Idempotent multipliers of Figa-Talamanca-Herz algebras

Document Type : Research Paper

Authors

1 Department of Mathematics, Behbahan Khatam Alanbia University of Technology, Behbahan, Iran

2 Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea

10.22075/ijnaa.2023.28296.3854

Abstract

For a locally compact group $G$ and $p\in(1,\infty)$, let $B_p(G)$ is the multiplier algebra of the Fig\`{a}-Talamanca-Herz algebra $A_p(G)$. For $p=2$ and $G$ amenable, the algebra $B(G):= B_2(G)$ is the usual Fourier-Stieltjes algebra. In this paper, we show that $A_p(G)$ is a Bochner-Schoenberg-Eberlin (BSE) algebra and every clopen subset of $G$ is a synthetic set for $A_p(G)$. Furthermore, we characterize idempotent elements of the Banach algebra $B_p(G)$. This result generalizes the Cohen-Host idempotent theorems for the case of Fig\`{a}-Talamanca-Herz algebras. Characterization of idempotent elements of $B_p(G)$ is of paramount importance to study homomorphisms in Fig\`{a}-Talamanca-Herz algebras.

Keywords

[1] P.J. Cohen, Homomorphisms and idempotents of group algebras, Bull. Amer. Math. Soc. 65 (1959), 120–122.
[2] P.J. Cohen, On a conjecture of Littlewood and idempotent measures, Amer. J. Math. 82 (1960), 191–212.
[3] P. Eymard, L’algebre de Fourier d’un groupe localement compact, Bull. Soc. Math. France 92 (1964), 181–236.
[4] B. Forrest, Amenability and bounded approximate identities in ideals of A(G), Illinois J. Math. 34 (1990), 1–25.
[5] C.S. Herz, The theory of p-spaces with an application to convolution operators, Trans. Amer. Math. Soc. 154 (1971), 69–82.
[6] C.S. Herz, Harmonic synthesis for subgroups, Ann. Inst. Fourier 23 (1973), 91–123.
[7] B. Host, Le theoreme des idempotents dans B(G), Bull. Soc. Math. France 114 (1986), 215–223.
[8] M. Ilie, On Fourier algebra homomorphisms, J. Funct. Anal. 213 (2004), 88–110.
[9] M. Ilie, Completely bounded homomorphisms of the Fourier algebras, J. Funct. Anal. 225 (2005), 480–499.
[10] R. Larsen, An Introduction to the Theory of Multipliers, Springer, Berlin, 1971.
[11] R. Larsen, Banach Algebra: An Introduction, Dekker, New York, 1973.
[12] J.-P. Pier, Amenable Locally Compact Groups, John Wiley & Sons Inc., New York, 1984.
[13] S.-E. Takahasi and O. Hatori, Commutative Banach algebras which satisfy a Bochner-Schoenberg-Eberlein type-theorem, Proc. Amer. Math. Soc. 110 (1990), no. 1, 149–158.
[14] A. Ulger, Multipliers with closed range on commutative semisimple Banach algebras, Studia Math. 153 (2002), 59–80.

Articles in Press, Corrected Proof
Available Online from 11 February 2024
  • Receive Date: 05 September 2022
  • Revise Date: 30 December 2023
  • Accept Date: 30 December 2023