Multipliers and Uniformly Continuous Functionals Over Fourier Algebras of Ultraspherical Hypergroups

Abstract

Let H be an ultraspherical hypergroup associated to a locally compact group G and let A(H) be the Fourier algebra of H. For a left Banach A(H)-submodule X of VN(H), define QX to be the norm closure of the linear span of the set {u f : u ∈ A(H), f ∈ X} in BA(H)(A(H), X ∗ ) ∗ . We will show that BA(H)(A(H), X ∗ ) is a dual Banach space with predual QX. Applications obtained on the multiplier algebra M(A(H)) of the Fourier algebra A(H). In particular, we prove that G is amenable if and only if M(A(H)) = Bλ(H). We also study the uniformly continuous functionals associated with the Fourier algebra A(H) and obtain some characterizations for H to be discrete. Finally, we establish a contractive and injective representation from Bλ(H) into B σ A(H) (Bλ(H)). As an application of this result we show that the induced representation Φ : Bλ(H) → B σ A(H) (Bλ(H)) is surjective if and only if G is amenable.