent foundational aspect and that the use of set theory in their solution was necessary. I will describe two of several problems on C*-algebras that I plan to solve and whose solutions appear to require set theory. The first one is Naimark's problem. It asks whether every C*-algebra with a unique (up to conjugacy) irreducible representation is isomorphic to the algebra of compact operators on some Hilbert space. A negative answer to this problem was shown to be relatively consistent with the standard axioms of set theory by C. Akemann and N. Weaver in 2002. I conjecture that the positive answer to Naimark's problem is also relatively consistent with the standard axioms of set theory and that it would lead to an interesting dichotomy in representation theory for arbitrary C*-algebras. The second problem, or rather circle of problems, is of a very different nature since it does not involve independence results. Elliott's program to classify nuclear C*-algebras by K-theoretic invariants is a major theme in the modern theory of operator algebras. In recent years it underwent a transformation after the discovery of counterexamples by M. Rordam and A. Toms and the introduction of new invariants such as the Cuntz semigroup. I propose to analyze the classification problem for separable C*-algebras using the abstract classification theory developed by Hjorth, Kechris, and others. This theory provides the means for comparing the complexity of classification problems and it can isolate concrete obstructions to classification by simple invariants such as countable groups.