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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Characterizations of elements with compact support in the dual spaces of $A_{p}(G)$-modules of $PM_{p}(G)$


Author: Tianxuan Miao
Journal: Proc. Amer. Math. Soc. 132 (2004), 3671-3678
MSC (2000): Primary 43A07
Published electronically: June 2, 2004
MathSciNet review: 2084090
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Abstract: For a locally compact group $G$ and $ 1 < p < \infty $, let $A_{p}(G)$ be the Figà-Talamanca-Herz algebra and let $PM_{p}(G)$ be its dual Banach space. For a Banach $A_{p}(G)$-module $X$ of $PM_{p}(G)$, we denote the norm closure of the subspace of the elements in $X^{*}$ with compact support by $A_{p,X}(G)$. We prove that an element $u$ of $X^{*}$ is in $A_{p,X}(G)$ if and only if for any $\epsilon > 0$, there exists a compact subset $K$ of $G$such that $\vert \langle u, f \rangle \vert < \epsilon $ for all $f\in X$ with $\Vert f\Vert \le 1$ and $supp \, (f)\subseteq G\sim K$. In particular, we have that an element $b$ of $W_{p}(G)$ is in $A_{p}(G)$ if and only if for any $\epsilon > 0$, there exists a compact subset $K$ of $G$ such that $\vert \langle u, f \rangle \vert < \epsilon $ for all $f\in L^{1}(G\sim K)$ with $\Vert f\Vert \le 1$. If $A_{p, X}(G)$ has an orthogonal complement $A_{p, X}^{s}(G)$ in $X^{*}$, we characterize $A_{p, X}^{s}(G)$ by the following condition: $u\in X^{*}$ is in $A_{p, X}^{s}(G)$ if and only if for any $\epsilon > 0$ and any compact subset $K$ of $G$, there exists some $f\in X$with $\Vert f\Vert \le 1$ and $supp\, (f)\subseteq G\sim K$ such that $\vert \langle u, f \rangle \vert > \Vert u\Vert - \epsilon $. Some results of Flory (1971) and Miao (1999) can be obtained from our main theorems by taking $p=2$ and $X$ as some $C^{*}$-subalgebras of $PM_{p}(G)$.


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Additional Information

Tianxuan Miao
Affiliation: Department of Mathematical Sciences, Lakehead University, Thunder Bay, Ontario, Canada P7E 5E1
Email: tmiao@mail.lakeheadu.ca

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07550-1
PII: S 0002-9939(04)07550-1
Keywords: Locally compact groups, amenable groups, Fourier algebra, Fourier-Stieltjes algebra, Lebesgue-type decomposition, Fig\`a-Talamanca-Herz algebra
Received by editor(s): January 22, 2003
Received by editor(s) in revised form: September 3, 2003
Published electronically: June 2, 2004
Additional Notes: This research is supported by an NSERC grant
Communicated by: Andreas Seeger
Article copyright: © Copyright 2004 American Mathematical Society