A Linearization of Connes' Embedding Problem

TL;DR

This paper links Connes' embedding problem for II_1-factors to a distributional statement involving sums of self-adjoint operators, utilizing a linearization approach and asymptotic freeness of Gaussian matrices.

Contribution

It introduces a linearization technique that simplifies Connes' embedding problem by relating it to operator distributions and employs random matrix theory for proof.

Findings

Connes' embedding problem is equivalent to a distributional statement.

Linearization reduces the problem to analyzing sums of self-adjoint operators.

Asymptotic second order freeness of Gaussian matrices is key to the proof.

Abstract

We show that Connes' embedding problem for II_1-factors is equivalent to a statement about distributions of sums of self-adjoint operators with matrix coefficients. This is an application of a linearization result for finite von Neumann algebras, which is proved using asymptotic second order freeness of Gaussian random matrices.