|dc.description.abstract||In this thesis we study ideals in the Fourier algebra, A(G), of a
locally compact group G.
For a locally compact abelian group G, necessary conditions for a
closed ideal in A(G) to be weakly complemented are given, and a complete
characterization of the complemented ideals in A(G) is given when
G is a discrete abelian group. The closed ideals in A(G) with
bounded approximate identities are also characterized for any
locally compact abelian group G.
When G is an arbitrary locally compact group, we exploit the
natural operator space structure that A(G) inherits as the predual
of the group von Neumann algebra, VN(G), to study ideals in
A(G). Using operator space techniques, necessary conditions for an
ideal in A(G) to be weakly complemented by a completely bounded
projection are given for amenable G, and the ideals in A(G)
possessing bounded approximate identities are completely
characterized for amenable G. Ideas from homological algebra are
then used to study the biprojectivity of A(G) in the category of
operator spaces. It is shown that A(G) is operator biprojective
if and only if G is a discrete group. This result is then used to
show that every completely complemented ideal in A(G) is
invariantly completely complemented when G is discrete.
We conclude by proving that for certain discrete groups G, there are complemented ideals in A(G) which fail to be complemented or weakly complemented by completely bounded projections.||en