We introduce and investigate using Hilbert modules the properties of the {\em Fourier algebra} $A(G)$ for a locally compact groupoid $G$. We establish a duality theorem for such groupoids in terms of multiplicative module maps. This includes as a special case the classical duality theorem for locally compact groups proved by P. Eymard.

Keywords:

Fourier algebra, locally compact groupoids, Hilbert modules, positive definite functions, completely bounded maps Fourier algebra, locally compact groupoids, Hilbert modules, positive definite functions, completely bounded maps

MSC Classifications:

43A32 show english descriptions Other transforms and operators of Fourier type 43A32 - Other transforms and operators of Fourier type

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