Lemma: Local quadratic approximation

If $f\in {C_c}^{\infty }(O;\mathbb{R})$, then $\overline{Q(f)}$ is affiliated with $\mathfrak{A}(O)$.

Proof. Choose a real compact smooth mollifier ${\rho }_{ϵ}$, normalized to integral one, and supported in a ball of radius $ϵ$. For sufficiently small $ϵ$, every translated mollifier meeting $\mathrm{supp}f$ is supported in $O$. Put

For fixed $ϵ$, the integral is first defined by its quadratic Fock kernels. Those kernels are Bochner integrals of rank-one and rank-two kernels over a compact parameter set.

Riemann sums are finite real linear combinations of

Each summand is affiliated with $\mathfrak{A}(O)$, by the spectral calculus for the local Segal field. Each finite sum is essentially self-adjoint on ${\mathcal{D}}_{\mathrm{fin}}$, by the same bounded-number argument as in Lemma lem:Qsa. The unitary product formula for these self-adjoint summands, whose sum is essentially self-adjoint on the common finite-particle core, gives its unitary group as a strong limit of products of their local unitary groups Trotter. Hence each finite sum is affiliated with $\mathfrak{A}(O)$.

For fixed $ϵ$, the Riemann-sum pair kernels converge in Hilbert–Schmidt norm, and the number-preserving kernels converge in operator norm. The associated operators therefore converge on ${\mathcal{D}}_{\mathrm{fin}}$. This yields strong-resolvent convergence of their self-adjoint closures, so $Q_{ϵ}(f)$ is affiliated with $\mathfrak{A}(O)$.

For completeness, the strong-resolvent implication used here has a short core argument. If self-adjoint $A_j,A$ satisfy $A_j\psi \to A\psi$ on a core of $A$, then

The range of $A-\mathrm{i}$ on that core is dense and the resolvents are uniformly bounded, so convergence holds on the whole Hilbert space.

Now remove the mollifier. On the mass shell the one-particle multiplier ${\hat{\rho }}_{ϵ}(p)$ is uniformly bounded and converges pointwise to one. Thus the pair kernels converge to $F_f$ in Hilbert–Schmidt norm by dominated convergence. The number-preserving kernels converge strongly to $B_f$, with uniform operator-norm bounds, because they are obtained by placing these bounded multipliers on the two sides of $B_f$. It follows that $Q_{ϵ}(f)\psi \to Q(f)\psi$ on ${\mathcal{D}}_{\mathrm{fin}}$. Lemma lem:Qsa and the same core argument give strong-resolvent convergence to $\overline{Q(f)}$.

The resolvents of all approximants belong to $\mathfrak{A}(O)$, and that von Neumann algebra is strongly closed. The limiting resolvents therefore belong to $\mathfrak{A}(O)$. This is affiliation of $\overline{Q(f)}$ with the original net.