Metrics on $C^{\ast }$-algebras of Étale groupoids from length functions

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.