reader’s convenience.

\[Voi-formula\] Let $(M,\tau)$ be a tracial $W^\ast$-probability space. If $X_1,\dots,X_n \in M$ are self-adjoint with $\Phi^\ast(X_1,\dots,X_n) < \infty$, then we have for $j=1,\dots,n$ that $${\mathbb{C}}\langle X_1,\dots,X_n\rangle \otimes {\mathbb{C}}\langle X_1,\dots,X_n\rangle \subseteq D(\partial_j^\ast),$$ i.e. $\partial_j^\ast$ is densely defined as well and thus $\partial_j$ is closable. More explicitly, we have for each $Y\in {\mathbb{C}}\langle X_1,\dots,X_n\rangle^{\otimes 2}$ the formula $$\partial_j^\ast(Y) = m_{\xi_j}(Y) - m_1({\operatorname{id}}\otimes\tau\otimes{\operatorname{id}})(\partial_j\otimes{\operatorname{id}}+{\operatorname{id}}\otimes\partial_j)(Y).$$

Here, for any $\eta\in L^2(M,\tau)$, we denote by $m_\eta$ the linear operator $m_\eta: M\otimes M \rightarrow L^2(M,\tau)$ defined on the algebraic tensor product $M\otimes M$, which is given by $m_\eta(a_1 \otimes a_2) := a_1\eta a_2$. And thus of course, $m_1(a_1\otimes a_2)=a_1a_2$.

The lemma above allows us to conclude that in particular $$\label{conjugate_formulas}
\begin{aligned}
\partial_j^\ast(P\otimes 1) &= P \xi_j - ({\operatorname{id}}\otimes\tau)(\partial_j P),\\
\partial_j^\ast(1\otimes P) &= \xi_j P - (\tau\otimes{\operatorname{id}})(\partial_j P)
\end{aligned}$$ holds for $j=1,\dots,n$ and any $P\in{\mathbb{C}}\langle X_1,\dots,X_n\rangle$.

\[Dab-estimates\] Let $(M,\tau)$ be a tracial $W^\ast$-probability space. If $X_1,\dots,X_n \in M$ are self-adjoint such that the condition $\Phi^\ast(X_1,\dots,X_n) < \infty$ is satisfied, then we have for each $P\in{\mathbb{C}}\langle X_1,\dots,X_n\rangle$ that $$\label{Dab-estimates-1}
\begin{aligned}
\|P \xi_j - ({\operatorname{id}}\otimes\tau)(\partial_j P)\|_2 &\leq \|\xi_j\|_2 \|P\|,\\
\|\xi_j P - (\tau\otimes{\operatorname{id}})(\partial_j P)\|_2 &\leq \|\xi_j\|_2 \|P\|
\end{aligned}$$ and therefore $$\label{Dab-estimates-2}
\begin{aligned}
\|({\operatorname{id}}\otimes\tau)(\partial_j P)\|_2 &\leq 2\|\xi_j\|_2 \|P\|,\\
\|(\tau\otimes{\operatorname{id}})(\partial_j P)\|_2 