About the Connes Embedding Conjecture---Algebraic approaches---

TL;DR

This paper explores algebraic perspectives on the Connes Embedding Conjecture, providing new proofs of its equivalence with related conjectures and problems in operator algebras and quantum information.

Contribution

It introduces novel algebraic proofs establishing the equivalence of the Connes Embedding Conjecture with several key conjectures and problems.

Findings

New proofs of equivalence between Connes Embedding and Kirchberg's Conjecture

Positivstellensatze for trace positive polynomials derived

Connections established with Tsirelson's Problem

Abstract

This is an expanded lecture note for "Masterclass on sofic groups and applications to operator algebras" (University of Copenhagen, 5-9 November 2012). It is about algebraic aspects of the Connes Embedding Conjecture. It contains new proofs of equivalence of the Connes Embedding Conjecture, Positivstellensatze for trace positive polynomials, Kirchberg's Conjecture, and Tsirelson's Problem.