Fourier-Stieltjes Algebras

Abstract

For locally compact groups, Fourier algebras and Fourier-Stieltjes algebras have proven to be useful dual objects. They encode the representation theory of the group that is the positive definitefunctions on the group, the information about the algebra of the group in the geometry of the Banach space structure, and the group appears as atopological subspace of the maximal ideal space of the algebra. Fourier-Stieltjes algebra and Fourier algebras of locally compact group are extended to an arbitrary measured groupoid. For alocally compact group, a continous unitary representation is an - representation if the matrix coefficient functions lie in.The - Fourier algebra is defined to be the set of matrix coefficient functions of - representation. Similarly, the- Fourier Stieltjes algebra is defined to be the weak*-closure of the Fourier algebra in the Fourier Stieltjes algebra.These are always ideals in the Fourier Stieltjes algebra containing the Fourier algebra.