The Connes Embedding Problem: A guided tour

Abstract

A. 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 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. 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. In fact, we outline two such proofs, one following the "traditional" route that goes via Kirchberg's QWEP problem in C ˚-algebra theory and Tsirelson's problem in quantum information theory and a second that uses basic ideas from logic. C 7.6. A Gödelian refutation of the CEP 62 7.7. The universal theory of R and the moment approximation problem 63 7.8. A negative solution to Tsirelson's problem from MIP ˚" RE 64 8. The enforceable II 1 factor (should it exist) 66 8.1. A different kind of game 66 8.2. Enforceable properties of tracial von Neumann algebras 66 8.3. Existentially closed tracial von Neumann algebras (and yet another reformulation of CEP) 68 8.4. CEP and enforceability 69 8.5. Properties of the enforceable II 1 factor (again, should it exist) 70 References 71