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

We study weakly compact left and right multipliers on the Banach algebra L(0)(infinity)(G)* of a locally compact group G. We prove that G is compact if and only if L(0)(infinity)(G)* has either a non-zero weakly compact left multiplier or a certain weakly compact right multiplier on L(0)(infinity)(G)*. We also give a description of weakly compact multipliers on L(0)(infinity)(G)* in terms of weakly completely continuous elements of L(0)(infinity)(G)*. Finally we show that G is finite if and only if there exists a multiplicative linear functional non L(0)(infinity)(G)* such that n is a weakly completely continuous element of L(0)(infinity)(G)*.