Multipliers and asymptotic behaviour of the Fourier algebra of nonamenable groups
HTML articles powered by AMS MathViewer
- by Claudio Nebbia PDF
- Proc. Amer. Math. Soc. 84 (1982), 549-554 Request permission
Abstract:
Let $G$ be a locally compact group and $A(G)$ the algebra of matrix coefficients of the regular representation. We prove that $G$ is amenable if and only if there exist functions $u \in A(G)$ which vanish at infinity at any arbitrarily slow rate. The "only if" part of the result was essentially known. With the additional hypothesis that $G$ be discrete, we deduce that $G$ is amenable if and only if every multiplier of the algebra $A(G)$ is a linear combination of positive definite functions. Again, the "only if" part of this result was known.References
- Marek Bożejko, A new group algebra and lacunary sets in discrete noncommutative groups, Studia Math. 70 (1981), no. 2, 165–175. MR 642192, DOI 10.4064/sm-70-2-165-175
- Michael Cowling, The Kunze-Stein phenomenon, Ann. of Math. (2) 107 (1978), no. 2, 209–234. MR 507240, DOI 10.2307/1971142
- Pierre Eymard, L’algèbre de Fourier d’un groupe localement compact, Bull. Soc. Math. France 92 (1964), 181–236 (French). MR 228628 —, Algèbres ${A_p}$ et convoluteurs de ${L^p}$ (Séminaire Bourbaki, 1969/70) Lecture Notes in Math., vol. 180, Springer-Verlag, Berlin and New York, 1971, pp. 55-72.
- Pierre Eymard, Initiation à la théorie des groupes moyennables, Analyse harmonique sur les groupes de Lie (Sém. Nancy-Strasbourg, 1973-75), Lecture Notes in Math., Vol. 497, Springer, Berlin, 1975, pp. 89–107 (French). MR 0394037
- Alessandro Figà-Talamanca, A remark on multipliers of the Fourier algebra of the free group, Boll. Un. Mat. Ital. A (5) 16 (1979), no. 3, 577–581 (English, with Italian summary). MR 551384 A. Figà-Talamanca and M. A. Picardello, Multiplicateurs de $A(G)$ qui ne sont pas dans $B(G)$, C. R. Acad. Sci. Paris Sér. A 277 (1973), 117-119.
- John E. Gilbert, Convolution operators on $L^{p}(G)$ and properties of locally compact groups, Pacific J. Math. 24 (1968), 257–268. MR 231942
- 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
- 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
- Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115, Springer-Verlag, Berlin-New York, 1979. Structure of topological groups, integration theory, group representations. MR 551496
- R. A. Kunze, $L_{p}$ Fourier transforms on locally compact unimodular groups, Trans. Amer. Math. Soc. 89 (1958), 519–540. MR 100235, DOI 10.1090/S0002-9947-1958-0100235-1
- Horst Leptin, Sur l’algèbre de Fourier d’un groupe localement compact, C. R. Acad. Sci. Paris Sér. A-B 266 (1968), A1180–A1182 (French). MR 239002
- H. Leptin, On locally compact groups with invariant means, Proc. Amer. Math. Soc. 19 (1968), 489–494. MR 239001, DOI 10.1090/S0002-9939-1968-0239001-7
- Massimo Angelo Picardello, Lacunary sets in discrete noncommutative groups, Boll. Un. Mat. Ital. (4) 8 (1973), 494–508 (English, with Italian summary). MR 0344804
- P. F. Renaud, Centralizers of Fourier algebra of an amenable group, Proc. Amer. Math. Soc. 32 (1972), 539–542. MR 293019, DOI 10.1090/S0002-9939-1972-0293019-8
Additional Information
- © Copyright 1982 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 84 (1982), 549-554
- MSC: Primary 43A07; Secondary 43A30
- DOI: https://doi.org/10.1090/S0002-9939-1982-0643747-3
- MathSciNet review: 643747