2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/68393In a series of papers, we have shown that from the representatio theory of a compact groupoid one can reconstruct the groupoid using the procedure similar to the Tannaka-Krein duality for compact groups. In this part we study continuous representations of compact groupoids. We show that irreducible representations have finite dimensional fibres. We prove the Schur's lemma, Gelfand-Raikov theorem and Peter-Weyl theorem for compact groupoids.15 pagesOperator AlgebrasRepresentation Theory43A65, 43A40text