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

Abstract

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 L ∞(G) ∗ . Finally, we prove that G is finite if and only if there exists a non-zero multiplicative linear functional µ on LUC(G) such that µ is a left weakly completely continuous elements of LUC(G) ∗ .