Let $\mathcal A$ be a unital complex $*$-algebra with identity $1_{\mathcal A}$, let $a_1,\dots,a_n \in \mathcal A$, and let $\varphi: \mathcal A \to \mathbb C$ be a state, meaning that $\varphi$ is linear, $\varphi(1_{\mathcal A}) = 1$, and $\varphi(c^*c) \geq 0$ for every $c \in \mathcal A$.
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Let $\mathbb C\langle X_1,\dots,X_n,X_1^*,\dots,X_n^*\rangle$ denote the free unital $*$-algebra of noncommutative $*$-polynomials, and define the joint $*$-law of $(a_1,\dots,a_n)$ with respect to $\varphi$ to be the linear functional
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\begin{align*}
\mu_{a_1,\dots,a_n}: \mathbb C\langle X_1,\dots,X_n,X_1^*,\dots,X_n^*\rangle \to \mathbb C
\end{align*}