The Connes Embedding Problem: A guided tour

• URL: http://arxiv.org/abs/2109.12682v1
• Date: Sun, 26 Sep 2021 19:45:36 GMT
• Title: The Connes Embedding Problem: A guided tour
• Authors: Isaac Goldbring
• Abstract summary: The Connes Embedding Problem (CEP) is a problem in the theory of tracial von Neumann algebras. 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.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $\operatorname{MIP}^*=\operatorname{RE}$. 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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