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# Representing Multipliers of the Fourier Algebra on Non-Commutative $L^p$ Spaces

Published:2011-03-25
Printed: Aug 2011
• Matthew Daws,
School of Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom
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## 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 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 $L^p$ spaces we construct and show that they are completely isometric to those considered recently by Forrest, Lee, and Samei. We improve a result of theirs about module homomorphisms. We suggest a definition of a Figa-Talamanca-Herz algebra built out of these non-commutative $L^p$ spaces, say $A_p(\widehat G)$. It is shown that $A_2(\widehat G)$ is isometric to $L^1(G)$, generalising the abelian situation.
 Keywords: multiplier, Fourier algebra, non-commutative $L^p$ space, complex interpolation
 MSC Classifications: 43A22 - Homomorphisms and multipliers of function spaces on groups, semigroups, etc. 43A30 - Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc. 46L51 - Noncommutative measure and integration 22D25 - $C^*$-algebras and $W^*$-algebras in relation to group representations [See also 46Lxx] 42B15 - Multipliers 46L07 - Operator spaces and completely bounded maps [See also 47L25] 46L52 - Noncommutative function spaces