Inductive limits of compact quantum metric spaces
Botao Long, Ghadir Sadeghi
TL;DR
The article develops a framework to equip the inductive limit of a sequence of compact quantum metric spaces with a Lip-norm by leveraging the inverse limit of their state spaces, yielding a compact quantum metric space structure on $A=\varinjlim (A_n,\phi_n)$. It proves that $\mathcal{S}(A)$ is affinely homeomorphic to the inverse limit of $\mathcal{S}(A_n)$ and defines the Lip-norm $L$ on $A$ via $L(a)=\sup_{\mu\neq\nu} \frac{|\mu(a)-\nu(a)|}{\rho(\mu,\nu)}$, where $\rho$ is the induced metric on $\mathcal{S}(A)$, ensuring $L$ is lower semicontinuous and that $(A,L)$ is a compact quantum metric space with the correct state-space topology. The paper then provides sufficient conditions under which two inductive limits are Lipschitz isomorphic, using common target spaces and uniformly bounded Lipschitz constants, thereby enabling comparisons across different inductive systems and applications to well-known classes such as AF algebras and Bunce–Deddens algebras. This contributes a relaxed and broadly applicable method for constructing quantum metric structures on inductive limits and for establishing Lipschitz-equivalence between different limits. The results bridge previous approaches and expand the repertoire of $C^*$-algebra classes that can be realized as compact quantum metric spaces in a controlled, metric-compatible way.
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 $φ_n:A_n\to A_{n+1}$ 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 $\varinjlim(A_n,φ_n)$ 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.