Inductive limits of compact quantum metric spaces

Abstract: A compact quantum metric space is a unital $C^*$-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 $^*$-homomorphism preserving Lipschitz elements for $n\geq 1$. We show that there exists a compact quantum metric space structure on the inductive limit $\varinjlim(A_n,\phi_n)$ 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.