Let $\mathcal{A}$ be a $C^* $-algebra, let $\pi : \mathcal{A} \to \mathcal{B}(H)$ be a representation of $\mathcal{A}$ on the space of bounded linear operators on a Hilbert space $H$ and let $\omega: \mathcal{A} \to \mathbb{C}$ be a state on $\mathcal{A} $. In order to "represent" $\omega $, I think we need $\pi $ to be injective (and thus an isomorphism), so that $\pi (\mathcal{A})$ is itself a $C^*$-algebra and the map
$$\tau : \pi (\mathcal{A}) \to \mathbb{C}$$
defined by $\tau \big( \pi (A) \big) = \omega (A)$, $\forall A \in \mathcal{A}$ is indeed a state. Furthermore, by the Hahn-Banach theorem, we can extend $\tau$ to a state on $\mathcal{B}(H)$.
Question 1: is the above construction of $\tau$ on $\mathcal{B}(H)$ correct?
Question 2: what are the properties of $\tau$ given $\omega$? For instance, when is $\tau$ normal?