C*-Algebra of Field Observables 𝔄

A $C^*$-algebra is a norm-closed *-subalgebra of bounded operators with $\|A^*A\|=\|A\|^2$.

In §6, $\mathfrak A$ is the uniform (operator-norm) closure of the algebra generated by Weyl unitaries $\exp(iR(z))$ based on finite-dimensional subspaces.

Key properties (paper use):

• $\mathfrak A$ is abstractly independent of the chosen representation (Segal).
• Symplectic $T$ act by *-automorphisms $\theta(T)$ on $\mathfrak A$.

Example: $B(\mathcal H)$ (all bounded operators) is a $C^*$-algebra.