  We establish several properties of the free Stein dimension, an invariant for
finitely generated unital tracial $*$-algebras. We give formulas for its
behaviour under direct sums and tensor products with finite dimensional
algebras. Among a given set of generators, we show that (approximate) algebraic
relations produce (non-approximate) bounds on the free Stein dimension.
Particular treatment is given to the case of separable abelian von Neumann
algebras, where we show that free Stein dimension is a von Neumann algebra
invariant. In addition, we show that under mild assumptions $L^2$-rigidity
implies free Stein dimension one. Finally, we use limits superior/inferior to
extend the free Stein dimension to a von Neumann algebra invariant -- which is
substantially more difficult to compute in general -- and compute it in several
cases of interest.
