Convergence of spectral truncations for compact metric groups

Abstract

We consider Gromov-Hausdorff convergence of state spaces for spectral truncations of a compact metric group $G$. 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 $\Lambda \subseteq \hat{G}$. 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 $G$.

9 pages