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The *-automorphism induced by sending Weyl operators V(z) to V(Tz)
For , is the unique *-automorphism of the CCR -algebra of observables such that
It induces an action on states by pullback: .
Key property (paper use):
For a symplectic Hilbert space , let denote the group of bounded invertible real-linear operators on . The symplectic group is
Let be a -algebra. A -automorphism of is a -isomorphism ; equivalently, it is a bijective complex-linear map that preserves multiplication and involution. Its inverse is again a -automorphism. Under composition, all such maps form the automorphism group . If is unital, every -automorphism preserves automatically. No topology on , continuity in an external parameter, or choice of an implementing operator is included in the definition.
In §6, the -algebra of field observables is the operator-norm closure of the -algebra generated by the Weyl unitaries based on finite-dimensional subspaces. As a concrete -algebra, it is a norm-closed -subalgebra of bounded operators and satisfies .