The free field: realization via unbounded operators and Atiyah property

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.