Compact Quantum Metric Spaces from Free Graph Algebras

Abstract

Starting with a vertex-weighted pointed graph (Γ, µ, v0), we form the free loop algebra S0 defined in Hartglass-Penneys' article on canonical C * -algebras associated to a planar algebra. Under mild conditions, S0 is a non-nuclear simple C * -algebra with unique tracial state. There is a canonical polynomial subalgebra A ⊂ S0 together with a Dirac number operator N such that (A, L 2 A, N ) is a spectral triple. We prove the Haagerup-type bound of Ozawa-Rieffel to verify (S0, 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 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) C * -algebra associated to a planar algebra. We conclude that the compact quantum metric spaces coming from the GJS C * -algebras of many infinite families of planar algebras converge in quantum Gromov-Hausdorff distance.