The CCR-algebra \(A_W\) is the \(C^*\)-algebra generated by \(W(a,b)= e^{-i(bq- ap)}\), \((a,b)\in\mathbb{R}^2\), and \(({\mathcal H},\pi)\) (where \({\mathcal H}\) is a Hilbert space, \(\pi\) a state) is the representation of \(A_W\). The authors generalize the Stone-von Neumann uniqueness theorem:
If the operator-valued function \((a,b)\to \pi(W(a,b))\) is strongly continuous, \(({\mathcal H},\pi)\) is unitary equivalent to the Schrödinger representation \(\pi_s\) based on the realization of the subalgebra generated by \(W(0,b)\). They prove the following GNS theorem (construction):
Suppose that \({\mathcal T}^2\equiv [0,1)\times [0,2\pi)\ni (a,b)\to \pi(W(a,b))y\in{\mathcal H}\) is \(\mu_x\)-measurable, with respect to \(\forall\mu_x\) relating to \((x,\pi(W)x)\), for \(\forall y\in H\). Then there exist a positive measure \(\mu\) and an isomorphism \({\mathcal U}\) from \({\mathcal H}\) onto \({\mathcal L}^2({\mathcal T}^2,\mu)\) giving \({\mathcal U}\pi(W(a,0)){\mathcal U}^{-1}\), \({\mathcal U}\pi(W(0, b)){\mathcal U}^{-1}\), under a condition on \(({\mathcal H},\pi)\). Momentum states \(\omega_p(W(a,b))= 0\) if \(b\neq 0\), \(=e^{ipa}\) for \(b=0\); \(p\in\mathbb{R}\), and Zac states \(\omega_{\zeta\gamma}(W(a,b))= e^{i\pi mn} e^{in\zeta} e^{i2\pi m\gamma}\) if \((a, b)= (n,2\pi m)\), are given as examples.