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 II$_1$ factor. The CEP has had interactions with a wide variety of areas of mathematics, including C*-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 $\operatorname{MIP}^*=\operatorname{RE}$. In these notes, we introduce all of the background material necessary to understand the proof of the negative solution of the CEP from $\operatorname{MIP}^*=\operatorname{RE}$. In fact, we outline two such proofs, one following the "traditional" route that goes via Kirchberg's QWEP problem in C*-algebra theory and Tsirelson's problem in quantum information theory and a second that uses basic ideas from logic.

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Connes 嵌入问题：导览

Connes 嵌入问题 (CEP) 是追踪冯诺依曼代数理论中的一个问题，它询问是否每个追踪冯诺依曼代数都嵌入到超有限 II$_1$ 因子的超幂中。CEP 与广泛的数学领域进行了交互，包括 C*-代数理论、几何群论、自由概率和非对易实代数几何（仅举几例）。在保持开放 40 多年之后，最近获得了一个否定解，作为量子复杂性理论中具有里程碑意义的结果的推论，称为 $\operatorname{MIP}^*=\operatorname{RE}$。在这些笔记中，我们介绍了从 $\operatorname{MIP}^*=\operatorname{RE}$ 理解 CEP 的负解的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“