Vol. 27, No. 3, 1968

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Harmonic analysis on groupoids

Joel John Westman

Vol. 27 (1968), No. 3, 621–632

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 12(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 2(Z) as a direct sum of certain simple H-algebras.

Mathematical Subject Classification
Primary: 22.65
Secondary: 46.00
Received: 5 July 1967
Published: 1 December 1968
Joel John Westman