Representing multipliers of the Fourier algebra on non-commutative L spaces

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 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 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 spaces, say Ap(Ĝ). It is shown that A2(Ĝ) is isometric to L (G), generalising the abelian situation. Subject classification: 43A22, 43A30, 46L51 (Primary); 22D25, 42B15, 46L07, 46L52 (Secondary).