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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: tmiao@mail.lakeheadu.ca
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Abstract

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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

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