Almost Representations of Groupoids on Banach Bundles

TL;DR

This paper generalizes the concept of almost representations from compact groups to proper groupoids with Haar systems, establishing new representation existence results and extending Tannaka duality to a topological groupoid context.

Contribution

It extends classical results on almost representations to proper groupoids and proves a topological Tannaka duality theorem for these structures.

Findings

Existence of continuous representations of proper groupoids on finite-rank Hilbert bundles

Generalization of Tannaka duality to topological groupoids

Extension of de la Harpe and Karoubi's results to a broader class of groupoids

Abstract

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.