# 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 $x\in G_{l}$,

 $\phi(x)=L(x)\xi(s(x),\eta(r(x))).$ (0.1)

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

## 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 locally compact groupoids, (2003).
 Title Fourier-Stieltjes algebra of a groupoid Canonical name FourierStieltjesAlgebraOfAGroupoid Date of creation 2013-03-22 18:16:11 Last modified on 2013-03-22 18:16:11 Owner bci1 (20947) Last modified by bci1 (20947) Numerical id 11 Author bci1 (20947) Entry type Definition Classification msc 55N33 Classification msc 55N20 Classification msc 55P10 Classification msc 55U40 Classification msc 42B10 Classification msc 42A38 Classification msc 43A25 Classification msc 43A30 Synonym Fourier-Stieltjes algebra of a groupoid Related topic FourierStieltjesTransform Related topic Distribution4 Defines Fourier-Stieltjes algebra