C*-Algebra of Field Observables 𝔄

In §6, the $C^*$ -algebra of field observables $\mathfrak A$ is the operator-norm closure of the $*$ -algebra generated by the Weyl unitaries $\exp(iR(z))$ based on finite-dimensional subspaces. As a concrete $C^*$ -algebra, it is a norm-closed $*$ -subalgebra of bounded operators and satisfies $\|A^*A\|=\|A\|^2$ .

Remarks

$\mathfrak A$ is abstractly independent of the chosen representation.
Each symplectic transformation $T$ induces a $*$ -automorphism $\theta(T)$ of $\mathfrak A$ .