Multipliers over Fourier algebras of ultraspherical hypergroups

TL;DR

This paper studies multipliers over Fourier algebras of ultraspherical hypergroups, characterizes certain dual spaces, and links properties like amenability and discreteness of hypergroups to algebraic structures.

Contribution

It introduces a new framework for analyzing multipliers on Fourier algebras of ultraspherical hypergroups and characterizes their dual spaces and algebraic properties.

Findings

B_{A(H)}(A(H), X^*) is a dual Banach space with predual Q_X

G is amenable iff M(A(H)) equals B_{λ}(H)

Characterizations of when an ultraspherical hypergroup is discrete

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 $Q_X$, 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.