  Let G be a locally compact group, and let A(G) and B(G) denote its Fourier
and Fourier-Stieltjes algebras. These algebras are dual objects of the group
and measure algebras, L^1(G) and M(G), in a sense which generalizes the
Pontryagin duality theorem on abelian groups. We wish to consider the
amenability properties of A(G) and B(G) and compare them to such properties for
L^1(G) and M(G). For us, ``amenability properties'' refers to amenability, weak
amenability, and biflatness, as well as some properties which are more suited
to special settings, such as the hyper-Tauberian property for semisimple
commutative Banach algebras. We wish to emphasize that the theory of operator
spaces and completely bounded maps plays an indispensable role when studying
A(G) and B(G). We also show some applications of amenability theory to problems
of complemented ideals and homomorphisms.
