Almost Representations of Groupoids on Banach Bundles

We extend an old result of de la Harpe and Karoubi, concerning almost representations of compact groups, to proper groupoids admitting continuous Haar measure systems. As an application, we establish the existence of sufficiently many continuous representations of such groupoids on finite-rank Hilbert bundles locally, and use this fact to prove a new generalization of the classical Tannaka duality theorem to groupoids in a purely topological setting.