Let $\mathcal A$ be a unital complex $*$-algebra with unit $1_{\mathcal A}$, and let $\varphi: \mathcal A \to \mathbb C$ be a state, meaning that $\varphi$ is complex-linear, $\varphi(1_{\mathcal A}) = 1$, and $\varphi(a^*a) \geq 0$ for every $a \in \mathcal A$. Then there exist a [Hilbert space](/page/Hilbert%20Space) $H_\varphi$, a dense complex linear subspace $\mathcal D_\varphi \subset H_\varphi$, a complex-[linear map](/page/Linear%20Map) $\pi_\varphi: \mathcal A \to \operatorname{End}_{\mathbb C}(\mathcal D_\varphi)$, and a vector $\xi_\varphi \in \mathcal D_\varphi$ such that $\pi_\varphi$ is unital and multiplicative, and for all $a,b,c \in \mathcal A$ one has

\begin{align*} \pi_\varphi(1_{\mathcal A}) = \operatorname{id}_{\mathcal D_\varphi}. \end{align*}

\begin{align*} \pi_\varphi(a)\pi_\varphi(b) = \pi_\varphi(ab). \end{align*}

\begin{align*} (\pi_\varphi(a)u,v)_{H_\varphi} = (u,\pi_\varphi(a^*)v)_{H_\varphi} \end{align*}

for every $u,v \in \mathcal D_\varphi$. Moreover, for every $a \in \mathcal A$,

\begin{align*} \varphi(a) = (\pi_\varphi(a)\xi_\varphi,\xi_\varphi)_{H_\varphi}. \end{align*}

The cyclic subspace satisfies $\pi_\varphi(\mathcal A)\xi_\varphi = \mathcal D_\varphi$, so $\pi_\varphi(\mathcal A)\xi_\varphi$ is dense in $H_\varphi$.

If, in addition, for every $a \in \mathcal A$ there exists a constant $C_a \geq 0$ such that $\|\pi_\varphi(a)u\|_{H_\varphi} \leq C_a\|u\|_{H_\varphi}$ for every $u \in \mathcal D_\varphi$, then each $\pi_\varphi(a)$ extends uniquely to an element of $\mathcal L(H_\varphi)$, and the resulting map $\pi_\varphi: \mathcal A \to \mathcal L(H_\varphi)$ is a unital $*$-representation by bounded operators.