considered non-commutative distributions. Thus one expects many ”smooth" properties for random variables $X_1,\dots,X_n$ for which either one of the quantities $\chi(X_1,\dots,X_n)$, $\chi^*(X_1,\dots,X_n)$, $\Phi(X_1,\dots,X_n)$, or $\Phi^*(X_1,\dots,X_n)$ is finite. In particular, it is commonly expected that such a finiteness implies that

-   there cannot exist non-trivial algebraic relations between the considered random variables;

-   and that such algebraic relations can also not hold locally on non-trivial Hilbert subspaces; more formally this means that there are no zero divisors in the affiliated von Neumann algebra.

Up to now there has been no proof of such general statements. We will show here such results. In the first preprint version of this paper this was done under the assumption of finite non-microstates free Fisher information $\Phi^*(X_1\dots,X_n)$. Inspired by this, Shlyakhtenko could prove in [@Shly2014], by combining our ideas with his earlier work in [@Connes-Shlyakhtenko], our results under the weaker assumption of maximal non-microstates free entropy dimension $\delta^\ast(X_1,\dots,X_n)=n$. In the light of this, we reexamined our original arguments and noticed that they can also be extended without much extra effort to this most general case.

Our work was originally inspired by the realization that in the usual approaches to conjugate variables one usually assumes that there 