Invariant $φ$-means on multiplier completion of Banach algebras with application to hypergroups

Abstract

Let A be a Banach algebra and let ϕ 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 ϕ-means on A * M are defined and studied. Under some conditions on A, we will show that the set of topologically left invariant ϕ-means on A * and on A * M have the same cardinality. We also study the left uniformly continuous functionals associated with these algebras. The main applications are concerned with the Fourier algebra of an ultraspherical hypergroup H. In particular, we obtain some characterizations of discreteness of H.