Algebraic reformulation of Connes embedding problem and the free group algebra

TL;DR

This paper presents an algebraic reformulation of the Connes embedding problem focusing on the *-algebra of the free group, simplifying the problem by restricting to quadratic polynomials in unitary generators.

Contribution

It introduces a modified algebraic approach that limits the scope to quadratic polynomials, potentially simplifying analysis of the Connes embedding problem.

Findings

Reformulation restricts to quadratic polynomials in unitary generators

Simplifies the algebraic framework of the Connes embedding problem

Provides a new perspective on the algebraic structure of free group algebras

Abstract

We give a modification of I. Klep and M. Schweighofer algebraic reformulation of Connes' embedding problem by considering *-algebra of the countably generated free group. This allows to consider only quadratic polynomials in unitary generators instead of arbitrary polynomials in self-adjoint generators.