Weakly amenable groups and amalgamated products

Bożejko, Marek; Picardello, Massimo A.

doi:10.1090/S0002-9939-1993-1119263-9

Weakly amenable groups and amalgamated products
HTML articles powered by AMS MathViewer

by Marek Bożejko and Massimo A. Picardello PDF

Proc. Amer. Math. Soc. 117 (1993), 1039-1046 Request permission

Abstract:

Denote by ${B_2}(G)$ the Herz-Schur multiplier algebra of a locally compact group $G$ and by ${B_{2,\lambda }}(G)$ the closure of the Fourier algebra in the topology of pointwise convergence boundedly in the norm of ${B_2}(G)$. $G$ is said to be weakly amenable if ${B_{2,\lambda }}(G) = {B_2}(G)$. We show that every amalgamated product of a countable collection of locally compact amenable groups over a compact open subgroup is weakly amenable. This improves and extends previous results that hold for amalgams of compact groups.

References

Roger Alperin, Locally compact groups acting on trees, Pacific J. Math. 100 (1982), no. 1, 23–32. MR 661438
Roger Alperin, Locally compact groups acting on trees and property $T$, Monatsh. Math. 93 (1982), no. 4, 261–265. MR 666827, DOI 10.1007/BF01295226
Marek Bożejko, Positive definite bounded matrices and a characterization of amenable groups, Proc. Amer. Math. Soc. 95 (1985), no. 3, 357–360. MR 806070, DOI 10.1090/S0002-9939-1985-0806070-2
Marek Bożejko, Uniformly bounded representations of free groups, J. Reine Angew. Math. 377 (1987), 170–186. MR 887407, DOI 10.1515/crll.1987.377.170

Herz-Schur multipliers and completely bounded multipliers of the Fourier algebra of a locally compact group

6

P. Cartier, Fonctions harmoniques sur un arbre, Symposia Mathematica, Vol. IX (Convegno di Calcolo delle Probabilità, INDAM, Rome, 1971) Academic Press, London, 1972, pp. 203–270 (French). MR 0353467
I. M. Chiswell, Length functions and free products with amalgamation groups, Proc. London Math. Soc. (3) 42 (1981), no. 1, 42–58. MR 602122, DOI 10.1112/plms/s3-42.1.42
Michael Cowling, Sur les coefficients des représentations unitaires des groupes de Lie simples, Analyse harmonique sur les groupes de Lie (Sém., Nancy-Strasbourg 1976–1978), II, Lecture Notes in Math., vol. 739, Springer, Berlin, 1979, pp. 132–178 (French). MR 560837
Michael Cowling and Uffe Haagerup, Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one, Invent. Math. 96 (1989), no. 3, 507–549. MR 996553, DOI 10.1007/BF01393695
Jean De Cannière and Uffe Haagerup, Multipliers of the Fourier algebras of some simple Lie groups and their discrete subgroups, Amer. J. Math. 107 (1985), no. 2, 455–500. MR 784292, DOI 10.2307/2374423

La proprieté

de Kazhdan pour les groupes localement compacts

175

Jacques Dixmier, Les $C^{\ast }$-algèbres et leurs représentations, Cahiers Scientifiques, Fasc. XXIX, Gauthier-Villars Éditeur, Paris, 1969 (French). Deuxième édition. MR 0246136
Masatoshi Enomoto and Yasuo Watatani, Reduced group $C^*$-algebras with the metric approximation property by positive maps, Proc. Japan Acad. Ser. A Math. Sci. 63 (1987), no. 8, 304–306. MR 931243
Alessandro Figà-Talamanca and Massimo A. Picardello, Harmonic analysis on free groups, Lecture Notes in Pure and Applied Mathematics, vol. 87, Marcel Dekker, Inc., New York, 1983. MR 710827
Alessandro Figà-Talamanca and Massimo A. Picardello, Restriction of spherical representations of $\textrm {PGL}_{2}(\textbf {Q}_{p})$ to a discrete subgroup, Proc. Amer. Math. Soc. 91 (1984), no. 3, 405–408. MR 744639, DOI 10.1090/S0002-9939-1984-0744639-3

Free topological groups

114

Theory of topological groups, norms and metrics on groups, complete groups, free topological groups

2

Uffe Haagerup, An example of a nonnuclear $C^{\ast }$-algebra, which has the metric approximation property, Invent. Math. 50 (1978/79), no. 3, 279–293. MR 520930, DOI 10.1007/BF01410082

Group

algebras without the completely bounded approximation property

Yasutaka Ihara, On discrete subgroups of the two by two projective linear group over ${\mathfrak {p}}$-adic fields, J. Math. Soc. Japan 18 (1966), 219–235. MR 223463, DOI 10.2969/jmsj/01830219
Carl Herz, Une généralisation de la notion de transformée de Fourier-Stieltjes, Ann. Inst. Fourier (Grenoble) 24 (1974), no. 3, xiii, 145–157 (French, with English summary). MR 425511
Viktor Losert, Properties of the Fourier algebra that are equivalent to amenability, Proc. Amer. Math. Soc. 92 (1984), no. 3, 347–354. MR 759651, DOI 10.1090/S0002-9939-1984-0759651-8
Massimo A. Picardello, Positive definite functions and $L^p$ convolution operators on amalgams, Pacific J. Math. 123 (1986), no. 1, 209–221. MR 834149
Massimo Picardello and Wolfgang Woess, Random walks on amalgams, Monatsh. Math. 100 (1985), no. 1, 21–33. MR 807295, DOI 10.1007/BF01383714
T. Pytlik and R. Szwarc, An analytic family of uniformly bounded representations of free groups, Acta Math. 157 (1986), no. 3-4, 287–309. MR 857676, DOI 10.1007/BF02392596
Jean-Pierre Serre, Trees, Springer-Verlag, Berlin-New York, 1980. Translated from the French by John Stillwell. MR 607504
Ryszard Szwarc, Groups acting on trees and approximation properties of the Fourier algebra, J. Funct. Anal. 95 (1991), no. 2, 320–343. MR 1092129, DOI 10.1016/0022-1236(91)90032-Z