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$_1$ 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.