be the full Fock space over $H$ with vacuum vector $\Omega$. For each $f \in H$, let $\ell(f) \in \mathcal{L}(\mathcal{F}(H))$ be the left creation operator, defined by $\ell(f)\Omega = f$ and $\ell(f)(h_1 \otimes \cdots \otimes h_m) = f \otimes h_1 \otimes \cdots \otimes h_m$ for $m \geq 1$ and $h_1,\dots,h_m \in H$.
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For each $i \in I$, let $B_i \subset \mathcal{L}(\mathcal{F}(H))$ be the unital algebra generated by the set
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\begin{align*}
\{\ell(f) + \ell(f)^* : f \in H_i\}.
\end{align*}
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Let $\varphi: \mathcal{L}(\mathcal{F}(H)) \to \mathbb{C}$ be the vacuum state defined by
Then the family of unital algebras $(B_i)_{i \in I}$ is freely independent with respect to $\varphi$: whenever $n \in \mathbb{N}$, $i_1,\dots,i_n \in I$ satisfy $i_k \neq i_{k+1}$ for $1 \leq k < n$, and $T_k \in B_{i_k}$ satisfy $\varphi(T_k)=0$ for every $1 \leq k \leq n$, one has