exist no algebraic relations between the considered variables. Though this is not necessary for the definition of conjugate variables themselves, more advanced arguments (which rely on the existence of non-commutative derivative operators) only work in the absence of such algebraic relations. As alluded to above one actually expected that the existence of conjugate variables (and thus the finiteness of $\Phi^*$) implies the absence of such relations. But since this has not been shown up to now, there was a kind of an annoying gap in the theory. This gap will be closed in Section \[algebraic\_relations\].

It turned out that our ideas for this could also be extended to the much deeper question whether relations could hold locally; instead of asking whether for a non-trivial polynomial $P$ we can have algebraically $P(X_1,\dots,X_n)=0$, we weaken this to the question whether $P(X_1,\dots,X_n)$ could be zero on an affiliated Hilbert subspace; if $u$ denotes the projection onto this subspace, then this is the question whether we can have a zero divisor $u$ for $P(X_1,\dots,X_n)$, i.e., an element $u$ in the von Neumann algebra generated by $X_1,\dots,X_n$ such that $P(X_1,\dots,X_n)u=0$. We will show that already the condition $\delta^\ast(X_1,\dots,X_n)=n$ for the non-microstates free entropy dimension excludes 