Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A characterization of discreteness for locally compact groups in terms of the Banach algebras $ A\sb{p}(G)$


Author: Edmond E. Granirer
Journal: Proc. Amer. Math. Soc. 54 (1976), 189-192
DOI: https://doi.org/10.1090/S0002-9939-1976-0387954-3
MathSciNet review: 0387954
Full-text PDF

Abstract | References | Additional Information

Abstract: The Banach algebra $ A$ is said to have the bounded power property if for any $ x \in A$, with $ \vert\vert x\vert{\vert _{sp}} = {\lim _{n \to \infty }}\vert\vert{x^n}\vert{\vert^{1/n}} \leqslant 1$, one has $ {\sup _n}\vert\vert{x^n}\vert\vert < \infty $. It has been shown by B. M. Schreiber [9, Theorem (8.6)] that, if $ G$ is a locally compact abelian group, then the Fourier algebra $ A(G) = {L^1}{(\Gamma )^ \wedge }$ has the bounded power property, if and only if $ G$ is discrete. We improve this result in the Theorem. Let $ G$ be an arbitrary locally compact group and $ 1 < p < \infty $. Then $ {A_p}(G)$ has the bounded power property if and only if $ G$ is discrete. Our proof, even for abelian $ G$ and $ p = 2$ (then $ {A_2}(G) = A(G)$ is the usual Fourier algebra of $ G$), is much simpler and entirely different from that of [9].


References [Enhancements On Off] (What's this?)

  • [1] P. J. Cohen, On homomorphisms of group algebras, Amer. J. Math. 82(1960), 213-226. MR24 #A3232. MR 0133398 (24:A3232)
  • [2] P. Eymard, L'algèbre de Fourier d'un groupe localement compact, Bull. Soc. Math. France 92(1964), 181-236. MR37 #4208. MR 0228628 (37:4208)
  • 1. -, Algèbres $ {A_p}$ et convoluteurs de $ {L^p}$, Séminaire Bourbaki, Vol. 1969/70; Exposé 367, Lecture Notes in Math., vol. 180, Springer-Verlag, Berlin and New York, 1971. MR42 #7461.
  • [4] J. E. Gilbert, On projections of $ {L^\infty }G$ onto translation-invariant subspaces, Proc. London Math. Soc. (3) 19(1969), 69-88. MR39 #6019. MR 0244705 (39:6019)
  • [5] C. Herz, Harmonic synthesis for subgroups, Ann. Inst. Fourier (Grenoble) 23(1973), 91-123. MR 0355482 (50:7956)
  • [6] -, Une généralisation de la notion de transformée de Fourier-Stieltjes, Ann. Inst. Fourier (Grenoble) 24 (1974), 145-157. MR 0425511 (54:13466)
  • [7] E. Hewitt and K. A. Ross, Abstract harmonic analysis. Vols. I, II, Die Grundlehren der máth. Wissenschaften, Bände 115, 152, Academic Press, New York; Springer-Verlag, Berlin and New York, 1963, 1970. MR28 #158; 41 #7378; erratum, 42, p. 1825. MR 551496 (81k:43001)
  • [8] W. Rudin, Fourier analysis on groups, Interscience Tracts in Pure and Appl. Math., no. 12, Interscience, New York and London, 1962. MR 27 #2808. MR 0152834 (27:2808)
  • [9] B. M. Schreiber, Measures with bounded convolution powers, Trans. Amer. Math. Soc. 151(1970), 405-431. MR41 #8931. MR 0264335 (41:8931)
  • [10] -, On the coset ring and strong Ditkin sets, Pacific J. Math. 32(1970), 805-812. MR41 #4140. MR 0259502 (41:4140)
  • [11] John L. Kelley, General topology, Van Nostrand, New York, 1955. MR16, 1136. MR 0070144 (16:1136c)


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0387954-3
Article copyright: © Copyright 1976 American Mathematical Society

American Mathematical Society