  We introduce a free probabilistic quantity called free Stein irregularity,
which is defined in terms of free Stein discrepancies. It turns out that this
quantity is related via a simple formula to the Murray--von Neumann dimension
of the closure of the domain of the adjoint of the non-commutative Jacobian
associated to Voiculescu's free difference quotients. We call this dimension
the free Stein dimension, and show that it is a $*$-algebra invariant. We
relate these quantities to the free Fisher information, the non-microstates
free entropy, and the non-microstates free entropy dimension. In the
one-variable case, we show that the free Stein dimension agrees with the free
entropy dimension, and in the multivariable case compute it in a number of
examples.

Operator Algebras