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Decomposition of $B(G)$


Author: Tianxuan Miao
Journal: Trans. Amer. Math. Soc. 351 (1999), 4675-4692
MSC (1991): Primary 43A07
DOI: https://doi.org/10.1090/S0002-9947-99-02328-4
Published electronically: July 20, 1999
MathSciNet review: 1608490
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Abstract: For any locally compact group $G$, let $A(G)$ and $B(G)$ be the Fourier and the Fourier-Stieltjes algebras of $G$, respectively. $B(G)$ is decomposed as a direct sum of $A(G)$ and $ B^{s}(G)$, where $ B^{s}(G)$ is a subspace of $B(G)$ consisting of all elements $ b\in B(G)$ that satisfy the property: for any $\epsilon > 0$ and any compact subset $ K\subset G$, there is an $ f\in L^{1}(G) $ with $\Vert f\Vert _{C^{*}(G)} \le 1$ and $ supp(f) \subset K^{c}$ such that $\vert \langle f, b \rangle \vert > \Vert b\Vert - \epsilon .$ $A(G)$ is characterized by the following: an element $b\in B(G)$ is in $A(G)$ if and only if, for any $\epsilon > 0,$ there is a compact subset $ K\subset G$ such that $ \vert \langle f, b \rangle \vert < \epsilon $ for all $ f\in L^{1}(G) $ with $\Vert f\Vert _{C^{*}(G)} \le 1 $ and $ supp(f) \subset K^{c}$. Note that we do not assume the amenability of $G$. Consequently, we have $\Vert 1 + a\Vert = 1 + \Vert a\Vert $ for all $a\in A(G)$ if $G$ is noncompact. We will apply this characterization of $B^{s}(G)$ to investigate the general properties of $B^{s}(G)$ and we will see that $B^{s}(G)$ is not a subalgebra of $B(G)$ even for abelian locally compact groups. If $G$ is an amenable locally compact group, then $B^{s}(G)$ is the subspace of $B(G)$ consisting of all elements $b\in B(G)$ with the property that for any compact subset $K\subseteq G$, $\Vert b\Vert = \sup \, \{ \, \Vert a b\Vert : \, a\in A(G), \; supp(a) \subseteq K^{c} \; \text{ and } \; \Vert a\Vert \le 1 \, \}$.


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  • 1. 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
  • 2. L. Baggett and K. Taylor, Groups with completely reducible regular representation, Proc. Amer. Math. Soc. 72 (1978), 593-600. MR 80b:22009
  • 3. L.J. Bunce, The Dunford-Pettis property in the predual of a Von Neumann algebra, Proc. Amer. Math. Soc. 116 (1992), 99-100. MR 92k:46100
  • 4. M. Cowling, An application of Littlewood-Paley theory in harmonic analysis, Math. Ann. 241 (1979), 83-96. MR 81f:43003
  • 5. J. Diestel, Sequences and series in Banach spaces, Graduate Texts in Math., Springer-Verlag, New York, 1984. MR 85i:46020
  • 6. -, A survey of results related to the Dunford-Pettis property, Contemp. Math. 2 (1980), 15-60. MR 82i:46023
  • 7. J. Dixmier, $C^{*}$-Algebras, North-Holland, New York, 1977. MR 56:16388
  • 8. 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
  • 9. V. Flory, On the Fourier algebra of a locally compact amenable group, Proc. Amer. Math. Soc. 29 (1971), 603-606. MR 44:371
  • 10. -, Eine Lebesgue-Zerlegung und funktorielle Eigenschaften der Fourier-Stieltjes Algebra, Inaugural Dissertation, University of Heidelberg (1972).
  • 11. E. E. Granirer and M. 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. MR 85f:43009
  • 12. F.P. Greenleaf, Invariant Means on Topological Groups and their Applications, Van Nostrand Reinhold, New York, 1969. MR 40:4776
  • 13. C. Herz, Harmonic synthesis for subgroups, Ann. Inst. Fourier (Grenoble) 23 (1973), 91-123. MR 50:7956
  • 14. E. Hewitt and K. Ross, Abstract harmonic analysis, Vol. I, Second Edition, Springer-Verlag, Berlin, 1979. MR 81k:43001
  • 15. A.T. Lau and A. Ülger, Some geometric properties on the Fourier and Fourier Stieltjes algebras of locally compact groups, Arens regularity and related problems, Trans. Amer. Math. Soc. 337 (1993), 321-359. MR 93j:22007
  • 16. G. J. Murphy, $C^{*}$-algebra and operator theory, Academic Press, 1990. MR 91m:46084
  • 17. T.W. Palmer, Classes of nonabelian, noncompact locally compact groups, Rocky Mountain J Math. 8 (1978), 683-741. MR 81j:22003
  • 18. A.L.T. Paterson, Amenability, Mathematical Surveys and Monographs, 29, Amer. Math. Soc., Providence, Rhode Island, 1988. MR 90e:43001
  • 19. J.P. Pier, Amenable Locally Compact Groups, Wiley, New York, 1984. MR 86a:43001
  • 20. H. Reiter, Classical Harmonic Analysis and Locally Compact Groups, Oxford, Clarendon, 1968. MR 46:5933
  • 21. K. Taylor, Geometry of Fourier algebras and locally compact groups with atomic representations, Math. Ann. 262 (1983), 183-190. MR 84h:43020

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Additional Information

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

DOI: https://doi.org/10.1090/S0002-9947-99-02328-4
Keywords: Locally compact groups, amenable groups, the Fourier algebra of a locally compact group, the Fourier-Stieltjes algebra of a locally compact group, the Lebesgue-type decomposition of the Fourier-Stieltjes algebra
Received by editor(s): April 29, 1997
Published electronically: July 20, 1999
Additional Notes: This research is supported by an NSERC grant
Article copyright: © Copyright 1999 American Mathematical Society

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