$M$. Thus, each of the non-commutative derivatives $$\hat{\partial}_j:\ {\mathbb{C}}\langle X_1,\dots,X_n\rangle \rightarrow {\mathbb{C}}\langle X_1,\dots,X_n\rangle \otimes {\mathbb{C}}\langle X_1,\dots,X_n\rangle,$$ whose existence is claimed in part (b) of Theorem \[main1\], is naturally induced under this identification by the corresponding non-commutative derivative $\partial_j$ on ${\mathbb{C}}\langle Z_1,\dots,Z_n\rangle$. Due to this strong relationship, we do not have to distinguish anymore between $\partial_j$ and $\hat{\partial}_j$.

We finish this section by noting that $\Phi^\ast(X_1,\dots,X_n)<\infty$ moreover excludes analytic relations. More precisely, this means that there is no non-zero non-commutative power series $P$, which is convergent on a polydisc $$D_R:=\{(Y_1,\dots,Y_n)\in M^n|\ \forall j=1,\dots,n:\ \|Y_j\| < R\}$$ for some $R>0$, such that $(X_1,\dots,X_n)\in D_R$ and $P(X_1,\dots,X_n)=0$. Based on Voiculescu’s original definition of the non-microstates free Fisher information and hence under the additional assumption that $X_1,\dots,X_n$ are algebraically free, this was shown by Dabrowski in [@Dab-free_stochastic_PDE Lemma 37].

Non-microstates free entropy dimension and zero divisors
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Inspired by the methods used in the proof of Theorem \[main1\], we address now the more general question of existence of zero divisors under the assumption of full non-microstates free entropy dimension.

First of all, we shall make more precise what we mean by this. We postpone the definition of the non-microstates free entropy dimension $\delta^\ast(X_1,\dots,X_n)$ and 