Weakly compact multipliers on Banach algebras related to a locally compact group

Abstract

We study weakly compact left and right multipliers on the Banach algebra L 1 (G) ⁄ of a locally compact group G. We prove that G is compact if and only if L 1 (G) ⁄ has either a non-zero weakly compact left multiplier or a certain weakly compact right multiplier on L 1 (G) ⁄ . We also give a description of weakly compact multipliers on L 1 (G) ⁄ in terms of weakly completely continuous elements of L 1 (G) ⁄ . Finally we show that G is flnite if and only if there exists a multiplicative linear functional n on L 1 (G) such that n is a weakly completely continuous element of L 1 (G) ⁄ .