Norm-completed quasi-local quantum spin algebra

Use the finite local matrix algebras and isometric factor inclusions of Local spin matrix algebras and their factor inclusions and Local spin factor inclusions preserve the operator norm. Identify $A\in {\mathcal{A}}_{\mathrm{\Lambda }}$ with its image in every larger finite-region algebra. The directed union

${\mathcal{A}}_{\mathrm{l}\mathrm{o}\mathrm{c}}:={\bigcup }_{\mathrm{\Lambda }\subseteq \mathrm{\Gamma },\mathrm{\mid }\mathrm{\Lambda }\mathrm{\mid }<\mathrm{\infty }}{\mathcal{A}}_{\mathrm{\Lambda }}$

is the algebra of strictly local observables. Its norm is $\mathrm{\parallel }A\mathrm{\parallel }=\mathrm{\parallel }A{\mathrm{\parallel }}_{{\mathcal{A}}_{\mathrm{\Lambda }}}$ for any finite localization region $\mathrm{\Lambda }$ of $A$; isometric inclusions make this independent of the chosen region. The quasi-local spin algebra is the norm completion

$\mathcal{A}:={\overline{{\mathcal{A}}_{\mathrm{l}\mathrm{o}\mathrm{c}}}}^{\mathrm{\parallel }\cdot \mathrm{\parallel }}\mathrm{.}$

The C*-algebra structure on this completion has product and adjoint extended continuously from the finite local matrix algebras. Countable Choice is assumed for the published existence theorem for metric completions; Quasi-local algebra completion and locality ↗ supplies the full extension argument using the finite C*-identity of Finite matrix C-star algebra and density-state duality.

No global Hilbert tensor product is selected. “Quasi-local” means norm completion of the directed algebra of finite-region observables; it does not mean a chosen representation as operators on an infinite tensor-product Hilbert space. A local operator $A\in {\mathcal{A}}_{\mathrm{\Lambda }}$ may have more than one localization region. Disjoint finite local factors commute in their common finite tensor algebra.