Convergence of Spectral Truncations for Compact Metric Groups

Abstract We study Gromov–Hausdorff convergence of state spaces for truncations of a compact metric group $G$ in momentum space. We work in the context of order-unit spaces and consider orthogonal projections $P_\Lambda $ in $L^{2}(G)$ corresponding to finite subsets of irreducible representations $\Lambda \subseteq \widehat G$ appearing in the Peter–Weyl decomposition of $G$. We then prove that the sequence of truncated state spaces $\{ S(P_\Lambda C(G) P_\Lambda )\}_\Lambda $ Gromov–Hausdorff converges to the original state space $S(C(G))$, when these are equipped with a metric associated to a Lip-norm which in turn is induced by the action of $G$.