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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 with the right von Neumann algebra of $G$. The resulting image is the idealiser of the image of the Fourier algebra. 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 noncommutative ${{L}^{p}}$ spaces we construct and show that they are completely isometric to those considered recently by Forrest, Lee, and Samei. We improve a result of theirs 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.
