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 $Q_X$ to be the norm closure of the linear span of the set $\{uf: u\in A(H), f\in X\}$  in $B_{A(H)}(A(H), X^*)^*$. We will show that  $B_{A(H)}(A(H), X^*)$ is a dual Banach space with predual $Q_X$. 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))= B_{\lambda}(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  $B_{\lambda}(H)$ into
$B^\sigma_{A(H)}(B_{\lambda}(H))$. As an application of this result we
show that the induced representation $\Phi: B_{\lambda}(H)\rightarrow  B^\sigma_{A(H)}(B_{\lambda}(H))$ is surjective if and only if $G$ is amenable.