A note on the perturbations of compact quantum metric spaces

In this short note, we consider the perturbation of compact quantum metric spaces. We first show that for two compact quantum metric spaces (A, P) and (B, Q) for which A and B are subspaces of an order-unit space C and P and Q are Lip-norms on A and B respectively, the quantum Gromov–Hausdorff distance between (A, P) and (B, Q) is small under certain conditions. Then some other perturbation results on compact quantum metric spaces derived from spectral triples are also given.