Multiplier completion of Banach algebras with application to quantum groups

Let A be a Banach algebra and let phi be a non-zero character on A. Suppose that A(M) is the closure of the faithful Banach algebra A in the multiplier norm. In this paper, topologically left invariant phi-means on A(M)* are defined and studied. Under some conditions on A, we will show that the set of topologically left invariant phi-means on A* and on A(M)* have the same cardinality. The main applications are concerned with the quantum group algebra L-1(G) of a locally compact quantum group G. In particular, we obtain some characterizations of compactness of G in terms of the existence of a non-zero (wealdy) compact left or right multiplier on L-M(1) (G) or on its bidual in some senses.