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 a groupoid, $G_l$

, 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 $x \in G_l$ ,

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

(0.1)

Therefore, $\phi$ belongs to $L^\infty{G_l} = L^\infty({G_l},\nu)$ .

Bibliography

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 locally compact groupoids, (2003).

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