Inductive limits of compact quantum metric spaces

A compact quantum metric space is a unital $C^{\ast }$-algebra equipped with a Lip-norm. Let $\{(A_n,L_n)\}$ be a sequence of compact quantum metric spaces, and let ${\phi }_n:A_n\to A_{n+1}$ be a unital ${}^{\ast }$-homomorphism preserving Lipschitz elements for $n\ge 1$. We show that there exists a compact quantum metric space structure on the inductive limit $\mathrm{}(A_n,{\phi }_n)$ by means of the inverse limit of the state spaces $\{\mathcal{S}(A_n)\}$. We also give some sufficient conditions that two inductive limits of compact quantum metric spaces are Lipschitz isomorphic.