= Quasi-local observable algebra
{title2=$\mathcal A=\overline{\bigcup_\Lambda\mathcal A_\Lambda}^{\|\cdot\|}$}
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.
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