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Abstract  We describe representations of groupoidC   ∗   ‐algebras on Hilbert modules over arbitraryC   ∗   ‐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 groupoidC   ∗   ‐algebras as crossed products for étale groupoids and transformation groupoids of group actions on spaces.

A universal property for groupoid C * ‐algebras. I