Harmonic analysis on groupoids

Abstract

This paper generalizes harmonic analysis on groups to obtain a theory of harmonic analysis on groupoids. A system of measures is obtained for a locally compact locally trivial groupoid, Z, analogous to left Haar measure for a locally compact group. Then a convolution and involution are defined on C0(Z) - the continuous complex valued functions on Z with compact support. Strongly continuous unitary representations of Z on certain fiber bundles, called representation bundles, are lifted to C0(Z) yielding ⋆ representations of C0(Z). A norm, || ||12, is defined on C0(Z), and the convolution, involution, and representations all extend to ℒ12(Z) = the || ||12 completion of CC{Z). The main example given is that of the groupoid Z = Z(G, H) that arises naturally from a Lie group G and a closed subgroup H. In this example, the repre entations of Z are related to induced representations of G. Finally, if Zee (=the group of elements in Z with left unit=right unit = e) is compact then we canonically represent ℒ2(Z) as a direct sum of certain simple H⋆-algebras. © 1968 by Pacific Journal of Mathematics.