A universal property for groupoid C‐algebras. I

We describe representations of groupoid C∗‐algebras on Hilbert modules over arbitrary C∗‐algebras by a universal property. For Hilbert space representations, our universal property is equivalent to Renault's integration–disintegration theorem. For a locally compact group, it is related to the automatic continuity of measurable group representations. It implies known descriptions of groupoid C∗‐algebras as crossed products for étale groupoids and transformation groupoids of group actions on spaces.