# The free field: realization via unbounded operators and Atiyah property

## TL;DR

This paper characterizes when the division closure of operators in a von Neumann algebra forms a free skew field, linking it to non-commutative distributions, free entropy, and dual systems, with implications for free probability and random matrices.

## Contribution

It provides a new characterization of free skew fields generated by operators in von Neumann algebras, connecting algebraic properties with free probability concepts.

## Key findings

- Division closure is a skew field iff it satisfies the strong Atiyah property.
- Generators form a free skew field iff no non-zero finite rank operators commute with all generators.
- Results include conditions for atoms in distributions and eigenvalue distributions in free probability.

## Abstract

Let $X_1,\dots,X_n$ be operators in a finite von Neumann algebra and consider their division closure in the affiliated unbounded operators. We address the question when this division closure is a skew field (aka division ring) and when it is the free skew field. We show that the first property is equivalent to the strong Atiyah property and that the second property can be characterized in terms of the non-commutative distribution of $X_1,\dots,X_n$. More precisely, $X_1,\dots,X_n$ generate the free skew field if and only if there exist no non-zero finite rank operators $T_1,\dots,T_n$ such that $\sum_i[T_i,X_i]=0$. Sufficient conditions for this are the maximality of the free entropy dimension or the existence of a dual system of $X_1,\dots,X_n$. Our general theory is not restricted to selfadjoint operators and thus does also include and recover the result of Linnell that the generators of the free group give the free skew field. We give also consequences of our result for the question of atoms in the distribution of rational functions in free variables or in the asymptotic eigenvalue distribution of matrices over polynomials in asymptotically free random matrices. This solves in particular a conjecture of Charlesworth and Shlyakhtenko.