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Transactions of the American Mathematical Society

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Herz-Schur multipliers
and weakly almost periodic functions
on locally compact groups


Author: Guangwu Xu
Journal: Trans. Amer. Math. Soc. 349 (1997), 2525-2536
MSC (1991): Primary 43A30, 43A60, 43A46; Secondary 22D05, 22D25
DOI: https://doi.org/10.1090/S0002-9947-97-01733-9
MathSciNet review: 1373647
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Abstract: For a locally compact group $G$ and $1<p<\infty $, let $A_{p}(G)$ be the Herz-Figà-Talamanca algebra and $B_{p}(G)$ the Herz-Schur multipliers of $G$, and $MA_{p}(G)$ the multipliers of $A_{p}(G)$. Let $W(G)$ be the algebra of continuous weakly almost periodic functions on $G$. In this paper, we show that (1), if $G$ is a noncompact nilpotent group or a noncompact [IN]-group, then $W(G)/B_{p}(G)^{-}$ contains a linear isometric copy of $l^{\infty }({\mathbb {N}})$; (2), for a noncommutative free group $F, B_{p}(F)$ is a proper subset of ${MA_{p}(F)\cap {W(F)}}$.


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  • 1. G. Bennett, Schur multipliers, Duke Math. J. 44 (1977), 603-639. MR 58:12490
  • 2. M. Bo$\dot z$ejko, Remark on Herz-Schur multipliers on free groups, Math. Ann. 258 (1981), 11-15. MR 83a:43001
  • 3. -, Positive definite bounded matrices and a characterization of amenable groups, Proc. Amer. Math. Soc. 95 (1985), 357-359. MR 87h:43006
  • 4. -, Positive and negative definite kernels on discrete groups, Lectures at Heidelberg Univ., 1987.
  • 5. M. Bo$\dot z$ejko and G. Fendler, Herz-Schur multipliers and completely bounded multipliers of Fourier algebra of a locally compact group, Boll. Un. Mat. Ital. (6)3-A (1984), 297-302. MR 86b:43009
  • 6. R. B. Burckel, Weakly almost periodic functions on semigroups, Gordon and Breach, New York, 1970. MR 41:8562
  • 7. C. Chou, Weakly almost periodic functions and Fourier-Stieltjes algebras of locally compact groups, Trans. Amer. Math. Soc. 274 (1982), 141-157. MR 84a:43008
  • 8. -, Weakly almost periodic functions and thin sets in discrete groups, Trans. Amer. Math. Soc. 321 (1990), 333-346. MR 90m:43023
  • 9. M. Cowling and U. Haagerup, Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one, Invent. Math. 96 (1989), 507-549. MR 90h:22008
  • 10. H. W. Davis, Generalized almost periodicity in groups, Trans. Amer. Math. Soc. 191 (1974), 329-352. MR 50:898
  • 11. 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
  • 12. P. Eymard, L'algèbre de Fourier d'un groupe localement compact, Bull. Soc. Math. France 92 (1964), 181-236. MR 37:4208
  • 13. G. Fendler, Herz Schur multipliers and coefficients of bounded representations, Thesis at Heidelberg Univ., 1987.
  • 14. A. Figà-Talamanca and M. A. Picardello, Harmonic analysis on free groups, Lecture Notes in Pure and Appl. Math., vol. 87, Marcel Dekker, New York, 1983. MR 85j:43001
  • 15. K. Furuta, Algebras $A_{p}$ and $B_{p}$ and amenability of locally compact groups, Hokkaido Math. J. 20 (1991), 579-591. MR 92m:43004
  • 16. E. E. Granirer, Some results on $A_{p}(G)$ submodules of $PM_{p}(G)$, Colloq. Math. 51 (1987), 155-163. MR 88f:43008
  • 17. -, On some spaces of linear functionals on the algebras $A_{p}(G)$ for locally compact groups, Colloq. Math. 52 (1987), 119-132. MR 88k:43006
  • 18. U. Haagerup, An example of a nonnuclear $C^{*}$-algebra which has the metric approximation property, Invent. Math. 50 (1979), 279-293. MR 80j:46094
  • 19. -, Group $C^{*}$-algebra without the completely bounded approximation property, Preprint.
  • 20. U. Haagerup and J. Kraus, Approximation properties for group $C^{*}$-algebras and group von Neumann algebras, Trans. Amer. Math. Soc. 344 (1994), 667-699. MR 94k:22008
  • 21. C. Herz, The theory of $p$-spaces with an application to convolution operators, Trans. Amer. Math. Soc. 154 (1971), 69-82. MR 42:7833
  • 22. -, Harmonic synthesis for subgroups, Ann. Inst. Fourier (Grenoble) 23 (1973), 91-123. MR 50:7956
  • 23. -, Une généralization de la notion de transformée de Fourier-Stieltjes, Ann. Inst. Fourier (Grenoble) 24 (1974), 145-157. MR 54:13466
  • 24. A. T. Lau and A. Ülger, Some geometric properties on the Fourier algebras and Fourier Stieltjes algebras of locally compact groups, Arens regularity and related problems, Trans. Amer. Math. Soc. 337 (1993), 321-359. MR 93g:22007
  • 25. M. Leinert, Faltungsoperatoren auf gewissen diskreten Gruppen, Studia Mat. 62 (1973), 149-158. MR 50:7954
  • 26. V. Losert, Properties of the Fourier algebra that are equivalent to amenability, Proc. Amer. Math. Soc. 92 (1984), 347-354. MR 86b:43010
  • 27. C. Nebbia, Multipliers and asymptotic behaviour of the Fourier algebra of nonamenable groups, Proc. Amer. Math. Soc. 84 (1982), 549-554. MR 83h:43002
  • 28. M. A. Picardello, Lacunary sets in discrete noncommutative groups, Bull. Un. Mat. Ital. 8 (1973), 494-508. MR 49:9543
  • 29. G. Pisier, Completely bounded maps between sets of Banach space operators, Indiana Univ. Math. J. 39 (1990), 249-277. MR 91k:47078
  • 30. -, Multipliers and lacunary sets in non-amenable groups, Amer. J. Math. 117 (1995), 337-376. MR 96e:46078
  • 31. D. Ramirez, Weakly almost periodic functions and Fourier-Stieltjes transforms, Proc. Amer. Math. Soc. 19 (1968), 1087-1088. MR 38:488
  • 32. W. Rudin, Weakly almost periodic functions and Fourier-Stieltjes transforms, Duke Math J. 26 (1959), 215-220. MR 21:1492

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Additional Information

Guangwu Xu
Affiliation: Department of Mathematics, State University of New York at Buffalo, Buffalo, New York 14214-3093
Address at time of publication: Department of Mathematical Sciences, University of Alberts, Edmonton, AB, T6G 2G1, Canada
Email: gxu@vega.math.ualberta.ca

DOI: https://doi.org/10.1090/S0002-9947-97-01733-9
Received by editor(s): November 29, 1994
Received by editor(s) in revised form: January 29, 1996
Article copyright: © Copyright 1997 American Mathematical Society

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