A continuous representation of the stabilizer (isotropy) group at a base point of a proper locally compact groupoid can always be extended to a continuous representation of the whole groupoid over some neighbourhood of that point, with no differentiability or local-triviality assumptions. The extension is obtained by first proving an almost-representation theorem: on a proper groupoid admitting continuous Haar measure systems, any pseudorepresentation on a Fell Banach bundle whose defect is small enough lies uniformly close to a genuine representation, produced by recursive averaging of the operator against a cutoff function. Learning this shows how properness plus Haar averaging substitutes for compactness, and how dropping local triviality in favour of Banach bundles yields enough finite-type unitary representations to separate arrows and to reconstruct the groupoid from its representation category.
This arxiv paper (June 2024) by Giorgio Trentinaglia studies representation theory of locally compact groupoids -- a core topological algebraic system that generalizes topological groups and semigrou…