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Representing Multipliers of the Fourier Algebra on Non-Commutative Lp Spaces

Published online by Cambridge University Press:  20 November 2018

Matthew Daws
Matthew Daws*
Affiliation:
School of Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom email: matt.daws@cantab.net
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Abstract

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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 noncommutative ${{L}^{p}}$ 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 ${{A}_{p}}(\hat{G})$. It is shown that ${{A}_{2}}(\hat{G})$ is isometric to ${{L}^{1}}(G)$, generalising the abelian situation.

Keywords

43A2243A3046L5122D2542B1546L0746L52multiplierFourier algebranon-commutative Lp spacecomplex interpolation
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 63 , Issue 4 , 01 August 2011 , pp. 798 - 825
Copyright
Copyright © Canadian Mathematical Society 2011

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