  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).
