  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)$.
