I am currently trying to do a GNS construction for the free 1D quantum mechanical particle. I am working with the limit of Weyl algebra $\mathfrak{U}_N$, that is all the elements
$ a = \sum_{qp} c(q,p) W(q, p)$
where q is in element of a 1D lattice with N evenly spaced points inside an interval $[-L/2, L/2[$ and p is in the reciprocal lattice and $c(q,p) \in \mathbb{C}$ are the coefficients. For convenience we set $L = 2 \pi$. Chaining elements can be done via
$ W(q,p) W(q',p') = e^{i(q'p-qp')/2} W(q+q', p+p') $
For the GNS construction I look at the inductive limit algebra
$ \mathfrak{U} = \overline{\bigcup_{N \in 2^\mathbb{N}} \mathfrak{U}_N} $
I now have to find the kernel for the (simple) state $\omega(W(q,p)) := \delta_{q, 0}$ which makes sense since $q\in \mathbb{N}$. An element of the closure should be
$ a = \sum_{q \in \mathbb{N}} \int_{-\pi}^\pi dp\ c(q,p) W(q,p) $
and the star operation acts as
$ a^\ast = \sum_{q \in \mathbb{N}} \int_{-\pi}^\pi dp\ c^\ast(q,p) W(-q,-p) $
The kernel $\mathcal{N} = \{ a \in \mathfrak{U} : \omega(a^\ast a) = 0 \}$ then reduces to
$ 0 = \omega(a^\ast a) = \sum_{q \in \mathbb{N}} |\int_{-\pi/2}^{\pi/2} dp\ c(q,p) e^{iqp/2}|^2 $
so each term in the sum must vanish. I expect the resulting Hilbert space to be
$ \mathcal{H} = \overline{\mathfrak{U}/\mathcal{N}} \cong L^2(S^1) $
motivated by physics. Does someone have a clever idea how to use the kernel condition to show that all element $[a] = a + \mathcal{N}$ in the quotient space can be written in a much simpler form, probably in a way that the index $p$ disappears completly, by cleverly using the above condition? It can be noted that is is also a Fourier transform in the second variable of c, but I do not see how that helps. The guess would be that
$ [a] \cong \sum_{q \in \mathbb{N}} a(q) W(q,0) $