The Connes Embedding Problem: A guided tour

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.