Compact and weakly compact multipliers on Fourier algebras of ultraspherical hypergroups

A locally compact group $G$ is discrete if and only if the Fourier algebra $A(G)$ has a non-zero (weakly) compact multiplier. We partially extend this result to the setting of ultraspherical hypergroups. Let $H$ be an ultraspherical hypergroup and let $A(H)$ denote the corresponding Fourier algebra. We will give several characterizations of discreteness of $H$ in the terms of the algebraic properties of $A(H)$. We also study Arens regularity of closed ideals of $A(H)$.