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 $L^p$ 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.

23 pages; added references, fixed typos

References in corpus (4)

• Noncommutative vector valued $L_p$-spaces and completely $p$-summing maps
• Projectivity of modules over Fourier algebras
• A Representation Theorem for Completely Contractive Dual Banach Algebras
• Operator Figa-Talamanca-Herz algebras

Cited by in corpus (1)

• Noncommutative Figa-Talamanca-Herz algebras for Schur multipliers