Logic and C*-algebras

逻辑和 C* 代数

• 批准号: RGPIN-2017-05650
• 负责人: Farah, Ilijas
• 金额: $ 2.7万
• 依托单位: York University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=650458
• 关键词: Logic; algebras

This is an interdisciplinary proposal at the interface between C*-algebras and logic. A C*-algebra is an algebra of bounded linear operators on a complex Hilbert space closed under the formation of adjoints and the norm topology. Hilbert space is the infinite-dimensional modification of our standard three-dimensional space. The study of C*-algebras began in the 1940s, and has since expanded to touch much of modern mathematics, including number theory, geometry, ergodic theory, mathematical physics, and topology.***Two areas of mathematical logic with connections to C*-algebras are set theory and model theory. Some well-known and long-standing open problems about C*-algebras were recently resolved using set theory and model theory. Moreover, in some cases it was proved that these problems have an inherent foundational aspect and that the use of logic in their solution was necessary. ***Most important questions that I will work on are the existence of a K-theory reversing automorphism of the Calkin algebra and Naimark's problem. The first question was asked by Brown, Douglas and Fillmore in 1977. I have proved that a negative answer is relatively consistent with the standard axioms of set theory, ZFC and (together with Phillips and Weaver) that a closely related problem of the existence of outer automorphisms of the Calkin algebra cannot be decided in ZFC. Naimark's problem 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. A confirmation of this conjecture would be a first step in extending Glimm's dichotomy to the realm of nonseparable C*-algebras.***Model theory studies the definable sets in mathematical structures and their first-order properties. The methods of continuous model theory were adapted to operator algebras less than a decade ago and much progress was made in understanding structure theory of massive algebras, such as ultrapowers and relative commutants. The latter algebras are more important and less understood, and I propose to investigate the exact formal properties which make them such an important tool in the study of operator algebras. ***A resolution of, or even a substantial progress on, any of these problems would provide new insight into the structure of C*-algebras. Mathematical logic (and set theory in particular) was developed in the `commutative' context and noncommutative problems pose new challenges. Further progress in applications of logic to operator algebras will require refinement of the existing techniques and development of new ones.

这是 C* 代数和逻辑之间的跨学科提案。 C* 代数是在伴随伴随和范数拓扑的形成下封闭的复希尔伯特空间上的有界线性算子的代数。希尔伯特空间是我们标准三维空间的无限维修改。 C* 代数的研究始于 20 世纪 40 年代，此后扩展到许多现代数学领域，包括数论、几何、遍历理论、数学物理和拓扑。***与 C* 代数相关的数理逻辑的两个领域是集合论和模型论。最近使用集合论和模型论解决了一些有关 C* 代数的众所周知且长期存在的开放问题。此外，在某些情况下，事实证明这些问题具有固有的基础方面，并且在解决问题时使用逻辑是必要的。 ***我将研究的最重要的问题是卡尔金代数反转自同构的 K 理论的存在性和奈马克问题。第一个问题是由 Brown、Douglas 和 Fillmore 在 1977 年提出的。我已经证明否定答案与集合论的标准公理 ZFC 相对一致，并且（与 Phillips 和 Weaver 一起）卡尔金代数的外自同构的存在性密切相关的问题不能在 ZFC 中决定。 Naimark 的问题询问是否每个具有唯一（直到共轭）不可约表示的 C* 代数都与某些希尔伯特空间上的紧算子代数同构。 C. Akemann 和 N. Weaver 在 2002 年证明这个问题的否定答案与集合论的标准公理相对一致。我推测奈马克问题的肯定答案也与集合论的标准公理相对一致。对这一猜想的证实是将 Glimm 的二分法扩展到不可分的 C* 代数领域的第一步。***模型理论研究数学结构中的可定义集合及其一阶性质。连续模型理论的方法在不到十年前就适用于算子代数，并且在理解超幂和相对交换子等大规模代数的结构理论方面取得了很大进展。后者的代数更重要，但理解较少，我建议研究确切的形式属性，这使得它们成为算子代数研究中如此重要的工具。 ***任何这些问题的解决，甚至是实质性进展，都将为 C* 代数的结构提供新的见解。数理逻辑（特别是集合论）是在“交换”背景下发展起来的，非交换问题提出了新的挑战。逻辑在算子代数中的应用的进一步进展将需要改进现有技术并开发新技术。