A universal property for groupoid C*-algebras. I

TL;DR

This paper introduces a universal property for groupoid C*-algebras that unifies various representation theories and connects to known descriptions like crossed products, enhancing understanding of their structure.

Contribution

It establishes a universal property characterizing representations of groupoid C*-algebras on Hilbert modules, generalizing Renault's theorem and linking to automatic continuity in group representations.

Findings

Universal property for groupoid C*-algebras on Hilbert modules

Equivalence to Renault's Integration-Disintegration Theorem for Hilbert space representations

Connections to crossed product descriptions for étale and transformation groupoids

Abstract

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 \'etale groupoids and transformation groupoids of group actions on spaces.