Representing Multipliers of the Fourier Algebra on Non-Commutative L-p 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 Lp 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 Lp 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 Lp spaces, say Ap(ˆG). It is shown that A2(ˆG) is isometric to L1(G), generalising the abelian situation.