Spectral continuity for étale groupoids with the Rapid decay property

We show that the reduced groupoid C*-algebras of continuous fields of étale groupoids satisfying the rapid decay property yield continuous fields of C*-algebras. This establishes a new sufficient criterion that applies in the non-amenable case where the full and reduced groupoid algebras may differ. Potential applications include convergence of spectra in inverse systems of finite-index subgroups and magnetic models on hyperbolic lattices.