Compact Quantum Metric Spaces from Free Graph Algebras
Abstract
Starting with a vertexweighted pointed graph $(\Gamma,\mu,v_0)$, we form the free loop algebra $\mathcal{S}_0$ defined in HartglassPenneys' article on canonical $\rm C^*$algebras associated to a planar algebra. Under mild conditions, $\mathcal{S}_0$ is a nonnuclear 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 Haageruptype bound of OzawaRieffel to verify $(\mathcal{S}_0, A, N)$ yields a compact quantum metric space in the sense of Rieffel. We give a weighted analog of BenjaminiSchramm convergence for vertexweighted pointed graphs. As our $\rm C^*$algebras are nonnuclear, we adjust the Lipnorm coming from $N$ to utilize the finite dimensional filtration of $A$. We then prove that convergence of vertexweighted pointed graphs leads to quantum GromovHausdorff convergence of the associated adjusted compact quantum metric spaces. As an application, we apply our construction to the GuionnetJonesShyakhtenko (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 GromovHausdorff distance.
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 arXiv:
 arXiv:2109.06985
 Bibcode:
 2021arXiv210906985A
 Keywords:

 Mathematics  Operator Algebras;
 46L37;
 46L87;
 46L09;
 46L54
 EPrint:
 15 pages, some figures