2021/09/14 by Konrad Aguilar, Aguilar, Konrad, Michael Hartglass +3 · 1 citation
Mathematics · #46L09 #46L37 #46L54 #46L87 #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Operator Algebras (math.OA)
paper · pdf · doi:10.48550/arxiv.2109.06985
openalex publication_date 2021/09/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Starting with a vertex-weighted pointed graph (Γ,μ,v0), we form the free loop algebra S0 defined in Hartglass-Penneys' article on canonical \rm C^*-algebras associated to a planar algebra. Under mild conditions, S0 is a non-nuclear simple \rm C^*-algebra with unique tracial state. There is a canonical polynomial subalgebra A⊂ S0 together with a Dirac number operator N such that (A, L2A,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 \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.