State, Pure State, Regular State (CCR context)

A state on a $C^*$-algebra $\mathfrak A$ is a linear functional $E$ with $E(I)=1$ and $E(A^*A)\ge0$.

Pure means not a nontrivial convex combination of other states.

In Shale’s CCR setting, regular means $z\mapsto E(A^*e^{iR(z)}B)$ is continuous on each finite-dimensional subspace.

Example: In a representation on $\mathcal H$, $E(X)=(Xx,x)$ for a unit vector $x$.