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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