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Preprint Number 2064
2064. Isaac Goldbring The Connes Embedding Problem: A guided tour E-mail: Submission date: 26 September 2021 Abstract: 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 MIP^*=RE. In these notes, we introduce all of the background material necessary to understand the proof of the negative solution of the CEP from MIP^*=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. Mathematics Subject Classification: Keywords and phrases: |

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