Metrics on $C^{\ast}$-algebras of Étale groupoids from length functions
Arnab Chattopadhyay, Md Amir Hossain, Soumalya Joardar
TL;DR
This work extends the quantum metric space framework to reduced C*-algebras of étale groupoids using length functions with rapid decay. It identifies that naive Lip-norms produce a pseudo-metric on the state space due to kernels containing $C(\mathcal{G}^{(0)})$, and introduces a Dirac-type operator $D = M_\ell$ with commutators $\Delta = [D,\lambda(\cdot)]$ to define higher-order Lip-norms $L_\ell^k(a)=\|\Delta^k(a)\|_{\mathrm{adj}}$. For transformation groupoids $\mathcal{G}=\Gamma\ltimes X$, the state space splits into fibres $S_\eta$ indexed by $\eta\in\mathcal{M}(X)$ with fibrewise finite diameter, and if $|X|$ is finite the fibre metrics metrize the weak$^*$-topology, yielding quasi-compact quantum metric spaces for $k>p$. Overall, the results provide a concrete, fibrewise quantum metric framework for groupoid C*-algebras, enabling analysis of convergence and isometry-invariance in noncommutative geometry.
Abstract
We show that for an étale groupoid with compact unit space, the natural Dirac type operator from a continuous length function produces a natural pseudo-metric on the state space of the corresponding reduced $C^{\ast}$-algebra. For a transformation groupoid with a continuous, proper length function with rapid decay, the state space decomposes into genuine metric spaces with a uniform finite diameter fibred over the state space of the compact unit space. Moreover, when the unit space of the transformation groupoid has finitely many points, the metric on each fibre metrizes the weak*-topology.