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

TL;DR

This paper proves that for spectral triples built from groups with polynomial growth, spectral truncations converge to the original structure in the quantum Gromov-Hausdorff sense, using Lip-norms from high order derivatives.

Contribution

It establishes quantum Gromov-Hausdorff convergence of spectral truncations for spectral triples associated with polynomial growth groups.

Findings

Spectral truncations approximate the original spectral triple in the quantum Gromov-Hausdorff sense.

Convergence holds when using Lip-norms derived from high order derivatives.

Results apply specifically to spectral triples constructed from groups with polynomial growth.

Abstract

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 $P_{\Lambda}$ of $D$ onto $[-\Lambda,\Lambda]$ 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 $P_{\Lambda}H$, we obtain a sequence of operator system spectral triples $\{(P_{\Lambda}\mathcal{A}P_{\Lambda},P_{\Lambda}H,P_{\Lambda}DP_{\Lambda})\}_{\Lambda}$. We prove that if the spectral triple is the one constructed using a discrete group with polynomial growth, then the sequence of operator systems $\{P_{\Lambda}\mathcal{A}P_{\Lambda}\}_{\Lambda}$ converges to $\mathcal{A}$ in the sense of quantum Gromov-Hausdorff convergence with respect to the Lip-norm coming from high order derivatives.