Quasi-local observable algebra ($\mathcal{A}={\overline{{\bigcup }_{\Lambda }{\mathcal{A}}_{\Lambda }}}^{\Vert \cdot \Vert }$)

For a lattice system, finite-region observable algebras ${\mathcal{A}}_{\Lambda }$ embed into larger ones by tensoring with the identity. Their union has a norm closure called the quasi-local observable algebra, a C-star algebra. It describes finite-support observables and norm limits of such observables. An operator flipping every spin of an infinite lattice need not belong to it: successive finite-region flips are not norm-Cauchy. Restrictions to this algebra can therefore lose relative-phase information retained by increasingly nonlocal finite-system observables.