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Register a new userCompact Quantum Metric Spaces from Free Graph Algebras
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Starting with a vertex-weighted pointed graph $(Gamma,mu,v_0)$, we form the free loop algebra $mathcal{S}_0$ defined in Hartglass-Penneys article on canonical $rm C^*$-algebras associated to a planar algebra. Under mild conditions, $mathcal{S}_0$ is a non-nuclear simple $rm C^*$-algebra with unique tracial state. There is a canonical polynomial subalgebra $Asubset mathcal{S}_0$ together with a Dirac number operator $N$ such that $(A, L^2A,N)$ is a spectral triple. We prove the Haagerup-type bound of Ozawa-Rieffel to verify $(mathcal{S}_0, A, N)$ 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.
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We show that any quantum family of maps from a non commutative space to a compact quantum metric space has a canonical quantum semi metric structure.
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Motivated by noncommutative geometry and quantum physics, the concept of `metric operator field is introduced. Roughly speaking, a metric operator field is a vector field on a set with values in self tensor product of a bundle of C*-algebras, satisfying properties similar to an ordinary metric (distance function). It is proved that to any such object there naturally correspond a Banach *-algebra that we call Lipschitz algebra, a class of probabilistic metrics, and (under some conditions) a (nontrivial) continuous field of C*-algebras in the sense of Dixmier. It is proved that for metric operator fields with values in von Neumann algebras the associated Lipschitz algebras are dual Banach spaces, and under some conditions, they are not amenable Banach algebras. Some examples and constructions are considered. We also discuss very briefly a possible application to quantum gravity.
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