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.
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.
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