The Connes embedding problem: A guided tour
The Connes embedding problem: A guided tour
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Connes 嵌入问题：导览
DOI：
10.1090/bull/1768
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发表时间：
2022
影响因子：
1.3
通讯作者：
Goldbring, Isaac
中科院分区：
文献类型：
--
作者：
Goldbring, Isaac
Connes 嵌入问题 (CEP) 是微量冯诺依曼代数理论中的一个问题，它询问是否每个微量冯诺依曼代数都嵌入到超有限 II 因子的超幂中。 CEP 与数学的各个领域都有相互作用，包括代数理论、几何群论、自由概率和非交换实代数几何等等。在保持开放 40 多年之后，最近获得了一个负解，作为量子复杂性理论中具有里程碑意义的结果的推论。在这些笔记中，我们介绍了理解 CEP 负解证明所需的所有背景材料。事实上，我们概述了两个这样的证明，一个遵循“传统”路线，通过基希伯格的代数理论中的 QWEP 问题和量子信息理论中的 Tsirelson 问题，第二个则使用逻辑的基本思想。参考
The Connes embedding problem (CEP) is a problem in the theory of tracial von Neumann algebras and asks whether or not every tracial von Neumann algebra embeds into an ultrapower of the hyperfinite IIfactor. The CEP has had interactions with a wide variety of areas of mathematics, including-algebra theory, geometric group theory, free probability, and noncommutative real algebraic geometry, to name a few. After remaining open for over 40 years, a negative solution was recently obtained as a corollary of a landmark result in quantum complexity theory known as. In these notes, we introduce all of the background material necessary to understand the proof of the negative solution of the CEP from. In fact, we outline two such proofs, one following the “traditional” route that goes via Kirchberg’s QWEP problem in-algebra theory and Tsirelson’s problem in quantum information theory and a second that uses basic ideas from logic. References