Isotony and spacelike commutativity of scalar local Weyl algebras

For real $f$, the essentially self-adjoint finite-particle scalar field has closure $\overline{{\mathrm{\Phi }}_{\mu }(f)}={\mathrm{\Phi }}_{\mathrm{S}\mathrm{e}\mathrm{g}\mathrm{a}\mathrm{l}}(\sqrt{2}{K}_{\mu }f)$ by the Segal normalization and [F4]. Thus [F4] identifies this closure as the self-adjoint generator of $t\mapsto W_{\mu }(tf)$. If $O_1\subseteq O_2$, each real test function supported in $O_1$ is also a real test function supported in $O_2$. Thus the generators and their adjoints for ${\mathfrak{A}}_{\mu }(O_1)$ belong to the generating set for ${\mathfrak{A}}_{\mu }(O_2)$; finite sums and products do too, and norm closure preserves inclusion. This proves isotony.

Let $f,g$ be real tests supported in the two spacelike-separated regions. By [F2] their field commutator is zero. Fact [F3], obtained by expanding the fields with the CCR, then gives $2i\mathrm{Im}⟨K_{\mu }f,K_{\mu }g{⟩}_{2}I=0$; hence the imaginary part vanishes. Substituting it in [F1] gives $W_{\mu }(f)W_{\mu }(g)=W_{\mu }(f+g)=W_{\mu }(g)W_{\mu }(f)$. Taking adjoints shows each generator from the first region commutes with each generator and adjoint from the second. Commutation is preserved under finite linear combinations and products; if bounded sequences $A_n\to A$ and $B_n\to B$ in operator norm satisfy $A_n{B}_n=B_n{A}_n$, then continuity of multiplication gives $AB=BA$. Taking the two norm closures proves the assertion for the full local algebras.