Regularity questions in free probability ask what can be learned about a tracial von Neumann algebra from probabilistic-flavoured qualities of a set of generators. Non-microstates free entropy theory was introduced by Voiculescu in the 1990's, and attempts to study properties arising from derivations defined on such a set of generators valued in the tensor product of the $L^2$ space they generate. The free Stein dimension is a related quantity, which measures the ease with which such derivations can be found. I will introduce this quantity and speak on some of its consequences to von Neumann algebras, with a focus on spaces of derivations.

This is joint work with Brent Nelson.