The Connes embedding problem: A guided tour
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- by Isaac Goldbring;
- Bull. Amer. Math. Soc. 59 (2022), 503-560
- DOI: https://doi.org/10.1090/bull/1768
- Published electronically: June 17, 2022
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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 $\mathrm {C}^*$-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 $\mathrm {C}^*$-algebra theory and Tsirelson’s problem in quantum information theory and a second that uses basic ideas from logic.References
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Bibliographic Information
- Isaac Goldbring
- Affiliation: Department of Mathematics, University of California, Irvine, 340 Rowland Hall (Bldg.# 400), Irvine, California 92697-3875
- MR Author ID: 858097
- Email: isaac@math.uci.edu
- Received by editor(s): October 4, 2021
- Published electronically: June 17, 2022
- Additional Notes: The author was partially supported by NSF grant DMS-2054477.
- © Copyright 2022 American Mathematical Society
- Journal: Bull. Amer. Math. Soc. 59 (2022), 503-560
- MSC (2020): Primary 46L10, 46L06, 81P45, 03C66
- DOI: https://doi.org/10.1090/bull/1768
- MathSciNet review: 4478032