Reading one paper, the [GNS construction](https://en.wikipedia.org/wiki/Gelfand%E2%80%93Naimark%E2%80%93Segal_construction) is mentioned as follows:

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> It is important to recall that a result (theorem) due to Gel'fand, Naimark and Segal (GNS) establishes that for any $\omega$ on $\mathcal{A}$ there always exists a representation $(f\_\omega, \mathfrak{h}\_\omega)$ of $\mathcal{A}$ and $\Phi\_\omega\in \mathfrak{h}\_\omega$ (usually called a *cyclic vector*) such that $f\_\omega(\mathcal{A})\Phi\_\omega$ is dense in $\mathfrak{h}\_\omega$ and $\omega(A)=\langle \Phi\_\omega | f\_{\omega}(\mathcal{A})|\Phi\_\omega\rangle$. Moreover the GNS result warrants that up to unitary equivalence, $(f\_\omega,\mathfrak{h}\_\omega)$ is the unique cyclic representation of $\mathcal{A}$.
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Now, considering the math there is a theorem and a corresponding proof. My point here is not to discuss these. My point here is to discuss the intuition about this construction from the Physics point of view.

So the first thing that makes me confused: in the $C^\ast$-algebra approach, I thought each state $\omega : \mathcal{A}\to \mathbb{R}$ was the counterpart of a ket $|\phi\rangle$ in the traditional approach.

We see in the GNS construction, though, that *each state $\omega$ induces one representation*. In other words, instead of having for each $\omega$ one ket, we have for each $\omega$ one whole Hilbert space.

More than that, we have that cyclic vector condition, which physically I don't understand.

So my question is: what is the intuition on the GNS construction from the Physics point of view? How does states $\omega$ from the algebraic approach relates to kets $|\psi\rangle$ (state vectors) in the traditional approach? What is that cyclic vector condition about from a physical perspective?