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 Cc(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 Cc(Z), yielding ∗
representations of Cc(Z). A norm, ∥∥12, is defined on Cc(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
representations 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.
