Acta Math. Hungar., 127 (3) (2010), 195–206. DOI: 10.1007/s10474-010-9117-7 First published online March 18, 2010 WEAKLY COMPACT MULTIPLIERS ON BANACH ALGEBRAS RELATED TO A LOCALLY COMPACT GROUP ∗ M. J. MEHDIPOUR 1 and R. NASR-ISFAHANI 2 1 Department of Mathematics, Shiraz University of Technology, Shiraz 71555, Iran e-mail: mehdipour@sutech.ac.ir 2 Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, Iran e-mail: isfahani@cc.iut.ac.ir (Received May 19, 2009; accepted September 15, 2009) Abstract. We study weakly compact left and right multipliers on the Banach algebra L ∞ 0 (G) * of a locally compact group G. We prove that G is compact if and only if L ∞ 0 (G) * has either a non-zero weakly compact left multiplier or a certain weakly compact right multiplier on L ∞ 0 (G) * . We also give a description of weakly compact multipliers on L ∞ 0 (G) * in terms of weakly completely continuous elements of L ∞ 0 (G) * . Finally we show that G is finite if and only if there exists a multiplicative linear functional n on L ∞ 0 (G) such that n is a weakly completely continuous element of L ∞ 0 (G) * . 1. Introduction Let A be a Banach algebra; a bounded operator T : A → A is called left (resp. right ) multiplier if T (ab)= T (a)b (resp. T (ab)= aT (b)) for all a, b ∈ A. An element a ∈ A is said left (resp. right) weakly completely continuous el- ement if the multiplier a T : b → ab (resp. T a : b → ba) is weakly compact. We denote by L wcc (A) (resp. R wcc (A)) the set of all left (resp. right) weakly completely continuous elements of A. Weakly compact left and right multipliers on the second dual algebras L 1 (G ) ∗∗ and M (G ) ∗∗ of a locally compact group G have been studied by Ghahramani and Lau in [6], [7] and [8]. In the same papers, they have ob- tained some results on the question of existence of non-zero weakly compact * This research was partially supported by the Center of Excellence for Mathematics at the Isfahan University of Technology. Key words and phrases: locally compact group, multiplier, weakly compact operator, weakly completely continuous element. 2000 Mathematics Subject Classification: 43A15, 43A20, 47B07, 47B48. 0236–5294/$ 20.00 c  2010 Akad´ emiai Kiad´o, Budapest