The Connes Embedding Problem: A guided tour

TL;DR

This paper explains the recent negative resolution of the Connes Embedding Problem, connecting it to quantum complexity theory and providing background on the proof methods involving operator algebras, quantum information, and logic.

Contribution

It introduces the background and outlines two different proofs of the negative solution to the Connes Embedding Problem, linking it to quantum complexity and logic.

Findings

The CEP has been negatively resolved.

Connections established between CEP, quantum complexity, and logic.

Two distinct proof strategies are presented.

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 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 C*-algebra theory and Tsirelson's problem in quantum information theory and a second that uses basic ideas from logic.