The free field: Realization via unbounded operators and Atiyah property

Let X1,…,Xn be operators in a finite von Neumann algebra and consider their division closure the affiliated unbounded operators. We address question when this is skew field (aka ring) it free field. show that first property equivalent to strong Atiyah second can characterized terms of noncommutative distribution X1,…,Xn. More precisely, generate if only there exist no non-zero rank T1,…,Tn such ∑i[Ti,Xi]=0. Sufficient conditions for are maximality entropy dimension or existence dual system Our general theory not restricted selfadjoint thus does also include recover result Linnell generators group give consequences our atoms rational functions variables asymptotic eigenvalue matrices over polynomials asymptotically random matrices. This solves particular conjecture Charlesworth Shlyakhtenko.