Weakly Completely Continuous Elements of the Banach Algebra LUC(G)*

In this paper, we study weakly compact left multipliers on the Banach algebra LUC(G). We show that G is compact if and only if there exists a non-zero weakly compact left multipliers on LUC(G). We also investigate the relation between positive left weakly completely continuous elements of the Banach algebras LUC(G) and L1(G). Finally, we prove that G is finite if and only if there exists a non-zero multiplicative linear functional ? on LUC(G) such that ? is a left weakly completely continuous elements of LUC(G).