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The Connes embedding problem: a guided tour. (English) Zbl 1505.46051

Summary: 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 \(\text{C}^*\)-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 \(\mathrm{MIP}^*=\mathrm{RE}\). In these notes, we introduce all of the background material necessary to understand the proof of the negative solution of the CEP from \(\mathrm{MIP}^*=\mathrm{RE}\). In fact, we outline two such proofs, one following the “traditional” route that goes via Kirchberg’s QWEP problem in \(\mathrm{C}^*\)-algebra theory and Tsirelson’s problem in quantum information theory and a second that uses basic ideas from logic.

MSC:

46L10 General theory of von Neumann algebras
46L06 Tensor products of \(C^*\)-algebras
81P45 Quantum information, communication, networks (quantum-theoretic aspects)
03C66 Continuous model theory, model theory of metric structures
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