The Connes embedding problem: A guided tour

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 _1 factor. The CEP has had interactions with a wide variety of areas of mathematics, including C ∗ \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 MIP ∗ = RE \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 MIP ∗ = RE \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 ∗ \mathrm {C}^* -algebra theory and Tsirelson’s problem in quantum information theory and a second that uses basic ideas from logic.