The Connes embedding problem: A guided tour

The Connes embedding problem (CEP) is a in the theory of tracial von Neumann algebras and asks whether or not every algebra embeds into an ultrapower hyperfinite II 1 _1 factor. CEP has had interactions with wide variety areas mathematics, including mathvariant="normal">C ? encoding="application/x-tex">\mathrm {C}^* -algebra theory, geometric group free probability, noncommutative real algebraic geometry, to name few. After remaining open for over 40 years, negative solution was recently obtained as corollary landmark result quantum complexity known alttext="upper M I P asterisk Baseline equals R E"> MIP = RE encoding="application/x-tex">\operatorname {MIP}^*=\operatorname {RE} . In these notes, we introduce all background material necessary understand proof from fact, outline two such proofs, one following “traditional” route that goes via Kirchberg’s QWEP Tsirelson’s information second uses basic ideas logic.