Let $d \in \mathbb{N}$, let $R > 0$, and let $\mathbb{C}\langle X_1,\dots,X_d\rangle$ be the unital complex $*$-algebra of noncommutative polynomials in self-adjoint generators $X_i^* = X_i$. Let $\mu: \mathbb{C}\langle X_1,\dots,X_d\rangle \to \mathbb{C}$ be a complex-linear functional satisfying $\mu(1)=1$ and

\begin{align*} \mu(P^*P) \geq 0 \end{align*}

for every $P \in \mathbb{C}\langle X_1,\dots,X_d\rangle$. Assume that $\mu$ has compact support bounded by $R$, in the sense that for every $i \in \{1,\dots,d\}$ and every $P \in \mathbb{C}\langle X_1,\dots,X_d\rangle$,

\begin{align*} \mu(P^*X_i^2P) \leq R^2 \mu(P^*P). \end{align*}

Then there exist a [Hilbert space](/page/Hilbert%20Space) $H$, bounded [self-adjoint operators](/page/Self-Adjoint%20Operators) $A_1,\dots,A_d \in \mathcal{L}(H)$, where $\mathcal{L}(H)$ denotes the algebra of bounded linear operators from $H$ to $H$, with $\|A_i\|_{\mathcal{L}(H)} \leq R$ for each $i$, and a unit vector $\Omega \in H$ such that, for every $P \in \mathbb{C}\langle X_1,\dots,X_d\rangle$,

\begin{align*} \mu(P) = (P(A_1,\dots,A_d)\Omega,\Omega)_H. \end{align*}