Correction of a proof in "Connes' embedding conjecture and sums of hermitian squares"

TL;DR

This paper clarifies the equivalence between Connes' embedding conjecture and a real algebraic reformulation, addressing a gap in previous proofs by utilizing the theory of real von Neumann algebras.

Contribution

It corrects a previous proof gap by establishing the equivalence between CEC and a real algebraic statement using real von Neumann algebra theory.

Findings

CEC is equivalent to a real algebraic reformulation

The gap in the previous proof is filled using real von Neumann algebras

The algebraic reformulation relates to trace positive polynomials

Abstract

We show that Connes' embedding conjecture (CEC) is equivalent to a real version of the same (RCEC). Moreover, we show that RCEC is equivalent to a real, purely algebraic statement concerning trace positive polynomials. This purely algebraic reformulation of CEC had previously been given in both a real and a complex version in a paper of the last two authors. The second author discovered a gap in this earlier proof of the equivalence of CEC to the real algebraic reformulation (the proof of the complex algebraic reformulation being correct). In this note, we show that this gap can be filled with help of the theory of real von Neumann algebras.