Operator Spaces and Ideals in Fourier Algebras
Brannan, Michael Paul
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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.