The free field: zero divisors, Atiyah property and realizations via unbounded operators

Abstract: We consider noncommutative rational functions as well as matrices in polynomials in noncommuting variables in two settings: in an algebraic context the variables are formal variables, and their rational functions generate the "free field"; in an analytic context the variables are given by operators from a finite von Neumann algebra and the question of rational functions is treated within the affiliated unbounded operators. Our main result shows that for a "good" class of operators - namely those for which the free entropy dimension is maximal - the analytic and the algebraic theory are isomorphic. This means in particular that any non-trivial rational function can be evaluated as an unbounded operator for any such good tuple and that those operators don't have zero divisors. On the matrix side, this means that matrices of polynomials which are invertible in the free field are also invertible as matrices over unbounded operators when we plug in our good operator tuples. We also address the question how this is related to the strong Atiyah property. The above yields a quite complete picture for the question of zero divisors (or atoms in the corresponding distributions) for operator tuples with maximal free entropy dimension. We give also some partial results for the question of existence and regularity of a density of the distribution.