Fourier-Stieltjes algebras of locally compact groupoids

This paper gives a first step toward extending the theory of Fourier-Stieltjes algebras from groups to groupoids. If G is a locally compact (second countable) groupoid, we show that B(G), the linear span of the Borel positive definite functions on G, is a Banach algebra when represented as an algebra of completely bounded maps on a C^*-algebra associated with G. This necessarily involves identifying equivalent elements of B(G). An example shows that the linear span of the continuous positive definite functions need not be complete. For groups, B(G) is isometric to the Banach space dual of C^*(G). For groupoids, the best analog of that fact is to be found in a representation of B(G) as a Banach space of completely bounded maps from a C^*-algebra associated with G to a C^*-algebra associated with the equivalence relation induced by G. This paper adds weight to the clues in the earlier study of Fourier-Stieltjes algebras that there is a much more general kind of duality for Banach algebras waiting to be explored.