Here, we stipulate that, whenever ${\mathcal{M}}$ is an ${\mathcal{A}}$-bimodule, the symbol $\sharp$ denotes the mapping $\sharp: {\mathcal{A}}\otimes{\mathcal{A}}\times {\mathcal{M}}\to {\mathcal{M}}$, which is determined by bilinear extension of $(a_1 \otimes a_2)\sharp m := a_1 \cdot m \cdot a_2$.

Throughout the following, let $(M,\tau)$ be a tracial $W^\ast$-probability space, which means that $M$ is a von Neumann algebra and $\tau$ is a faithful normal tracial state on $M$. For selfadjoint $X_1,\dots,X_n\in M$ we denote by ${\operatorname{vN}}(X_1,\dots,X_n)\subset M$ the von Neumann subalgebra of $M$ which is generated by $X_1,\dots,X_n$ and by $L^2(X_1,\dots,X_n,\tau)\subset L^2(M,\tau)$ the $L^2$-space which is generated by $X_1,\dots,X_n$ with respect to the inner product given by $\langle P,Q\rangle:=\tau(PQ^*)$.

In the following, we will denote by ${\operatorname{ev}}_X$, for a given $n$-tuple $X=(X_1,\dots,X_n)$ of self-adjoint elements of $M$, the homomorphism $${\operatorname{ev}}_X:\ {\mathbb{C}}\langle Z_1,\dots,Z_n\rangle \rightarrow {\mathbb{C}}\langle X_1,\dots,X_n\rangle \subset M$$ given by evaluation at $X=(X_1,\dots,X_n)$, i.e. the homomorphism ${\operatorname{ev}}_X$ is determined by $Z_i \mapsto X_i$.

For reasons of clarity, we put $P(X) := {\operatorname{ev}}_X(P)$ for any $P\in {\mathbb{C}}\langle Z_1,\dots,Z_n\rangle$ and $Q(X) := ({\operatorname{ev}}_X \otimes {\operatorname{ev}}_X)(Q)$ for any $Q\in{\mathbb{C}}\langle Z_1,\dots,Z_n\rangle^{\otimes 2}$.

\[conjugate\_system\] Let $X_1,\dots,X_n\in M$ be self-adjoint elements. If there are elements $\xi_1,\dots,\xi_n\in L^2(M,\tau)$, such that $$\label{conjugate_relations}
(\tau\otimes\tau)((\partial_jP)(X_1,\dots,X_n)) = \tau(\xi_j P(X_1,\dots,X_n))$$ is satisfied for each non-commutative polynomial 