Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-20T10:42:52.531Z Has data issue: false hasContentIssue false
  • English
  • Français

Predual of the Multiplier Algebra of Ap(G) and Amenability

Published online by Cambridge University Press:  20 November 2018

Tianxuan Miao
Tianxuan Miao*
Affiliation:
Department of Mathematical Sciences, Lakehead University, Thunder Bay, Ontario, P7E 5E1 e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For a locally compact group $G$ and $1\,<\,p\,<\,\infty $, let ${{A}_{p}}\left( G \right)$ be the Herz-Figà-Talamanca algebra and let $P{{M}_{p}}\left( G \right)$ be its dual Banach space. For a Banach ${{A}_{p}}\left( G \right)$-module $X$ of $P{{M}_{p}}\left( G \right)$, we prove that the multiplier space $\text{M}\left( {{A}_{p}}\left( G \right),{{X}^{*}} \right)$ is the dual Banach space of ${{Q}_{X}}$, where ${{Q}_{X}}$ is the norm closure of the linear span ${{A}_{p}}\left( G \right)X\,\text{of}\,u\,f\,\text{for}\,u\,\in \,{{A}_{p}}\left( G \right)\,\text{and}\,f\,\in \,X$ in the dual of $\text{M}\left( {{A}_{p}}\left( G \right),{{X}^{*}} \right)$. If $p\,=\,2$ and $P{{F}_{p}}\left( G \right)\subseteq X$, then ${{A}_{p}}\left( G \right)X$ is closed in $X$ if and only if $G$ is amenable. In particular, we prove that the multiplier algebra $M{{A}_{p}}\left( G \right)\,\text{of}\,{{A}_{p}}\left( G \right)$ is the dual of $Q$, where $Q$ is the completion of ${{L}^{1}}\left( G \right)$ in the $||\cdot |{{|}_{M}}$-norm. $Q$ is characterized by the following: $f\,\in \,Q$ if an only if there are ${{u}_{i}}\,\in \,{{A}_{p}}\left( G \right)$ and ${{f}_{i}}\in P{{F}_{p}}\left( G \right)\left( i=1,2,... \right)$ with $\sum\nolimits_{i=1}^{\infty }{||}\,{{u}_{i}}\,|{{|}_{{{A}_{p}}\left( G \right)}}||fi|{{|}_{P{{F}_{p}}\left( G \right)}}\,<\,\infty $ such that $f=\sum{_{i=1}^{\infty }\,{{u}_{i}}{{f}_{i}}}$ on $M{{A}_{p}}\left( G \right)$. It is also proved that if ${{A}_{p}}\left( G \right)$ is dense in $M{{A}_{p}}\left( G \right)$ in the associated ${{w}^{*}}$-topology, then the multiplier norm and $||\cdot |{{|}_{{{A}_{p}}\left( G \right)}}$-norm are equivalent on ${{A}_{p}}\left( G \right))$ if and only if $G$ is amenable.

Keywords

43A07Locally compact groupsamenable groupsmultiplier algebraHerz algebra
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 56 , Issue 2 , 01 April 2004 , pp. 344 - 355
Copyright
Copyright © Canadian Mathematical Society 2004

References

[1] Cowling, M. and Haagerup, U., Completely bounded Multipliers of the Fourier algebra of a simple Lie group of real rank one. Invent. Math. 96(1989), 507–549.Google Scholar
[2] De Cannière, J. and Haagerup, U., Multipliers of the Fourier algebra of some simple Lie groups and their discrete subgroups. Amer. J. Math. 107(1984), 455–500.Google Scholar
[3] Eymard, P., L'algèbre de Fourier d'un groupe localement compact. Bull. Soc. Math. France 92(1964), 181–236.Google Scholar
[4] Furuta, K., Algebras Ap and Bp and the amenability of locally compact groups. Hokkaido Math. J. 20(1991), 579–591.Google Scholar
[5] Granirer, E. E., An application of the Radon Nikodym property in harmonic analysis. Boll. Un. Mat. Ital. B (5) 18(1981), 663–671.Google Scholar
[6] Granirer, E. E. and Leinert, , On some topologies which coincide on the unit sphere of the Fourier-Stieltjes algebra B(G) and of the measure algebra M(G) . Rocky Mountain J. Math. 11(1981), 459–472.Google Scholar
[7] Haagerup, U. and Kraus, J., Approximation properties for group C*-algebras and group von Neumann algebras. Trans. Amer.Math. Soc. (2) 344(1994), 667–699.Google Scholar
[8] Herz, C., Harmonic synthesis for subgroups. Ann. Inst. Fourier, Grenoble (3) 23(1973), 91–123.Google Scholar
[9] Lau, A. T. and Losert, V., The C*-algebra generated by operators with compact support on a locally compact group. J. Funct. Anal. 112(1993), 1–30.Google Scholar
[10] Losert, V., Properties of the Fourier algebra that are equivalent to amenability. Proc. Amer. Math. Soc. 92(1984), 347–354.Google Scholar
[11] Pier, J. P., Amenable locally compact groups. Wiley, New York, 1984.Google Scholar
[12] Rudin, W., Functional analysis (Second Edition). McGraw-Hill, Inc., New York, 1991.Google Scholar
[13] Wojtaszczyk, P., Banach spaces for analysts. Cambridge University Press, Cambridge, New York, 1991.Google Scholar
[14] Xu, G., Amenability and Uniformly continuous Functionals on the Algebras Ap (G) for Discrete Groups. Proc. Amer. Math. Soc. (11) 123(1995), 3425–3429.Google Scholar