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

Details Representing multipliers of the Fourier algebra on non-commutative $L^p$ spaces Representing multipliers of the Fourier algebra on non-commutative $L^p$ spaces

2009-06-28

arxiv.org/abs/0906.5128v2

Canadian Journal of Mathematics 63 (2011) 798-825

Mathematics

Functional Analysis

23 pages; added references, fixed typos

Scientific paper

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.

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