Compact quantum metric spaces from free graph algebras

Starting with a vertex-weighted pointed graph [Formula: see text], we form the free loop algebra text] defined in Hartglass–Penneys’ article on canonical text]-algebras associated to planar algebra. Under mild conditions, is non-nuclear simple text]-algebra unique tracial state. There polynomial subalgebra together Dirac number operator such that spectral triple. We prove Haagerup-type bound of Ozawa–Rieffel verify yields compact quantum metric space sense Rieffel. give weighted analog Benjamini–Schramm convergence for graphs. As our are non-nuclear, adjust Lip-norm coming from utilize finite dimensional filtration text]. then graphs leads Gromov–Hausdorff adjusted spaces. an application, apply construction Guionnet–Jones–Shyakhtenko (GJS) conclude spaces GJS many infinite families algebras converge distance.