Tannaka-krein Duality for Compact Groupoids Iii, Duality Theory

This is the last in a series of papers in which we generalized the Tannaka-Krein duality to compact groupoids. In [A1] we studied the representation theory of compact groupoids. In particular, we showed that irreducible representations have finite dimensional fibres. We also proved the Schur’s lemma, Gelfand-Raikov theorem and Peter-Weyl theorem for compact groupoids. In [A2] we studied the Fourier and Fourier-Plancherel transforms and their inverse transforms on compact groupoids. In this part we show how to recover a compact groupoid from its representation theory. This is done along the lines of the Tannaka duality for compact groups. We refer the interested reader to [JS] for a clear exposition of this theory. All over this paper we assume that G is compact and the Haar system on G is normalized. We put X = G.