Fourier-Stieltjes algebra of a groupoid

Definition 0.1.

The Fourier-Stieltjes algebra of a groupoid, $G_{l}$ .In ref. [3]), A.L.T. Paterson defined the Fourier-Stieltjes algebra of agroupoid, $G_{l}$ , as the space of coefficients $\\phi=(\\xi,\\eta)$ , where $\\xi,\\eta$ are
$L^{\\infty}$ -sections for some measurable $G_{l}$ -Hilbert bundle $(\\mu,\\Re,L)$ . Thus, for $x\\in G_{l}$ ,

\\phi(x)=L(x)\\xi(s(x),\\eta(r(x))).

(0.1)

Therefore, $\\phi$ belongs to $L^{\\infty}{G_{l}}=L^{\\infty}({G_{l}},\u)$ .

References

1 A. Ramsay and M. E. Walter, Fourier-Stieltjes algebras of locally compact groupoids,J. Functional Anal. 148: 314-367 (1997).
2 A. L. T. Paterson, The Fourier algebra for locally compact groupoids., Preprint, (2001).
3 A. L. T. Paterson, The Fourier-Stieltjes and Fourier algebras for locallycompact groupoids, (2003).