Example of GNS construction I can't find any pedagogical and illustrative example of "step-by-step" GNS construction in the literature. What I mean by illustrative? - writing explicitly the functional $\rho: A \rightarrow \mathbb{C}$ (where $A$ denotes some specific algebra) and then for example constructing the quotient space.
For example let's start with Weyl algebra of the following form
$$ W(f,h) = e^{i(f x + h p)},  \quad W^*(f, h) = W (-f,-h)$$
with appropiate multiplication law. Also $p$ and $x$ can be thought as position and momentum respectively. How to perform a GNS construction to obtain "standard quantum mechanics"? I mean how to choose $\rho$? What will be resulting the Hilbert space - $L^2(\mathbb{R})?$ How elements of algebra will be represented as bounded operators on the Hilbert space?
 A: As a warm-up, let $\mathcal A:= \mathrm{Mat}_{2\times 2}(\mathbb C)$ be our algebra, equipped with the $*$-operation
$$\pmatrix{a&b\\c&d}^* := \pmatrix{\overline a&\overline c\\\overline b&\overline d}$$
where the line denotes complex conjugation.  It's easy to verify that this satisfies the requirements of a $C^*$-algebra.  A state is then defined to be a map $\phi:\mathcal A\rightarrow \mathbb C$ with the following properties:

*

*$\phi(\lambda \mathbf u)=\lambda \phi(\mathbf u)$ for all $\lambda \in \mathbb C$ and $\mathbf u\in \mathcal A$ (i.e. $\phi$ is linear)

*$\phi(\mathbf u^* \mathbf u) \geq 0$ for all $\mathbf u\in \mathcal A$ (i.e. $\phi$ is positive)

*$\phi(\mathbf 1) = 1$
In the following example, we consider the state $\phi : \pmatrix{a & b \\ c & d}\mapsto a$, which straightforwardly satisfies the rules above. From there, we may proceed with the GNS construction.
The first step is to define a sesquilinear form on $\mathcal A$. Given our $\phi$, we define
$$\langle \mathbf u,\mathbf v\rangle_\phi := \phi(\mathbf u^*\mathbf v)$$
$$\implies \left<\pmatrix{a_1&b_1\\c_1&d_1},\pmatrix{a_2&b_2\\c_2&d_2}\right>_\phi = a_1^* a_2+b_1^* b_2$$
From here, we identify the vector subspace $\mathcal S\subset \mathcal A$ defined by the condition than for all $\mathbf u\in \mathcal S$, $\Vert \mathbf u \Vert_\phi \equiv \sqrt{\langle\mathbf u,\mathbf u\rangle_\phi} = 0$.  In this case,
$$\mathcal S = \left\{\pmatrix{0&b \\ 0 & d} \ \bigg| \ b,d\in \mathbb C \right\}$$
The quotient space $\mathscr H := \mathcal A / \mathcal S$, equipped with the inner product $\langle [\mathbf u],[\mathbf v]\rangle_{\mathscr H} := \langle \mathbf u,\mathbf v\rangle_\phi$, constitutes a Hilbert space.  Note that for any $\mathbf x,\mathbf y\in \mathcal S$, $\langle\mathbf u + \mathbf x,\mathbf v+\mathbf y\rangle_\phi = \langle \mathbf u,\mathbf v\rangle_\phi$ so this is a well-defined operation.  For concreteness, we may represent each equivalence class in $\mathscr H$ by the member whose upper-right and lower-right entries are zero; that is,
$$\left[\pmatrix{a&b\\c&d}\right] \sim \pmatrix{a & 0 \\ c & 0} $$
Next, define the representation $\pi: \mathcal A \rightarrow \mathrm{End}(\mathscr H)$ via $\pi(\mathbf u): [\mathbf v] \mapsto [\mathbf u \mathbf v]$.  For example,
$$\pi\left(\pmatrix{\alpha&\beta\\ \gamma&\delta}\right)\left[\pmatrix{a & b \\ c& d}\right] = \left[\pmatrix{\alpha a + \beta c & \alpha c + \beta d\\ \gamma a + \delta c & \gamma b + \delta d}\right] \sim \pmatrix{\alpha a + \beta c &0\\ \gamma a + \delta c&0}$$
Finally, we identify $\xi := [\mathbf 1] \sim \pmatrix{1&0\\0&0}$ as the unit-norm cyclic vector which generates $\mathscr H$ (that is, any $\psi\in \mathcal H$ can be expressed as $\pi(\mathbf u) \xi$ for some $\mathbf u\in \mathscr A$).  Observing that $\phi(\mathbf u) = \langle \pi(\mathbf u) \xi, \xi\rangle_\mathscr H$, we note that the action of our state on an observable reproduces the familiar expression for the expected value of $\pi(\mathbf u)$ from elementary quantum mechanics; in that way, we may understand $\xi$ to be the "state vector" corresponding to $\phi$.
Clearly, $\mathscr H \simeq \mathbb C^2$ equipped with the familiar inner product, and $\pi(\mathbf u)$ simply means operating on an element of $\mathbb C^2$ with $\mathbf u$ via matrix multiplication.  As a result, the GNS construction has taken us from the $C^*$-algebra $\mathrm{Mat}_{2\times 2}(\mathbb C)$ to the Hilbert space $\mathscr H \simeq \mathbb C^2$, as promised.
It's worth pointing out that our particular choice of state happened to be pure; in this context, we could define a pure state as one such that the induced representation $\pi$ is irreducible.  Consider the following alternative choice:
$$\phi':\pmatrix{a&b\\c&d}\mapsto \frac{1}{2}(a+d) \qquad \mathcal S'=\{\mathbf 0\} \qquad \mathscr H' \simeq \mathrm{Mat}_{2\times 2}(\mathbb C) \qquad \pi'(\mathbf u) \mathbf v = \mathbf u \mathbf v$$
$$\xi' = \pi'(\mathbf 1) = \frac{1}{2}\pmatrix{1&0\\0&1}$$
We see that this representation is not irreducible because it has non-trivial invariant subspaces $\pmatrix{a&0\\c&0}$ and $\pmatrix{0&b\\0&d}$; as such, $\mathscr H$ is essentially two copies of $\mathbb C^2$.  This state $\phi'$ is mixed, and $\xi'$ is the corresponding density matrix.

When time permits, I will update this answer to explicitly reference the Weyl algebra. In the meantime, hopefully this provides a helpful demonstration of the salient points of the GNS construction.
