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

Daws, Matthew

Canadian journal of mathematics, Tome 63 (2011) no. 4, pp. 798-825

Résumé

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.

Détail

DOI : 10.4153/CJM-2011-020-2

Mots-clés : 43A22, 43A30, 46L51, 22D25, 42B15, 46L07, 46L52, multiplier, Fourier algebra, non-commutative L p space, complex interpolation