Compact quantum metric spaces from free graph algebras

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,LA,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 GromovHausdorff 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.

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