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.
