Quantum Gromov-Hausdorff convergence of spectral truncations for groups with polynomial growth

For a unital spectral triple $(\mathcal{A},H,D)$, we study when its truncation converges to itself. The spectral truncation is obtained by using the spectral projection $P_{\mathrm{\Lambda }}$ of $D$ onto $[-\mathrm{\Lambda },\mathrm{\Lambda }]$ to deal with the case where only a finite range of energy levels of a physical system is available. By restricting operators in $\mathcal{A}$ and $D$ to $P_{\mathrm{\Lambda }}H$, we obtain a sequence of operator system spectral triples $\{(P_{\mathrm{\Lambda }}\mathcal{A}{P}_{\mathrm{\Lambda }},P_{\mathrm{\Lambda }}H,P_{\mathrm{\Lambda }}D{P}_{\mathrm{\Lambda }}){\}}_{\mathrm{\Lambda }}$. We prove that if the spectral triple is the one constructed using a discrete group with polynomial growth, then the sequence of operator systems $\{P_{\mathrm{\Lambda }}\mathcal{A}{P}_{\mathrm{\Lambda }}{\}}_{\mathrm{\Lambda }}$ converges to $\mathcal{A}$ in the sense of quantum Gromov-Hausdorff convergence with respect to the Lip-norm coming from high order derivatives.