shows now that the validity of is already sufficient to exclude algebraic relations.

\[key-prop-2\] Let $(M,\tau)$ be a tracial $W^\ast$-probability space and let self-adjoint $X_1,\dots,X_n\in M$ be given, such that holds.

Then the non-commutative random variables $X_1,\dots,X_n$ do not satisfy any non-trivial algebraic relation, i.e., if $P(X_1,\dots,X_n) = 0$ holds for any non-commutative polynomial $P \in {\mathbb{C}}\langle Z_1,\dots,Z_n\rangle$, then we must have $P=0$.

For $j=1,\dots,n$, we may define $\Delta_j := ((\tau\circ{\operatorname{ev}}_X)\otimes{\operatorname{id}}) \circ \partial_j$, which gives a linear mapping $$\Delta_j:\ {\mathbb{C}}\langle Z_1,\dots,Z_n\rangle \rightarrow {\mathbb{C}}\langle Z_1,\dots,Z_n\rangle.$$ The assumption yields then that for any $P\in{\mathbb{C}}\langle Z_1,\dots,Z_n\rangle$ the implication $$\label{reduction-reduced}
P(X) = 0 \quad\Longrightarrow\quad \forall j=1,\dots,n:\ (\Delta_j P)(X) = 0.$$ holds true with $X=(X_1,\dots,X_n)$.

Assume now that there is any non-zero polynomial $P\in{\mathbb{C}}\langle Z_1,\dots,Z_n\rangle$ such that $P(X_1,\dots,X_n)=0$ holds, for instance $$P(Z_1,\dots,Z_n) = a_0 + \sum_{k=1}^d \sum_{i_1,\dots,i_k = 1}^n a_{i_1,\dots,i_k} Z_{i_1} \dots Z_{i_k},$$ where $d\geq1$ denotes the total degree of $P$. We choose any summand of highest degree $$a_{i_1,\dots,i_d} Z_{i_1} \dots Z_{i_d}$$ of $P$. Since $\Delta_{i_d} \dots \Delta_{i_1}$ is clearly zero on constants, any monomial of degree strictly less than $d$, and furthermore on any monomial of degree $d$, where the variables do not appear in the prescribed order, we see that $\Delta_{i_d} \dots \Delta_{i_1} 