Research Training Network in Model Theory
Publications > Preprint server > Preprint Number 2064

Preprint Number 2064

Previous Next Preprint server

2064. Isaac Goldbring
The Connes Embedding Problem: A guided tour

Submission date: 26 September 2021


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.

Mathematics Subject Classification:

Keywords and phrases:

Full text arXiv 2109.12682: pdf, ps.

Last updated: September 29 2021 18:47 Please send your corrections to: