Representing multipliers of the fourier algebra on non-commutative L p Spaces

Abstract

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 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 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 A p(G). It is shown that A2 (G) is isometric to L 1 (G), generalising the abelian situation. © Canadian Mathematical Society 2011.