Characterization of Weakly Compact Multipliers on Left φ-Amenable Banach Algebras

Let A be a left Phi-amenable Banach algebra, where Phi denotes a nonzero character on A. In this paper, we show that the existence of a nonzero compact or weakly compact right multiplier on A is equivalent to the concept of left Phi-contractibility of A. As an important class of Banach algebras, we employ Lau algebras. Commutative Lau algebras are left epsilon-amenable, where epsilon is the identity of A & lowast;; the dual space of A. As an application, we characterize the existence of a nonzero (weakly) compact multiplier on the Fourier algebra A(H) of an ultraspherical hypergroup H, and the Fourier algebra A(G), where G is a locally compact group.