Multipliers 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 $B_{A(H)}(A(H),X^{\ast }{)}^{\ast }$. We will show that $B_{A(H)}(A(H),X^{\ast })$ is a dual Banach space with predual $Q_X$, we characterize $Q_X$ in terms of elements in $A(H)$ and $X$. 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 $M(A(H))=B_{\lambda }(H)$, where $B_{\lambda }(H)$ is the reduced Fourier-Stieltjes algebra of $H$. Finally, we investigate some characterizations for an ultraspherical hypergroup to be discrete.