Transactions of the American Mathematical Society

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

Author(s): Guangwu Xu
Journal: Trans. Amer. Math. Soc. 349 (1997), 2525-2536.
MSC (1991): Primary 43A30, 43A60, 43A46; Secondary 22D05, 22D25
Retrieve article in: PDF DVI PostScript
This article is available free of charge

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 ${MA_{p}(F)\cap {W(F)}}$ .

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 -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

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 43A30, 43A60, 43A46, 22D05, 22D25

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS