I was reading Valter Moretti's book on Spectral Theory and Quantum Mechanics, and saw 2 definitions of a quantum state:
1.Let $\mathcal{H}$ be a Hilbert space. A positive, trace-class linear map $\rho:\mathcal{H} \to \mathcal{H}$ with unit trace is called a $\mathbf{state}$.
2.Let $\mathfrak{A}$ be the $C^{*}$-algebra with unit of the quantum theory. Then a $\mathbf{state}$ is a positive linear functional $\omega:\mathfrak{A} \to \mathbb{C}$, which is normalized, i.e. \begin{equation} \omega(a^{*}a) \geq 0 \; \; \forall a \in \mathfrak{A}, \; \; \omega(\mathbb{I})=1. \end{equation}
Are these two notions equivalent? The passage $2 \to 1$ I think should be done via GNS representation, but it's not entirely clear to me how. Conversely, how does one go $1 \to 2$?
I do understand that in QFT, the algebraic approach is more general, but in QM my intuition tells they should be equivalent cause of Stone-von-Neumann, however I haven't seen an explicit proof.