Measurable Schur Multipliers and Completely Bounded Multipliers of the Fourier Algebras

Let G be a locally compact group L^p(G) be the usual L^p-space for 1 =< p =< infty and A(G) be the Fourier algebra of G. Our goal is to study, in a new abstract context, the completely bounded multipliers of A(G), which we denote M_{cb}A(G). We show that M_{cb}A(G) can be characterised as the ``invariant part’’ of the space of (completely) bounded normal L^{infty}(G)-bimodule maps on B(L^2(G)), the space of bounded operator on L^2(G). In doing this we develop a function theoretic description of the normal L^{infty}(X,mu)-bimodule maps on B(L^2(X,mu)), which we denote by V^{infty}(X,\mu), and name the measurable Schur multipliers of (X,mu). Our approach leads to many new results, some of which generalise results hitherto known only for certain classes of groups. Those results which we develop here are a uniform approach to the functorial properties of M_{cb}A(G), and a concrete description of a standard predual of M_{cb}A(G).