Logic and C*-algebras

这是在C*-代数和逻辑的交界处提出的一个跨学科的建议。C*-代数是复Hilbert空间上的有界线性算子的代数，在伴随和范数拓扑的形成下是闭的。希尔伯特空间是我们的标准三维空间的无限维修正。C*-代数的研究始于20世纪40年代，此后已扩展到涉及许多现代数学，包括数论、几何、遍历理论、数学物理和拓扑学。 与C*-代数有关的两个数理逻辑领域是集合论和模型论。最近利用集合论和模型论解决了关于C*-代数的一些著名的和长期存在的公开问题。此外，在某些情况下，事实证明，这些问题有其内在的基本方面，在解决这些问题时必须使用逻辑。 我要研究的最重要的问题是K理论的存在性，它逆转了Calkin代数的自同构和Naimark问题。第一个问题是布朗、道格拉斯和菲尔莫尔在1977年提出的。我已经证明了否定答案与集合论、ZFC的标准公理相对一致，并且(连同Phillips和Weaver)一个密切相关的关于Calkin代数的外自同构的存在的问题不能在ZFC中决定。Naimark的问题是，每个具有唯一(直到共轭)不可约表示的C*-代数是否同构于某个Hilbert空间上的紧算子代数。对这个问题的否定回答被证明与C.Akemann和N.Weaver在2002年提出的集合论标准公理相对一致。我猜测，对奈马克问题的肯定答案也与集合论的标准公理相对一致。这一猜想的证实将是将Glimm的二分法扩展到不可分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。不到十年前，连续模型理论的方法被应用于算子代数，在理解大规模代数的结构理论，如超幂和相对换位子方面取得了很大的进展。后一类代数更重要，但了解得更少，我建议研究使它们成为研究算子代数的重要工具的确切形式性质。 这些问题的任何一个的解决，甚至是实质性的进展，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”的背景下发展起来的，非交换问题提出了新的挑战。在将逻辑应用于算子代数方面的进一步进展将需要改进现有技术和开发新的技术。

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.