Compact quantum metric spaces from free probability

Mathematics > Operator Algebras [Submitted on 17 Jun 2026] Title:Compact quantum metric spaces from free probability View PDF HTML (experimental)Abstract:We study quantum metric space structures on operator algebras arising from free probability, namely those associated to $q$-Gaussians and free Gibbs laws for convex potentials. We note that even for free semicirculars, Voiculescu's dual system does not produce a quantum metric space structure that recovers the weak-$*$ topology on the state space. However, for $q$-Gaussians, we can define a compact quantum metric space using length-like functions by the same method as has already been used for hyperbolic groups, quantum groups of rapid decay, free products, and free graph algebras. Next, motivated by the free transport results for free Gibbs laws, we describe a universal way of defining Lip-norms in terms of a generating set, which behaves well under changes of coordinates. We show using semigroup regularization that this Lip-norm defines a quantum metric space structure for $q$-Gaussians, and then transfer this property to free Gibbs laws for convex potentials using free transport. References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.