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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 $PM_{p}(G)$.

References [Enhancements On Off] (What's this?)

  • 1. C. A. Akemann and M. E. Walter, The Riemann-Lebesgue property for arbitrary locally compact groups, Duke Math. J. 43 (1976), 225-236. MR 53:3599
  • 2. G. Arsac, Sur l'espace de Banach engendr$\acute e$ par les coefficients d'une repr$\acute e$sentation unitaire, Publ. Dép. Math. (Lyon) 13 (1976), 1-101. MR 56:3180
  • 3. J. De Cannière and U. Haagerup, Multipliers of the Fourier algebra of some simple Lie groups and their discrete subgroups, Amer. J. Math. 107 (1984), 455-500. MR 86m:43002
  • 4. P. Eymard, L'alg$\grave e$bre de Fourier d'un groupe localement compact, Bull. Soc. Math. France, 92 (1964), 181-236. MR 37:4208
  • 5. V. Flory, On the Fourier algebra of a locally compact amenable group, Proc. Amer. Math. Soc. 29 (1971), 603-606. MR 44:371
  • 6. -, Eine Lebesgue-Zerlegung und funktorielle Eigenschaften der Fourier-Stieltjes Algebra, Inaugural Dissertation, University of Heidelberg (1972).
  • 7. E. E. Granirer, An application of the Radon-Nikodým property in harmonic analysis, Bollettino Un. Mat. Ital (5) 18-B (1981), 663 - 671. MR 83b:43004
  • 8. C. Herz, Harmonic synthesis for subgroups, Ann. Inst. Fourier, Grenoble 23, No. 3 (1973), 91 - 123. MR 50:7956
  • 9. E. Kaniuth, A. T. Lau and G. Schlichting, Lebesgue type decomposition of subspaces of Fourier-Stieltjes algebras, Trans. Amer. Math. Soc. 355 (2003), 1467 - 1490. MR 2004c:43004
  • 10. T. Miao, Decomposition of $B(G)$, Trans. Amer. Math. Soc. 351, No. 11 (1999), 4675 - 4692. MR 2000a:43006
  • 11. -, Predual of the multiplier algebra of $A_{p}(G)$and amenability, preprint.
  • 12. J. P. Pier, Amenable locally compact groups, Wiley, New York, 1984. MR 86a:43001

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Additional Information

Tianxuan Miao
Affiliation: Department of Mathematical Sciences, Lakehead University, Thunder Bay, Ontario, Canada P7E 5E1

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