  We show that the multiplier algebra of the Fourier algebra on a locally
compact group $G$ can be isometrically represented on a direct sum on
non-commutative $L^p$ spaces associated to the right von Neumann algebra of
$G$. If these spaces are given their canonical Operator space structure, then
we get a completely isometric representation of the completely bounded
multiplier algebra. We make a careful study of the non-commutative $L^p$ spaces
we construct, and show that they are completely isometric to those considered
recently by Forrest, Lee and Samei; we improve a result about module
homomorphisms. We suggest a definition of a Figa-Talamanca--Herz algebra built
out of these non-commutative $L^p$ spaces, say $A_p(\hat G)$. It is shown that
$A_2(\hat G)$ is isometric to $L^1(G)$, generalising the abelian situation.
