# Representing multipliers of the Fourier algebra on non-commutative $L^p$ spaces - Mathematics > Functional Analysis

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Abstract: We show that the multiplier algebra of the Fourier algebra on a locallycompact group $G$ can be isometrically represented on a direct sum onnon-commutative $L^p$ spaces associated to the right von Neumann algebra of$G$. If these spaces are given their canonical Operator space structure, thenwe get a completely isometric representation of the completely boundedmultiplier algebra. We make a careful study of the non-commutative $L^p$ spaceswe construct, and show that they are completely isometric to those consideredrecently by Forrest, Lee and Samei; we improve a result about modulehomomorphisms. We suggest a definition of a Figa-Talamanca-Herz algebra builtout of these non-commutative $L^p$ spaces, say $A p\hat G$. It is shown that$A 2\hat G$ is isometric to $L^1G$, generalising the abelian situation.

Author: ** Matthew Daws**

Source: https://arxiv.org/