Compact Quantum Metric Spaces from Free Graph Algebras

TL;DR

This paper constructs compact quantum metric spaces from free graph algebras, demonstrating convergence properties and applying the framework to planar algebra-related C*-algebras, advancing the understanding of non-nuclear quantum metric spaces.

Contribution

It introduces a method to produce compact quantum metric spaces from free loop algebras associated with graphs, including convergence results and applications to planar algebra C*-algebras.

Findings

Established Haagerup-type bounds for these algebras

Proved convergence of graph sequences implies quantum Gromov-Hausdorff convergence

Applied the framework to Guionnet-Jones-Shyakhtenko C*-algebras

Abstract

Starting with a vertex-weighted pointed graph $(\Gamma,\mu,v_0)$, we form the free loop algebra $\mathcal{S}_0$ defined in Hartglass-Penneys' article on canonical $\rm C^*$-algebras associated to a planar algebra. Under mild conditions, $\mathcal{S}_0$ is a non-nuclear simple $\rm C^*$-algebra with unique tracial state. There is a canonical polynomial subalgebra $A\subset \mathcal{S}_0$ together with a Dirac number operator $N$ such that $(A, L^2A,N)$ is a spectral triple. We prove the Haagerup-type bound of Ozawa-Rieffel to verify $(\mathcal{S}_0, A, N)$ yields a compact quantum metric space in the sense of Rieffel. We give a weighted analog of Benjamini-Schramm convergence for vertex-weighted pointed graphs. As our $\rm C^*$-algebras are non-nuclear, we adjust the Lip-norm coming from $N$ to utilize the finite dimensional filtration of $A$. We then prove that convergence of vertex-weighted pointed graphs leads to quantum Gromov-Hausdorff convergence of the associated adjusted compact quantum metric spaces. As an application, we apply our construction to the Guionnet-Jones-Shyakhtenko (GJS) $\rm C^*$-algebra associated to a planar algebra. We conclude that the compact quantum metric spaces coming from the GJS $\rm C^*$-algebras of many infinite families of planar algebras converge in quantum Gromov-Hausdorff distance.