A Survey on Connes' Embedding Conjecture

TL;DR

This survey reviews Connes' Embedding Conjecture, highlighting its significance in Operator Algebras and summarizing various approaches and equivalent formulations developed by mathematicians.

Contribution

It provides a comprehensive overview of the diverse methods and equivalent statements related to Connes' Embedding Conjecture in Operator Algebras.

Findings

Multiple equivalent formulations of the conjecture are presented.

The conjecture's deep connections to various subfields are summarized.

The survey clarifies the current state and open problems related to the conjecture.

Abstract

In a very celebrated paper A. Connes has formulated a conjecture which is now one of the most important open problem in Operator Algebras. This importance comes from the works of many mathematicians who have found some unexpected equivalent statements showing as this conjecture is transversal to almost all the sub-specialization of Operator Algebras. In this survey I would like to give a more or less detailed description of all these approaches.